Playing to Lose

Analysis of Stage Game

1 Win Probability

Two players i and j exert efforts x_i ≥ 0 and x_j ≥ 0. Player i's win probability is:

-->	W_i(x_i, x_j) := (x_i + ε)/(x_i + x_j + 2 · ε);

\begin{matrix} (%o1) & W_{i} (x_{i}, x_{j}) := \frac{x_{i} + ε}{x_{i} + x_{j} + 2 ε} \end{matrix}

Where ε is > 0 is a small parameter which guarantees positive probabilities of winning for both players.

2 Expected Utility and Best Response

When competing against player j we define player i's expected utility as:

-->	EU_i(x_i, x_j, S_i, F_i) := (1 − (x_j + ε)/(x_i + x_j + 2 · ε))·(S_i) + ((x_j + ε)/(x_i + x_j + 2 · ε)) · (F_i) − x_i;

\begin{matrix} (%o2) & {EU}_{i} (x_{i}, x_{j}, S_{i}, F_{i}) := (1 - \frac{x_{j} + ε}{x_{i} + x_{j} + 2 ε}) S_{i} + \frac{x_{j} + ε}{x_{i} + x_{j} + 2 ε} F_{i} - x_{i} \end{matrix}

Where S_i is player i's prize, when she wins the contest and F_i is her prize, when she loses the contest.

Next we define the expected utility of a bystander o, who does not participate in the stage game contest, but receives S_o, when player i wins and F_0, when she loses:

-->	EU_o(x_i, x_j, S_o, F_o) := (1 − (x_j + ε)/(x_i + x_j + 2 · ε))·(S_o) + ((x_j + ε)/(x_i + x_j + 2 · ε)) · (F_o);

\begin{matrix} (%o3) & {EU}_{o} (x_{i}, x_{j}, S_{o}, F_{o}) := (1 - \frac{x_{j} + ε}{x_{i} + x_{j} + 2 ε}) S_{o} + \frac{x_{j} + ε}{x_{i} + x_{j} + 2 ε} F_{o} \end{matrix}

If we can assume (S_i - F_i) > (x_j + 2ε)²/(x_j + ε), player i's best response is:

-->	BR_i(x_j, S_i, F_i) := sqrt((S_i − F_i) · (x_j + ε)) − (x_j + 2 · ε);

\begin{matrix} (%o4) & {BR}_{i} (x_{j}, S_{i}, F_{i}) := \sqrt{(S_{i} - F_{i}) (x_{j} + ε)} - (x_{j} + 2 ε) \end{matrix}

If that condition holds, an alternative way to write out player i's expected utility is then:

-->	EU_i2(x_j, S_i, F_i) :=S_i + x_j + 2 · ε − 2 · sqrt((S_i − F_i) · (x_j + ε));

\begin{matrix} (%o5) & EU_i2 (x_{j}, S_{i}, F_{i}) := S_{i} + x_{j} + 2 ε + - 2 \sqrt{(S_{i} - F_{i}) (x_{j} + ε)} \end{matrix}

Analogously, we can then define the bystander's expected utility as:

-->	EU_o2(x_j, S_o, F_o, S_i, F_i) := S_o + (x_j + ε)/sqrt((S_i − F_i) · (x_j + ε)) · (F_o − S_o);

\begin{matrix} (%o6) & EU_o2 (x_{j}, S_{o}, F_{o}, S_{i}, F_{i}) := S_{o} + \frac{x_{j} + ε}{\sqrt{(S_{i} - F_{i}) (x_{j} + ε)}} (F_{o} - S_{o}) \end{matrix}

3 Cases

We delve into the different cases:

3.1 Case I: One player prefers winning, the other losing

S_i ≤ F_i and F_j < S_j (a) or S_i > F_i und F_j ≥ S_j (b)

3.1.1 Case Ia: Player j prefers winning, i losing

-->	x_i1a : 0;

\begin{matrix} x_i1a & 0 \end{matrix}

-->	x_j1a : BR_i(x_i1a, S_j, F_j);

\begin{matrix} x_j1a & \sqrt{(S_{j} - F_{j}) ε} - 2 ε \end{matrix}

-->	EU_i1a : radcan(EU_i(x_i1a, x_j1a, S_i, F_i));

\begin{matrix} EU_i1a & \frac{(S_{i} - F_{i}) \sqrt{S_{j} - F_{j}} \sqrt{ε} + F_{i} S_{j} - F_{i} F_{j}}{S_{j} - F_{j}} \end{matrix}

-->	EU_j1a : radcan(EU_i(x_j1a, x_i1a, S_j, F_j));

\begin{matrix} EU_j1a & 2 ε - 2 \sqrt{S_{j} - F_{j}} \sqrt{ε} + S_{j} \end{matrix}

3.1.2 Case Ib: Player i prefers winning, j losing

-->	x_j1b : 0;

\begin{matrix} x_j1b & 0 \end{matrix}

-->	x_i1b : BR_i(x_j1b, S_i, F_i);

\begin{matrix} x_i1b & \sqrt{(S_{i} - F_{i}) ε} - 2 ε \end{matrix}

-->	EU_i1b : radcan(EU_i(x_i1b, x_j1b, S_i, F_i));

\begin{matrix} EU_i1b & 2 ε - 2 \sqrt{S_{i} - F_{i}} \sqrt{ε} + S_{i} \end{matrix}

-->	EU_j1b : radcan(EU_i(x_j1b, x_i1b, S_j, F_j));

\begin{matrix} EU_j1b & \frac{\sqrt{S_{i} - F_{i}} (S_{j} - F_{j}) \sqrt{ε} + F_{j} S_{i} - F_{i} F_{j}}{S_{i} - F_{i}} \end{matrix}

3.2 Case II: Neither player wants to win

S_i ≤ F_i and S_j ≤ F_j

-->

x_i2 : 0;

\begin{matrix} x_i2 & 0 \end{matrix}

-->

x_j2 : 0;

\begin{matrix} x_j2 & 0 \end{matrix}

-->	EU_i2 : radcan(EU_i(x_i2, x_j2, S_i, F_i));

\begin{matrix} EU_i2 & \frac{S_{i} + F_{i}}{2} \end{matrix}

-->	EU_j2 : radcan(EU_i(x_j2, x_i2, S_2, F_2));

\begin{matrix} EU_j2 & \frac{S_{2} + F_{2}}{2} \end{matrix}

3.3 Case III: Both players want to win

S_i > F_i and S_j > F_j

-->	x_i3 : ((S_i − F_i)^2 · (S_j − F_j)) / (S_i − F_i + S_j − F_j)^2 − ε;

\begin{matrix} x_i3 & \frac{{(S_{i} - F_{i})}^{2} (S_{j} - F_{j})}{{(S_{j} + S_{i} - F_{j} - F_{i})}^{2}} - ε \end{matrix}

-->	x_j3 : ((S_i − F_i) · (S_j − F_j)^2) / (S_i − F_i + S_j − F_j)^2 − ε;

\begin{matrix} x_j3 & \frac{(S_{i} - F_{i}) {(S_{j} - F_{j})}^{2}}{{(S_{j} + S_{i} - F_{j} - F_{i})}^{2}} - ε \end{matrix}

-->	EU_i3 : F_i + (S_i − F_i)^3 / (S_j − F_j + S_i − F_i)^2 + ε;

\begin{matrix} EU_i3 & ε + \frac{{(S_{i} - F_{i})}^{3}}{{(S_{j} + S_{i} - F_{j} - F_{i})}^{2}} + F_{i} \end{matrix}

-->	EU_j3 : F_j + (S_j − F_j)^3 / (S_j − F_j + S_i − F_i)^2 + ε;

\begin{matrix} EU_j3 & ε + \frac{{(S_{j} - F_{j})}^{3}}{{(S_{j} + S_{i} - F_{j} - F_{i})}^{2}} + F_{j} \end{matrix}

-->	EU_o3 : (S_i − F_i)/(S_i + S_j − F_i − F_j) · S_o + (S_j − F_j)/(S_i + S_j − F_i − F_j) · F_o;

\begin{matrix} EU_o3 & \frac{(S_{i} - F_{i}) S_{o}}{S_{j} + S_{i} - F_{j} - F_{i}} + \frac{F_{o} (S_{j} - F_{j})}{S_{j} + S_{i} - F_{j} - F_{i}} \end{matrix}

Analysis of Full Game

1 Setup

We set up V_1, V_2, and E:

-->

V_1 : 1;

\begin{matrix} V_1 & 1 \end{matrix}

-->

V_2 : 0;

\begin{matrix} V_2 & 0 \end{matrix}

-->	E : 1/3 + 1/3 · V_3;

\begin{matrix} E & \frac{V_{3}}{3} + \frac{1}{3} \end{matrix}

We define expected effort, when ε → 0:

-->	ex_eff_0 : ((223888806293)/(339116361570));

\begin{matrix} ex_eff_0 & \frac{223888806293}{339116361570} \end{matrix}

We set up the assumptions about our variables:

-->	assume(ε > 0, ε < 1 − V_2, ε < 1 − V_3, ε < 1 − E, ε < V_3 − V_2, ε < E − V_2, V_2 < 1, V_2 ≥ 0, E < 1, E > 1/3, V_3 > 0, V_3 < 1, V_3 > V_2, V_3 < 1, V_3 > 0);

\begin{matrix} (%o28) & [ε > 0, ε < 1, - ε - V_{3} + 1 > 0, - ε - \frac{V_{3}}{3} + \frac{2}{3} > 0, V_{3} > ε, - ε + \frac{V_{3}}{3} + \frac{1}{3} > 0, redundant, redundant, V_{3} < 2, redundant, redundant, V_{3} < 1, redundant, redundant, redundant] \end{matrix}

2 Analysis of the Nodes

2.1 Node d:

We define the prizes:

-->	S_2d : V_2;

\begin{matrix} S_2d & 0 \end{matrix}

-->	F_2d : V_3;

\begin{matrix} F_2d & V_{3} \end{matrix}

-->	S_3d : V_2;

\begin{matrix} S_3d & 0 \end{matrix}

-->	F_3d : V_3;

\begin{matrix} F_3d & V_{3} \end{matrix}

-->	S_2d − F_2d;

\begin{matrix} (%o33) & - V_{3} \end{matrix}

-->	S_3d − F_3d;

\begin{matrix} (%o34) & - V_{3} \end{matrix}

2.1.1 Cases A - B:

S_2d ≤ F_2d and S_3d ≤ F_3d, hence case II.

We calculate best responses, expected utilities and through ex_x_d the expected effort in the node:

-->

x_2d : 0;

\begin{matrix} x_2d & 0 \end{matrix}

-->	radcan(x_2d);

\begin{matrix} (%o36) & 0 \end{matrix}

-->	radcan(limit(x_2d, ε, 0, plus));

\begin{matrix} (%o37) & 0 \end{matrix}

-->

x_3d: 0;

\begin{matrix} x_3d & 0 \end{matrix}

-->	radcan(x_3d);

\begin{matrix} (%o39) & 0 \end{matrix}

-->	radcan(limit(x_3d, ε, 0, plus));

\begin{matrix} (%o40) & 0 \end{matrix}

-->	ex_x_d : x_2d + x_3d;

\begin{matrix} ex_x_d & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	EU_1d : V_1;

\begin{matrix} EU_1d & 1 \end{matrix}

-->	radcan(limit(EU_1d, ε, 0, plus));

\begin{matrix} (%o43) & 1 \end{matrix}

-->	EU_2d : 1/2 · (V_3 + V_2);

\begin{matrix} EU_2d & \frac{V_{3}}{2} \end{matrix}

-->	radcan(limit(EU_2d, ε, 0, plus));

\begin{matrix} (%o45) & \frac{V_{3}}{2} \end{matrix}

-->	EU_3d : 1/2 · (V_3 + V_2);

\begin{matrix} EU_3d & \frac{V_{3}}{2} \end{matrix}

-->	radcan(limit(EU_3d, ε, 0, plus));

\begin{matrix} (%o47) & \frac{V_{3}}{2} \end{matrix}

2.2 Node e:

We define the prizes:

-->

S_2e : E;

\begin{matrix} S_2e & \frac{V_{3}}{3} + \frac{1}{3} \end{matrix}

-->	F_2e : V_3;

\begin{matrix} F_2e & V_{3} \end{matrix}

-->	S_3e : V_1;

\begin{matrix} S_3e & 1 \end{matrix}

-->

F_3e : E;

\begin{matrix} F_3e & \frac{V_{3}}{3} + \frac{1}{3} \end{matrix}

And prize differences:

-->	S_2e − F_2e;

\begin{matrix} (%o52) & \frac{1}{3} - \frac{2 V_{3}}{3} \end{matrix}

-->	S_3e − F_3e;

\begin{matrix} (%o53) & \frac{2}{3} - \frac{V_{3}}{3} \end{matrix}

2.2.1 Case A:

S_2e > F_2e and S_3e > F_3e, hence case III.

We calculate best responses:

-->	x_2ea : x_i3, S_i = S_2e, F_i = F_2e, F_j = F_3e, S_j = S_3e;

\begin{matrix} x_2ea & \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{2} (\frac{2}{3} - \frac{V_{3}}{3})}{{(1 - V_{3})}^{2}} - ε \end{matrix}

-->	radcan(x_2ea);

\begin{matrix} (%o55) & - (\frac{(27 {V_{3}}^{2} - 54 V_{3} + 27) ε + 4 {V_{3}}^{3} - 12 {V_{3}}^{2} + 9 V_{3} - 2}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	radcan(limit(x_2ea, ε, 0, plus));

\begin{matrix} (%o56) & - (\frac{4 {V_{3}}^{3} - 12 {V_{3}}^{2} + 9 V_{3} - 2}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	x_3ea : x_j3, S_i = S_2e, F_i = F_2e, F_j = F_3e, S_j = S_3e;

\begin{matrix} x_3ea & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) {(\frac{2}{3} - \frac{V_{3}}{3})}^{2}}{{(1 - V_{3})}^{2}} - ε \end{matrix}

-->	radcan(x_3ea);

\begin{matrix} (%o58) & - (\frac{(27 {V_{3}}^{2} - 54 V_{3} + 27) ε + 2 {V_{3}}^{3} - 9 {V_{3}}^{2} + 12 V_{3} - 4}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	radcan(limit(x_3ea, ε, 0, plus));

\begin{matrix} (%o59) & - (\frac{2 {V_{3}}^{3} - 9 {V_{3}}^{2} + 12 V_{3} - 4}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

We calculate expected effort in this node:

-->	ex_x_ea : radcan(limit(x_2ea + x_3ea, ε, 0, plus));

\begin{matrix} ex_x_ea & - (\frac{2 {V_{3}}^{2} - 5 V_{3} + 2}{9 V_{3} - 9}) \end{matrix}

We calculate expected utilities:

-->	EU_1ea : EU_o3, S_i = S_2e, F_i = F_2e, F_j = F_3e, S_j = S_3e, S_o = E, F_o = V_2;

\begin{matrix} EU_1ea & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	radcan(limit(EU_1ea, ε, 0, plus));

\begin{matrix} (%o62) & \frac{2 {V_{3}}^{2} + V_{3} - 1}{9 V_{3} - 9} \end{matrix}

-->	EU_2ea : EU_i3, S_i = S_2e, F_i = F_2e, F_j = F_3e, S_j = S_3e;

\begin{matrix} EU_2ea & ε + V_{3} + \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} \end{matrix}

-->	radcan(limit(EU_2ea, ε, 0, plus));

\begin{matrix} (%o64) & \frac{19 {V_{3}}^{3} - 42 {V_{3}}^{2} + 21 V_{3} + 1}{27 {V_{3}}^{2} - 54 V_{3} + 27} \end{matrix}

-->	EU_3ea : EU_j3, S_i = S_2e, F_i = F_2e, F_j = F_3e, S_j = S_3e;

\begin{matrix} EU_3ea & ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3} \end{matrix}

-->	radcan(limit(EU_3ea, ε, 0, plus));

\begin{matrix} (%o66) & \frac{8 {V_{3}}^{3} - 3 {V_{3}}^{2} - 21 V_{3} + 17}{27 {V_{3}}^{2} - 54 V_{3} + 27} \end{matrix}

2.2.2 Case B:

S_2e ≤ F_2e and S_3e > F_3e, hence case I.

We calculate best responses:

-->	x_2eb : 0;

\begin{matrix} x_2eb & 0 \end{matrix}

-->	radcan(x_2eb);

\begin{matrix} (%o68) & 0 \end{matrix}

-->	radcan(limit(x_2eb, ε, 0, plus));

\begin{matrix} (%o69) & 0 \end{matrix}

-->	x_3eb: BR_i(x_2eb, S_3e, F_3e);

\begin{matrix} x_3eb & \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - 2 ε \end{matrix}

-->	radcan(x_3eb);

\begin{matrix} (%o71) & \frac{\sqrt{2 - V_{3}} \sqrt{ε} - 2 \sqrt{3} ε}{\sqrt{3}} \end{matrix}

-->	radcan(limit(x_3eb, ε, 0, plus));

\begin{matrix} (%o72) & 0 \end{matrix}

We calculate expected effort in this node:

-->	ex_x_eb : radcan(x_2eb + x_3eb), ε = 0;

\begin{matrix} ex_x_eb & 0 \end{matrix}

We calculate expected utilities:

-->	EU_1eb : V_2 + sqrt(ε/(V_1 − E)) · (E − V_2);

\begin{matrix} EU_1eb & \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	radcan(limit(EU_1eb, ε, 0, plus));

\begin{matrix} (%o75) & 0 \end{matrix}

-->	EU_2eb : EU_i1a, S_i = S_2e, F_i = F_2e, S_j = S_3e, F_j = F_3e;

\begin{matrix} EU_2eb & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}}{\frac{2}{3} - \frac{V_{3}}{3}} \end{matrix}

-->	radcan(EU_2eb);

\begin{matrix} (%o77) & \frac{\sqrt{2 - V_{3}} (2 V_{3} - 1) \sqrt{ε} + \sqrt{3} {V_{3}}^{2} - 2 \sqrt{3} V_{3}}{\sqrt{3} V_{3} - 2 \sqrt{3}} \end{matrix}

-->	radcan(limit(EU_2eb, ε, 0, plus));

\begin{matrix} (%o78) & V_{3} \end{matrix}

-->	EU_3eb : EU_j1a, S_i = S_2e, F_i = F_2e, S_j = S_3e, F_j = F_3e;

\begin{matrix} EU_3eb & 2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} + 1 \end{matrix}

-->	radcan(limit(EU_3eb, ε, 0, plus));

\begin{matrix} (%o80) & 1 \end{matrix}

2.3 Knoten f:

We define the prizes:

-->	S_2f : V_1;

\begin{matrix} S_2f & 1 \end{matrix}

-->

F_2f : E;

\begin{matrix} F_2f & \frac{V_{3}}{3} + \frac{1}{3} \end{matrix}

-->

S_3f : E;

\begin{matrix} S_3f & \frac{V_{3}}{3} + \frac{1}{3} \end{matrix}

-->	F_3f : V_3;

\begin{matrix} F_3f & V_{3} \end{matrix}

And prize differences:

-->	S_2f − F_2f;

\begin{matrix} (%o85) & \frac{2}{3} - \frac{V_{3}}{3} \end{matrix}

-->	S_3f − F_3f;

\begin{matrix} (%o86) & \frac{1}{3} - \frac{2 V_{3}}{3} \end{matrix}

2.3.1 Case A:

S_2f > F_2f and S_3f > F_3f, hence case III.

We calculate best responses:

-->	x_2fa : x_i3, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} x_2fa & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) {(\frac{2}{3} - \frac{V_{3}}{3})}^{2}}{{(1 - V_{3})}^{2}} - ε \end{matrix}

-->	radcan(x_2fa);

\begin{matrix} (%o88) & - (\frac{(27 {V_{3}}^{2} - 54 V_{3} + 27) ε + 2 {V_{3}}^{3} - 9 {V_{3}}^{2} + 12 V_{3} - 4}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	radcan(limit(x_2fa, ε, 0, plus));

\begin{matrix} (%o89) & - (\frac{2 {V_{3}}^{3} - 9 {V_{3}}^{2} + 12 V_{3} - 4}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	x_3fa : x_j3, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} x_3fa & \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{2} (\frac{2}{3} - \frac{V_{3}}{3})}{{(1 - V_{3})}^{2}} - ε \end{matrix}

-->	radcan(x_3fa);

\begin{matrix} (%o91) & - (\frac{(27 {V_{3}}^{2} - 54 V_{3} + 27) ε + 4 {V_{3}}^{3} - 12 {V_{3}}^{2} + 9 V_{3} - 2}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

-->	radcan(limit(x_3fa, ε, 0, plus));

\begin{matrix} (%o92) & - (\frac{4 {V_{3}}^{3} - 12 {V_{3}}^{2} + 9 V_{3} - 2}{27 {V_{3}}^{2} - 54 V_{3} + 27}) \end{matrix}

We calculate expected effort in this node:

-->	ex_x_fa : radcan(limit(x_2fa + x_3fa, ε, 0, plus));

\begin{matrix} ex_x_fa & - (\frac{2 {V_{3}}^{2} - 5 V_{3} + 2}{9 V_{3} - 9}) \end{matrix}

We calculate expected utilities:

-->	EU_1fa : EU_o3, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f, S_o = V_2, F_o = E;

\begin{matrix} EU_1fa & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	radcan(limit(EU_1fa, ε, 0, plus));

\begin{matrix} (%o95) & \frac{2 {V_{3}}^{2} + V_{3} - 1}{9 V_{3} - 9} \end{matrix}

-->	EU_2fa : EU_i3, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} EU_2fa & ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3} \end{matrix}

-->	radcan(limit(EU_2fa, ε, 0, plus));

\begin{matrix} (%o97) & \frac{8 {V_{3}}^{3} - 3 {V_{3}}^{2} - 21 V_{3} + 17}{27 {V_{3}}^{2} - 54 V_{3} + 27} \end{matrix}

-->	EU_3fa : EU_j3, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} EU_3fa & ε + V_{3} + \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} \end{matrix}

-->	radcan(limit(EU_3fa, ε, 0, plus));

\begin{matrix} (%o99) & \frac{19 {V_{3}}^{3} - 42 {V_{3}}^{2} + 21 V_{3} + 1}{27 {V_{3}}^{2} - 54 V_{3} + 27} \end{matrix}

2.3.2 Case B:

S_2f > F_2f and S_3f ≤ F_3f, hence case I.

We calculate best responses:

-->	x_3fb : 0;

\begin{matrix} x_3fb & 0 \end{matrix}

-->	radcan(x_3fb);

\begin{matrix} (%o101) & 0 \end{matrix}

-->	radcan(limit(x_3fb, ε, 0, plus));

\begin{matrix} (%o102) & 0 \end{matrix}

-->	x_2fb : BR_i(x_3fb, S_2f, F_2f);

\begin{matrix} x_2fb & \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - 2 ε \end{matrix}

-->	radcan(x_2fb);

\begin{matrix} (%o104) & \frac{\sqrt{2 - V_{3}} \sqrt{ε} - 2 \sqrt{3} ε}{\sqrt{3}} \end{matrix}

-->	radcan(limit(x_2fb, ε, 0, plus));

\begin{matrix} (%o105) & 0 \end{matrix}

We calculate expected effort in this node:

-->	ex_x_fb : radcan(limit(x_2fb + x_3fb, ε, 0, plus));

\begin{matrix} ex_x_fb & 0 \end{matrix}

We calculate expected utilities:

-->	EU_1fb : V_2 + sqrt(ε/(V_1 − E)) · (E − V_2);

\begin{matrix} EU_1fb & \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	radcan(limit(EU_1fb, ε, 0, plus));

\begin{matrix} (%o108) & 0 \end{matrix}

-->	EU_2fb : EU_i1b, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} EU_2fb & 2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} + 1 \end{matrix}

-->	radcan(limit(EU_2fb, ε, 0, plus));

\begin{matrix} (%o110) & 1 \end{matrix}

-->	EU_3fb : EU_j1b, S_i = S_2f, F_i = F_2f, F_j = F_3f, S_j = S_3f;

\begin{matrix} EU_3fb & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}}{\frac{2}{3} - \frac{V_{3}}{3}} \end{matrix}

-->	radcan(limit(EU_3fb, ε, 0, plus));

\begin{matrix} (%o112) & V_{3} \end{matrix}

2.4 Node g:

We define the prizes:

-->	S_2g : V_1;

\begin{matrix} S_2g & 1 \end{matrix}

-->	F_2g : V_2;

\begin{matrix} F_2g & 0 \end{matrix}

-->	S_3g : V_1;

\begin{matrix} S_3g & 1 \end{matrix}

-->	F_3g : V_2;

\begin{matrix} F_3g & 0 \end{matrix}

And prize differences:

-->	S_2g − F_2g;

\begin{matrix} (%o117) & 1 \end{matrix}

-->	S_3g − F_3g;

\begin{matrix} (%o118) & 1 \end{matrix}

2.4.1 Case A - B:

S_2g > F_2g and S_3g > F_3g, hence case III.

We calculate best responses:

-->	x_2g : (V_1 − V_2)/4 − ε;

\begin{matrix} x_2g & \frac{1}{4} - ε \end{matrix}

-->	radcan(x_2g);

\begin{matrix} (%o120) & - (\frac{4 ε - 1}{4}) \end{matrix}

-->	radcan(limit(x_2g, ε, 0, plus));

\begin{matrix} (%o121) & \frac{1}{4} \end{matrix}

-->	x_3g : (V_1 − V_2)/4 − ε;

\begin{matrix} x_3g & \frac{1}{4} - ε \end{matrix}

-->	radcan(x_3g);

\begin{matrix} (%o123) & - (\frac{4 ε - 1}{4}) \end{matrix}

-->	radcan(limit(x_3g, ε, 0, plus));

\begin{matrix} (%o124) & \frac{1}{4} \end{matrix}

We calculate expected effort in this node:

-->	ex_x_g : radcan(limit(x_2g + x_3g, ε, 0, plus));

\begin{matrix} ex_x_g & \frac{1}{2} \end{matrix}

We calculate expected utilities:

-->	EU_1g : V_3;

\begin{matrix} EU_1g & V_{3} \end{matrix}

-->	radcan(limit(EU_1g, ε, 0, plus));

\begin{matrix} (%o127) & V_{3} \end{matrix}

-->	EU_2g : 1/4 · V_1 + 3/4 · V_2 + ε;

\begin{matrix} EU_2g & ε + \frac{1}{4} \end{matrix}

-->	radcan(limit(EU_2g, ε, 0, plus));

\begin{matrix} (%o129) & \frac{1}{4} \end{matrix}

-->	EU_3g : 1/4 · V_1 + 3/4 · V_2 + ε;

\begin{matrix} EU_3g & ε + \frac{1}{4} \end{matrix}

-->	radcan(limit(EU_3g, ε, 0, plus));

\begin{matrix} (%o131) & \frac{1}{4} \end{matrix}

2.5 Node b:

2.5.1 Case A:

We define the prizes:

-->	S_1ba : EU_1d;

\begin{matrix} S_1ba & 1 \end{matrix}

-->	F_1ba : EU_1ea;

\begin{matrix} F_1ba & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	S_3ba : EU_3ea;

\begin{matrix} S_3ba & ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3} \end{matrix}

-->	F_3ba : radcan(EU_3d);

\begin{matrix} F_3ba & \frac{V_{3}}{2} \end{matrix}

And prize differences:

-->	S_1ba − F_1ba;

\begin{matrix} (%o136) & 1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	radcan(S_1ba − F_1ba);

\begin{matrix} (%o137) & - (\frac{2 {V_{3}}^{2} - 8 V_{3} + 8}{9 V_{3} - 9}) \end{matrix}

-->	S_3ba − F_3ba;

\begin{matrix} (%o138) & ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3} \end{matrix}

-->	radcan(S_3ba − F_3ba);

\begin{matrix} (%o139) & \frac{(54 {V_{3}}^{2} - 108 V_{3} + 54) ε - 11 {V_{3}}^{3} + 48 {V_{3}}^{2} - 69 V_{3} + 34}{54 {V_{3}}^{2} - 108 V_{3} + 54} \end{matrix}

S_1ba > F_1ba and S_3ba > F_3ba, hence case III.

We calculate best responses:

-->	x_1ba : x_i3, S_i = S_1ba, F_i = F_1ba, F_j = F_3ba, S_j = S_3ba;

\begin{matrix} x_1ba & \frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - ε \end{matrix}

-->	radcan(x_1ba);

\begin{matrix} (%o141) & - (\frac{(8748 {V_{3}}^{4} - 34992 {V_{3}}^{3} + 52488 {V_{3}}^{2} - 34992 V_{3} + 8748) ε^{3} + (- (7452 {V_{3}}^{5}) + 49896 {V_{3}}^{4} - 130896 {V_{3}}^{3} + 168480 {V_{3}}^{2} - 106596 V_{3} + 26568) ε^{2} + (1155 {V_{3}}^{6} - 10584 {V_{3}}^{5} + 40050 {V_{3}}^{4} - 80220 {V_{3}}^{3} + 89883 {V_{3}}^{2} - 53532 V_{3} + 13260) ε + 88 {V_{3}}^{7} - 1088 {V_{3}}^{6} + 5736 {V_{3}}^{5} - 16720 {V_{3}}^{4} + 29120 {V_{3}}^{3} - 30336 {V_{3}}^{2} + 17536 V_{3} - 4352}{(8748 {V_{3}}^{4} - 34992 {V_{3}}^{3} + 52488 {V_{3}}^{2} - 34992 V_{3} + 8748) ε^{2} + (- (7452 {V_{3}}^{5}) + 49896 {V_{3}}^{4} - 130896 {V_{3}}^{3} + 168480 {V_{3}}^{2} - 106596 V_{3} + 26568) ε + 1587 {V_{3}}^{6} - 14904 {V_{3}}^{5} + 57762 {V_{3}}^{4} - 118236 {V_{3}}^{3} + 134811 {V_{3}}^{2} - 81180 V_{3} + 20172}) \end{matrix}

-->	radcan(limit(x_1ba, ε, 0, plus));

\begin{matrix} (%o142) & - (\frac{88 {V_{3}}^{5} - 736 {V_{3}}^{4} + 2440 {V_{3}}^{3} - 4016 {V_{3}}^{2} + 3296 V_{3} - 1088}{1587 {V_{3}}^{4} - 8556 {V_{3}}^{3} + 17190 {V_{3}}^{2} - 15252 V_{3} + 5043}) \end{matrix}

-->	x_3ba : x_j3, S_i = S_1ba, F_i = F_1ba, F_j = F_3ba, S_j = S_3ba;

\begin{matrix} x_3ba & \frac{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) {(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}^{2}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - ε \end{matrix}

-->	radcan(x_3ba);

\begin{matrix} (%o144) & - (\frac{(26244 {V_{3}}^{5} - 131220 {V_{3}}^{4} + 262440 {V_{3}}^{3} - 262440 {V_{3}}^{2} + 131220 V_{3} - 26244) ε^{3} + (- (16524 {V_{3}}^{6}) + 125388 {V_{3}}^{5} - 390744 {V_{3}}^{4} + 641520 {V_{3}}^{3} - 586116 {V_{3}}^{2} + 282852 V_{3} - 56376) ε^{2} + (2385 {V_{3}}^{7} - 24849 {V_{3}}^{6} + 109998 {V_{3}}^{5} - 267930 {V_{3}}^{4} + 387405 {V_{3}}^{3} - 332181 {V_{3}}^{2} + 156312 V_{3} - 31140) ε + 242 {V_{3}}^{8} - 3080 {V_{3}}^{7} + 17060 {V_{3}}^{6} - 53768 {V_{3}}^{5} + 105602 {V_{3}}^{4} - 132560 {V_{3}}^{3} + 104048 {V_{3}}^{2} - 46784 V_{3} + 9248}{(26244 {V_{3}}^{5} - 131220 {V_{3}}^{4} + 262440 {V_{3}}^{3} - 262440 {V_{3}}^{2} + 131220 V_{3} - 26244) ε^{2} + (- (22356 {V_{3}}^{6}) + 172044 {V_{3}}^{5} - 542376 {V_{3}}^{4} + 898128 {V_{3}}^{3} - 825228 {V_{3}}^{2} + 399492 V_{3} - 79704) ε + 4761 {V_{3}}^{7} - 49473 {V_{3}}^{6} + 217998 {V_{3}}^{5} - 527994 {V_{3}}^{4} + 759141 {V_{3}}^{3} - 647973 {V_{3}}^{2} + 304056 V_{3} - 60516}) \end{matrix}

-->	radcan(limit(x_3ba, ε, 0, plus));

\begin{matrix} (%o145) & - (\frac{242 {V_{3}}^{6} - 2112 {V_{3}}^{5} + 7644 {V_{3}}^{4} - 14744 {V_{3}}^{3} + 16050 {V_{3}}^{2} - 9384 V_{3} + 2312}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ba : radcan(limit(x_1ba + x_3ba + x_1ba/(x_1ba + x_3ba) · ex_x_d + x_3ba/(x_1ba + x_3ba) · ex_x_ea, ε, 0, plus));

\begin{matrix} ex_x_ba & - (\frac{44 {V_{3}}^{4} - 247 {V_{3}}^{3} + 516 {V_{3}}^{2} - 481 V_{3} + 170}{207 {V_{3}}^{3} - 765 {V_{3}}^{2} + 927 V_{3} - 369}) \end{matrix}

We calculate expected utilities:

-->	EU_1ba : EU_i3, S_i = S_1ba, F_i = F_1ba, F_j = F_3ba, S_j = S_3ba;

\begin{matrix} EU_1ba & \frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	radcan(limit(EU_1ba, ε, 0, plus));

\begin{matrix} (%o148) & \frac{770 {V_{3}}^{6} - 2295 {V_{3}}^{5} - 3729 {V_{3}}^{4} + 23758 {V_{3}}^{3} - 37404 {V_{3}}^{2} + 25197 V_{3} - 6289}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129} \end{matrix}

-->	EU_2ba : EU_o3, S_i = S_1ba, F_i = F_1ba, F_j = F_3ba, S_j = S_3ba, S_o = EU_2d, F_o = EU_2ea;

\begin{matrix} EU_2ba & \frac{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) V_{3}}{2 (ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})} + \frac{(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3}) (ε + V_{3} + \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}})}{ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3}} \end{matrix}

-->	radcan(limit(EU_2ba, ε, 0, plus));

\begin{matrix} (%o150) & \frac{371 {V_{3}}^{5} - 1766 {V_{3}}^{4} + 3104 {V_{3}}^{3} - 2383 {V_{3}}^{2} + 655 V_{3} + 17}{621 {V_{3}}^{4} - 2916 {V_{3}}^{3} + 5076 {V_{3}}^{2} - 3888 V_{3} + 1107} \end{matrix}

-->	EU_3ba : EU_j3, S_i = S_1ba, F_i = F_1ba, F_j = F_3ba, S_j = S_3ba;

\begin{matrix} EU_3ba & \frac{{(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{V_{3}}{2} \end{matrix}

-->	radcan(limit(EU_3ba, ε, 0, plus));

\begin{matrix} (%o152) & \frac{6476 {V_{3}}^{7} - 46735 {V_{3}}^{6} + 137823 {V_{3}}^{5} - 209993 {V_{3}}^{4} + 168562 {V_{3}}^{3} - 58737 {V_{3}}^{2} - 2305 V_{3} + 4913}{14283 {V_{3}}^{6} - 105570 {V_{3}}^{5} + 323001 {V_{3}}^{4} - 523692 {V_{3}}^{3} + 474633 {V_{3}}^{2} - 228042 V_{3} + 45387} \end{matrix}

2.5.2 Case B:

We define the prizes:

-->	S_1bb : EU_1d;

\begin{matrix} S_1bb & 1 \end{matrix}

-->	F_1bb : EU_1eb;

\begin{matrix} F_1bb & \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	S_3bb : EU_3eb;

\begin{matrix} S_3bb & 2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} + 1 \end{matrix}

-->	F_3bb : radcan(EU_3d);

\begin{matrix} F_3bb & \frac{V_{3}}{2} \end{matrix}

And prize differences:

-->	S_1bb − F_1bb;

\begin{matrix} (%o157) & 1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	S_3bb − F_3bb;

\begin{matrix} (%o158) & 2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1 \end{matrix}

S_1bb > F_1bb and S_3bb > F_3bb for small enough ε, hence case III.

We calculate best responses:

-->	x_1bb : x_i3, S_i = S_1bb, F_i = F_1bb, F_j = F_3bb, S_j = S_3bb;

\begin{matrix} x_1bb & \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2} (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} - ε \end{matrix}

-->	radcan(x_1bb);

\begin{matrix} (%o160) & - (\frac{(32 3^{\frac{3}{2}} - 16 3^{\frac{3}{2}} V_{3}) ε^{3} + \sqrt{2 - V_{3}} \sqrt{ε} ((48 V_{3} - 240) ε^{2} + (- (4 {V_{3}}^{2}) + 172 V_{3} - 184) ε - 12 {V_{3}}^{2} - 12 V_{3} + 72) + (20 \sqrt{3} {V_{3}}^{2} - 200 \sqrt{3} V_{3} + 284 \sqrt{3}) ε^{2} + (- (\sqrt{3} {V_{3}}^{3}) + 46 \sqrt{3} {V_{3}}^{2} - 94 \sqrt{3} V_{3} + 4 3^{\frac{3}{2}}) ε - 2 3^{\frac{3}{2}} {V_{3}}^{2} + 8 3^{\frac{3}{2}} V_{3} - 8 3^{\frac{3}{2}}}{(32 3^{\frac{3}{2}} - 16 3^{\frac{3}{2}} V_{3}) ε^{2} + \sqrt{2 - V_{3}} \sqrt{ε} ((48 V_{3} - 240) ε - 12 {V_{3}}^{2} + 108 V_{3} - 240) + (28 \sqrt{3} {V_{3}}^{2} - 184 \sqrt{3} V_{3} + 292 \sqrt{3}) ε - 3^{\frac{3}{2}} {V_{3}}^{3} + 10 3^{\frac{3}{2}} {V_{3}}^{2} - 32 3^{\frac{3}{2}} V_{3} + 32 3^{\frac{3}{2}}}) \end{matrix}

-->	radcan(limit(x_1bb, ε, 0, plus));

\begin{matrix} (%o161) & - (\frac{2 V_{3} - 4}{{V_{3}}^{2} - 8 V_{3} + 16}) \end{matrix}

-->	x_3bb : x_j3, S_i = S_1bb, F_i = F_1bb, F_j = F_3bb, S_j = S_3bb;

\begin{matrix} x_3bb & \frac{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) {(2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}^{2}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} - ε \end{matrix}

-->	radcan(x_3bb);

\begin{matrix} (%o163) & - (\frac{\sqrt{2 - V_{3}} ((144 V_{3} - 288) ε^{3} + (- (180 {V_{3}}^{2}) + 504 V_{3} - 396) ε^{2} + (33 {V_{3}}^{3} - 42 {V_{3}}^{2} - 192 V_{3} + 288) ε - 9 {V_{3}}^{3} + 54 {V_{3}}^{2} - 108 V_{3} + 72) + \sqrt{ε} ((32 3^{\frac{3}{2}} {V_{3}}^{2} - 128 3^{\frac{3}{2}} V_{3} + 128 3^{\frac{3}{2}}) ε^{2} + (- (52 \sqrt{3} {V_{3}}^{3}) + 52 3^{\frac{3}{2}} {V_{3}}^{2} - 8 3^{\frac{5}{2}} V_{3} - 64 \sqrt{3}) ε + 3^{\frac{3}{2}} {V_{3}}^{4} + 3^{\frac{5}{2}} {V_{3}}^{3} - 14 3^{\frac{5}{2}} {V_{3}}^{2} + 100 3^{\frac{3}{2}} V_{3} - 8 3^{\frac{7}{2}})}{\sqrt{2 - V_{3}} ((144 V_{3} - 288) ε^{2} + (- (84 {V_{3}}^{2}) + 552 V_{3} - 876) ε + 9 {V_{3}}^{3} - 90 {V_{3}}^{2} + 288 V_{3} - 288) + \sqrt{ε} ((16 3^{\frac{3}{2}} {V_{3}}^{2} - 112 3^{\frac{3}{2}} V_{3} + 160 3^{\frac{3}{2}}) ε - 4 3^{\frac{3}{2}} {V_{3}}^{3} + 44 3^{\frac{3}{2}} {V_{3}}^{2} - 152 3^{\frac{3}{2}} V_{3} + 160 3^{\frac{3}{2}})}) \end{matrix}

-->	radcan(limit(x_3bb, ε, 0, plus));

\begin{matrix} (%o164) & \frac{{V_{3}}^{2} - 4 V_{3} + 4}{{V_{3}}^{2} - 8 V_{3} + 16} \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_bb : radcan(limit(x_1bb + x_3bb + x_1bab/(x_1bb + x_3bb) · ex_x_d + x_3bb/(x_1bb + x_3bb) · ex_x_eb, ε, 0, plus));

\begin{matrix} ex_x_bb & \frac{V_{3} - 2}{V_{3} - 4} \end{matrix}

We calculate expected utilities:

-->	EU_1bb : EU_i3, S_i = S_1bb, F_i = F_1bb, F_j = F_3bb, S_j = S_3bb;

\begin{matrix} EU_1bb & \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	radcan(limit(EU_1bb, ε, 0, plus));

\begin{matrix} (%o167) & \frac{4}{{V_{3}}^{2} - 8 V_{3} + 16} \end{matrix}

-->	EU_2bb : EU_o3, S_i = S_1bb, F_i = F_1bb, F_j = F_3bb, S_j = S_3bb, S_o = EU_2d, F_o = EU_2eb;

\begin{matrix} EU_2bb & \frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} \end{matrix}

-->	radcan(limit(EU_2bb, ε, 0, plus));

\begin{matrix} (%o169) & \frac{{V_{3}}^{2} - 3 V_{3}}{V_{3} - 4} \end{matrix}

-->	EU_3bb : EU_j3, S_i = S_1bb, F_i = F_1bb, F_j = F_3bb, S_j = S_3bb;

\begin{matrix} EU_3bb & \frac{{(2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{V_{3}}{2} \end{matrix}

-->	radcan(limit(EU_3bb, ε, 0, plus));

\begin{matrix} (%o171) & - (\frac{{V_{3}}^{2} - 2 V_{3} - 4}{{V_{3}}^{2} - 8 V_{3} + 16}) \end{matrix}

2.6 Node c:

2.6.1 Case A:

We define the prizes:

-->	S_1ca : EU_1fa;

\begin{matrix} S_1ca & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	F_1ca : EU_1g;

\begin{matrix} F_1ca & V_{3} \end{matrix}

-->	S_3ca : EU_3g;

\begin{matrix} S_3ca & ε + \frac{1}{4} \end{matrix}

-->	F_3ca : EU_3fa;

\begin{matrix} F_3ca & ε + V_{3} + \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} \end{matrix}

And prize differences:

-->	D_1ca : radcan(S_1ca − F_1ca);

\begin{matrix} D_1ca & - (\frac{7 {V_{3}}^{2} - 10 V_{3} + 1}{9 V_{3} - 9}) \end{matrix}

This expression switches sign within the interval and thus also player 1's behavior. We calculate the threshold:

-->	D_1cazp : solve(num(D_1ca)=0, V_3);

\begin{matrix} D_1cazp & [V_{3} = - (\frac{3 \sqrt{2} - 5}{7}), V_{3} = \frac{3 \sqrt{2} + 5}{7}] \end{matrix}

-->	allroots(num(D_1ca));

\begin{matrix} (%o178) & [V_{3} = 0.10819418755438785, V_{3} = 1.3203772410170405] \end{matrix}

-->	D_3ca : radcan(S_3ca − F_3ca);

\begin{matrix} D_3ca & - (\frac{76 {V_{3}}^{3} - 195 {V_{3}}^{2} + 138 V_{3} - 23}{108 {V_{3}}^{2} - 216 V_{3} + 108}) \end{matrix}

This expression switches sign within the interval and thus also player 3's behavior. We calculate the threshold:

-->	D3_cazp : solve(num(D_3ca)=0, V_3);

\begin{matrix} D3_cazp & [V_{3} = (\frac{- 1}{2} - \frac{\sqrt{3} %i}{2}) {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}} + \frac{729 (\frac{\sqrt{3} %i}{2} + \frac{- 1}{2})}{5776 {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}}} + \frac{65}{76}, V_{3} = (\frac{\sqrt{3} %i}{2} + \frac{- 1}{2}) {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}} + \frac{729 (\frac{- 1}{2} - \frac{\sqrt{3} %i}{2})}{5776 {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}}} + \frac{65}{76}, V_{3} = {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}} + \frac{729}{5776 {(\frac{27 \sqrt{23} %i}{2888} + \frac{189}{438976})}^{\frac{1}{3}}} + \frac{65}{76}] \end{matrix}

-->	allroots(num(D_3ca));

\begin{matrix} (%o181) & [V_{3} = 0.2410695879551234, V_{3} = 0.8529889228002085, V_{3} = 1.4717309629288786] \end{matrix}

From this subcases A.1, A.2, A.3 follow:

2.6.1.1 Subcase A.1:

S_1ca > F_1ca and S_3ca > F_3ca, hence case III.

We calculate best responses:

-->	x_1ca1 : x_i3, S_i = S_1ca, F_i = F_1ca, F_j = F_3ca, S_j = S_3ca;

\begin{matrix} x_1ca1 & \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) {(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{2}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}} - ε \end{matrix}

-->	radcan(x_1ca1);

\begin{matrix} (%o183) & - (\frac{(76800 {V_{3}}^{6} - 383040 {V_{3}}^{5} + 736803 {V_{3}}^{4} - 679980 {V_{3}}^{3} + 302490 {V_{3}}^{2} - 56700 V_{3} + 3675) ε + 14896 {V_{3}}^{7} - 80780 {V_{3}}^{6} + 170904 {V_{3}}^{5} - 176788 {V_{3}}^{4} + 91712 {V_{3}}^{3} - 22308 {V_{3}}^{2} + 2392 V_{3} - 92}{76800 {V_{3}}^{6} - 383040 {V_{3}}^{5} + 736803 {V_{3}}^{4} - 679980 {V_{3}}^{3} + 302490 {V_{3}}^{2} - 56700 V_{3} + 3675}) \end{matrix}

-->	radcan(limit(x_1ca1, ε, 0, plus));

\begin{matrix} (%o184) & - (\frac{14896 {V_{3}}^{7} - 80780 {V_{3}}^{6} + 170904 {V_{3}}^{5} - 176788 {V_{3}}^{4} + 91712 {V_{3}}^{3} - 22308 {V_{3}}^{2} + 2392 V_{3} - 92}{76800 {V_{3}}^{6} - 383040 {V_{3}}^{5} + 736803 {V_{3}}^{4} - 679980 {V_{3}}^{3} + 302490 {V_{3}}^{2} - 56700 V_{3} + 3675}) \end{matrix}

-->	x_3ca1 : x_j3, S_i = S_1ca, F_i = F_1ca, F_j = F_3ca, S_j = S_3ca;

\begin{matrix} x_3ca1 & \frac{{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2} (\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}} - ε \end{matrix}

-->	radcan(x_3ca1);

\begin{matrix} (%o186) & - (\frac{(230400 {V_{3}}^{7} - 1379520 {V_{3}}^{6} + 3359529 {V_{3}}^{5} - 4250349 {V_{3}}^{4} + 2947410 {V_{3}}^{3} - 1077570 {V_{3}}^{2} + 181125 V_{3} - 11025) ε + 40432 {V_{3}}^{8} - 265240 {V_{3}}^{7} + 715183 {V_{3}}^{6} - 1020862 {V_{3}}^{5} + 828259 {V_{3}}^{4} - 381892 {V_{3}}^{3} + 95197 {V_{3}}^{2} - 11638 V_{3} + 529}{230400 {V_{3}}^{7} - 1379520 {V_{3}}^{6} + 3359529 {V_{3}}^{5} - 4250349 {V_{3}}^{4} + 2947410 {V_{3}}^{3} - 1077570 {V_{3}}^{2} + 181125 V_{3} - 11025}) \end{matrix}

-->	radcan(limit(x_3ca1, ε, 0, plus));

\begin{matrix} (%o187) & - (\frac{40432 {V_{3}}^{8} - 265240 {V_{3}}^{7} + 715183 {V_{3}}^{6} - 1020862 {V_{3}}^{5} + 828259 {V_{3}}^{4} - 381892 {V_{3}}^{3} + 95197 {V_{3}}^{2} - 11638 V_{3} + 529}{230400 {V_{3}}^{7} - 1379520 {V_{3}}^{6} + 3359529 {V_{3}}^{5} - 4250349 {V_{3}}^{4} + 2947410 {V_{3}}^{3} - 1077570 {V_{3}}^{2} + 181125 V_{3} - 11025}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ca1 : radcan(limit(x_1ca1 + x_3ca1 + x_1ca1/(x_1ca1 + x_3ca1) · ex_x_fa + x_3ca1/(x_1ca1 + x_3ca1) · ex_x_g, ε, 0, plus));

\begin{matrix} ex_x_ca1 & - (\frac{280 {V_{3}}^{4} - 926 {V_{3}}^{3} + 969 {V_{3}}^{2} - 374 V_{3} + 43}{576 {V_{3}}^{3} - 1206 {V_{3}}^{2} + 720 V_{3} - 90}) \end{matrix}

We calculate expected utilities:

-->	EU_1ca1 : EU_i3, S_i = S_1ca, F_i = F_1ca, F_j = F_3ca, S_j = S_3ca;

\begin{matrix} EU_1ca1 & ε + V_{3} + \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}} \end{matrix}

-->	radcan(limit(EU_1ca1, ε, 0, plus));

\begin{matrix} (%o190) & \frac{20112 {V_{3}}^{7} - 98672 {V_{3}}^{6} + 186129 {V_{3}}^{5} - 167988 {V_{3}}^{4} + 72974 {V_{3}}^{3} - 13284 {V_{3}}^{2} + 729 V_{3} + 16}{25600 {V_{3}}^{6} - 127680 {V_{3}}^{5} + 245601 {V_{3}}^{4} - 226660 {V_{3}}^{3} + 100830 {V_{3}}^{2} - 18900 V_{3} + 1225} \end{matrix}

-->	EU_2ca1 : EU_o3, S_i = S_1ca, F_i = F_1ca, F_j = F_3ca, S_j = S_3ca, S_o = EU_2fa, F_o = EU_2g;

\begin{matrix} EU_2ca1 & \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} \end{matrix}

-->	radcan(limit(EU_2ca1, ε, 0, plus));

\begin{matrix} (%o192) & \frac{896 {V_{3}}^{5} - 932 {V_{3}}^{4} - 4183 {V_{3}}^{3} + 8213 {V_{3}}^{2} - 4505 V_{3} + 479}{5760 {V_{3}}^{4} - 20124 {V_{3}}^{3} + 24084 {V_{3}}^{2} - 10980 V_{3} + 1260} \end{matrix}

-->	EU_3ca1 : EU_j3, S_i = S_1ca, F_i = F_1ca, F_j = F_3ca, S_j = S_3ca;

\begin{matrix} EU_3ca1 & 2 ε + V_{3} + \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}} \end{matrix}

-->	radcan(limit(EU_3ca1, ε, 0, plus));

\begin{matrix} (%o194) & \frac{502208 {V_{3}}^{8} - 3039632 {V_{3}}^{7} + 7362152 {V_{3}}^{6} - 8979671 {V_{3}}^{5} + 5597723 {V_{3}}^{4} - 1475966 {V_{3}}^{3} - 19594 {V_{3}}^{2} + 58213 V_{3} - 5689}{921600 {V_{3}}^{7} - 5518080 {V_{3}}^{6} + 13438116 {V_{3}}^{5} - 17001396 {V_{3}}^{4} + 11789640 {V_{3}}^{3} - 4310280 {V_{3}}^{2} + 724500 V_{3} - 44100} \end{matrix}

2.6.1.2 Subcase A.2:

S_1ca ≤ F_1ca and S_3ca > F_3ca, hence case I.

We calculate best responses:

-->	x_1ca2 : 0;

\begin{matrix} x_1ca2 & 0 \end{matrix}

-->	radcan(x_1ca2);

\begin{matrix} (%o196) & 0 \end{matrix}

-->	radcan(limit(x_1ca2, ε, 0, plus));

\begin{matrix} (%o197) & 0 \end{matrix}

-->	x_3ca2 : radcan(BR_i(x_1ca2, S_3ca, F_3ca));

\begin{matrix} x_3ca2 & \frac{(4 3^{\frac{3}{2}} - 4 3^{\frac{3}{2}} V_{3}) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε}}{2 3^{\frac{3}{2}} V_{3} - 2 3^{\frac{3}{2}}} \end{matrix}

-->	radcan(x_3ca2);

\begin{matrix} (%o199) & \frac{(4 3^{\frac{3}{2}} - 4 3^{\frac{3}{2}} V_{3}) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε}}{2 3^{\frac{3}{2}} V_{3} - 2 3^{\frac{3}{2}}} \end{matrix}

-->	radcan(limit(x_3ca2, ε, 0, plus));

\begin{matrix} (%o200) & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ca2 : radcan(limit(x_1ca2 + x_3ca2 + ex_x_g, ε, 0, plus));

\begin{matrix} ex_x_ca2 & \frac{1}{2} \end{matrix}

We calculate expected utilities:

-->	EU_1ca2 : radcan(EU_i(x_1ca2, x_3ca2, S_1ca, F_1ca));

\begin{matrix} EU_1ca2 & \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} \end{matrix}

-->	radcan(limit(EU_1ca2, ε, 0, plus));

\begin{matrix} (%o203) & V_{3} \end{matrix}

-->	EU_2ca2 : radcan(EU_o(x_1ca2, x_3ca2, EU_2fa, EU_2g));

\begin{matrix} EU_2ca2 & - (\frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) \end{matrix}

-->	radcan(limit(EU_2ca2, ε, 0, plus));

\begin{matrix} (%o205) & \frac{1}{4} \end{matrix}

-->	EU_3ca2 : radcan(EU_i2(x_1ca2, S_3ca, F_3ca));

\begin{matrix} EU_3ca2 & - (\frac{(4 3^{\frac{5}{2}} - 4 3^{\frac{5}{2}} V_{3}) ε + 4 \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} - 3^{\frac{3}{2}} V_{3} + 3^{\frac{3}{2}}}{4 3^{\frac{3}{2}} V_{3} - 4 3^{\frac{3}{2}}}) \end{matrix}

-->	radcan(limit(EU_3ca2, ε, 0, plus));

\begin{matrix} (%o207) & \frac{1}{4} \end{matrix}

2.6.1.3 Subcase A.3:

S_1ca ≤ F_1ca and S_3ca ≤ F_3ca, hence case I.

We calculate best responses:

-->	x_1ca3 : 0;

\begin{matrix} x_1ca3 & 0 \end{matrix}

-->	radcan(x_1ca3);

\begin{matrix} (%o209) & 0 \end{matrix}

-->	radcan(limit(x_1ca3, ε, 0, plus));

\begin{matrix} (%o210) & 0 \end{matrix}

-->	x_3ca3 : 0;

\begin{matrix} x_3ca3 & 0 \end{matrix}

-->	radcan(x_3ca3);

\begin{matrix} (%o212) & 0 \end{matrix}

-->	radcan(limit(x_3ca3, ε, 0, plus));

\begin{matrix} (%o213) & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ca3 : radcan(limit(x_1ca3 + x_3ca3 + 1/2 · ex_x_fa + 1/2 · ex_x_g, ε, 0, plus));

\begin{matrix} ex_x_ca3 & - (\frac{4 {V_{3}}^{2} - 19 V_{3} + 13}{36 V_{3} - 36}) \end{matrix}

We calculate expected utilities:

-->	EU_1ca3 : radcan(1/2 · S_1ca + 1/2 · F_1ca);

\begin{matrix} EU_1ca3 & \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} \end{matrix}

-->	radcan(limit(EU_1ca3, ε, 0, plus));

\begin{matrix} (%o216) & \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} \end{matrix}

-->	EU_2ca3 : radcan(1/2 · EU_2fa + 1/2 · EU_2g);

\begin{matrix} EU_2ca3 & \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

-->	radcan(limit(EU_2ca3, ε, 0, plus));

\begin{matrix} (%o218) & \frac{32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

-->	EU_3ca3 : radcan(1/2 · F_3ca + 1/2 · S_3ca);

\begin{matrix} EU_3ca3 & \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 76 {V_{3}}^{3} - 141 {V_{3}}^{2} + 30 V_{3} + 31}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

-->	radcan(limit(EU_3ca3, ε, 0, plus));

\begin{matrix} (%o220) & \frac{76 {V_{3}}^{3} - 141 {V_{3}}^{2} + 30 V_{3} + 31}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

2.6.2 Case B:

We define the prizes:

-->	S_1cb : EU_1fb;

\begin{matrix} S_1cb & \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	F_1cb : EU_1g;

\begin{matrix} F_1cb & V_{3} \end{matrix}

-->	S_3cb : EU_3g;

\begin{matrix} S_3cb & ε + \frac{1}{4} \end{matrix}

-->	F_3cb : EU_3fb;

\begin{matrix} F_3cb & \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}}{\frac{2}{3} - \frac{V_{3}}{3}} \end{matrix}

-->	S_1cb − F_1cb;

\begin{matrix} (%o225) & \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - V_{3} \end{matrix}

-->	S_3cb − F_3cb;

\begin{matrix} (%o226) & ε - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}}{\frac{2}{3} - \frac{V_{3}}{3}} + \frac{1}{4} \end{matrix}

S_1cb ≤ F_1cb and S_3cb ≤ F_3cb for small enough ε, hence case II.

We calculate best responses:

-->	x_1cb : 0;

\begin{matrix} x_1cb & 0 \end{matrix}

-->	radcan(x_1cb);

\begin{matrix} (%o228) & 0 \end{matrix}

-->	radcan(limit(x_1cb, ε, 0, plus));

\begin{matrix} (%o229) & 0 \end{matrix}

-->	x_3cb : 0;

\begin{matrix} x_3cb & 0 \end{matrix}

-->	radcan(x_3cb);

\begin{matrix} (%o231) & 0 \end{matrix}

-->	radcan(limit(x_3cb, ε, 0, plus));

\begin{matrix} (%o232) & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_cb : radcan(limit(x_1cb + x_3cb + 1/2 · ex_x_g + 1/2 · ex_x_fb, ε, 0, plus));

\begin{matrix} ex_x_cb & \frac{1}{4} \end{matrix}

We calculate expected utilities:

-->	EU_1cb : radcan(1/2 · (S_1cb + F_1cb));

\begin{matrix} EU_1cb & \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} \end{matrix}

-->	radcan(limit(EU_1cb, ε, 0, plus));

\begin{matrix} (%o235) & \frac{V_{3}}{2} \end{matrix}

-->	EU_2cb : radcan(1/2 · (V_1 + sqrt(ε/(V_1 − E)) · (E − V_1) − sqrt((V_1 − E) · ε) + 2 · ε + 1/4 · V_1 + 3/4 · V_2));

\begin{matrix} EU_2cb & \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} \end{matrix}

-->	radcan(limit(EU_2cb, ε, 0, plus));

\begin{matrix} (%o237) & \frac{5}{8} \end{matrix}

-->	EU_3cb : radcan(1/2 · (F_3cb + S_3cb));

\begin{matrix} EU_3cb & \frac{(4 \sqrt{3} V_{3} - 8 \sqrt{3}) ε + \sqrt{2 - V_{3}} (8 V_{3} - 4) \sqrt{ε} + 4 \sqrt{3} {V_{3}}^{2} - 7 \sqrt{3} V_{3} - 2 \sqrt{3}}{8 \sqrt{3} V_{3} - 16 \sqrt{3}} \end{matrix}

-->	radcan(limit(EU_3cb, ε, 0, plus));

\begin{matrix} (%o239) & \frac{4 V_{3} + 1}{8} \end{matrix}

2.7 Node a:

2.7.1 Case A:

We define some prizes:

-->	S_1aa : EU_1ba;

\begin{matrix} S_1aa & \frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} \end{matrix}

-->	F_2aa : radcan(EU_2ba);

\begin{matrix} F_2aa & \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} \end{matrix}

Note, that depending on the corresponding subcase, the rest of the relevant prizes will be defined in the next sections:

2.7.1.1 Subcase A.1:

We define the remaining prizes in this subcase:

-->	F_1aa1 : EU_1ca1;

\begin{matrix} F_1aa1 & ε + V_{3} + \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}} \end{matrix}

-->	S_2aa1 : EU_2ca1;

\begin{matrix} S_2aa1 & \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} \end{matrix}

And prize differences:

-->	D_1aa1 : radcan(S_1aa − F_1aa1);

\begin{matrix} D_1aa1 & - (\frac{(378520128 {V_{3}}^{12} - 3961467648 {V_{3}}^{11} + 18549606204 {V_{3}}^{10} - 51214203720 {V_{3}}^{9} + 92454780564 {V_{3}}^{8} - 114323459616 {V_{3}}^{7} + 98550133272 {V_{3}}^{6} - 59078894832 {V_{3}}^{5} + 24118055208 {V_{3}}^{4} - 6424461216 {V_{3}}^{3} + 1038244716 {V_{3}}^{2} - 90005256 V_{3} + 3152196) ε^{2} + (- (322443072 {V_{3}}^{13}) + 4243777920 {V_{3}}^{12} - 25150566636 {V_{3}}^{11} + 88638976512 {V_{3}}^{10} - 206575899804 {V_{3}}^{9} + 334726527384 {V_{3}}^{8} - 385755867720 {V_{3}}^{7} + 317720693376 {V_{3}}^{6} - 185118730344 {V_{3}}^{5} + 74324851344 {V_{3}}^{4} - 19626749388 {V_{3}}^{3} + 3159323136 {V_{3}}^{2} - 273466044 V_{3} + 9573336) ε + 76041232 {V_{3}}^{14} - 1228864768 {V_{3}}^{13} + 9058349431 {V_{3}}^{12} - 40308639614 {V_{3}}^{11} + 120746963429 {V_{3}}^{10} - 256983065348 {V_{3}}^{9} + 399552981054 {V_{3}}^{8} - 459291560028 {V_{3}}^{7} + 390322255902 {V_{3}}^{6} - 242448052400 {V_{3}}^{5} + 107394521255 {V_{3}}^{4} - 32498250854 {V_{3}}^{3} + 6246453805 {V_{3}}^{2} - 668981068 V_{3} + 29847844}{(671846400 {V_{3}}^{11} - 6710065920 {V_{3}}^{10} + 29918186244 {V_{3}}^{9} - 78403031460 {V_{3}}^{8} + 133711605360 {V_{3}}^{7} - 155093117040 {V_{3}}^{6} + 124037279640 {V_{3}}^{5} - 67770563544 {V_{3}}^{4} + 24460982640 {V_{3}}^{3} - 5447729520 {V_{3}}^{2} + 656756100 V_{3} - 32148900) ε^{2} + (- (572313600 {V_{3}}^{12}) + 7258740480 {V_{3}}^{11} - 41342059476 {V_{3}}^{10} + 139564033884 {V_{3}}^{9} - 310256556336 {V_{3}}^{8} + 476877910248 {V_{3}}^{7} - 517260865248 {V_{3}}^{6} + 396358326432 {V_{3}}^{5} - 210971007864 {V_{3}}^{4} + 75043503000 {V_{3}}^{3} - 16597857420 {V_{3}}^{2} + 1995783300 V_{3} - 97637400) ε + 121881600 {V_{3}}^{13} - 1874393280 {V_{3}}^{12} + 13066767801 {V_{3}}^{11} - 54580377573 {V_{3}}^{10} + 152082412128 {V_{3}}^{9} - 297680858244 {V_{3}}^{8} + 419559545286 {V_{3}}^{7} - 428999197158 {V_{3}}^{6} + 316063021896 {V_{3}}^{5} - 164109798216 {V_{3}}^{4} + 57551160465 {V_{3}}^{3} - 12642253605 {V_{3}}^{2} + 1516221000 V_{3} - 74132100}) \end{matrix}

-->	lim_D_1aa1 : radcan(limit(D_1aa1, ε, 0, plus));

\begin{matrix} lim_D_1aa1 & - (\frac{76041232 {V_{3}}^{12} - 924699840 {V_{3}}^{11} + 5055385143 {V_{3}}^{10} - 16388299682 {V_{3}}^{9} + 34972224129 {V_{3}}^{8} - 51540970104 {V_{3}}^{7} + 53500204122 {V_{3}}^{6} - 39126863124 {V_{3}}^{5} + 19813986918 {V_{3}}^{4} - 6684652232 {V_{3}}^{3} + 1399964655 {V_{3}}^{2} - 159783306 V_{3} + 7461961}{121881600 {V_{3}}^{11} - 1386866880 {V_{3}}^{10} + 7031773881 {V_{3}}^{9} - 20905814529 {V_{3}}^{8} + 40332058488 {V_{3}}^{7} - 52729366176 {V_{3}}^{6} + 47313846630 {V_{3}}^{5} - 28826345934 {V_{3}}^{4} + 11502251640 {V_{3}}^{3} - 2795407920 {V_{3}}^{2} + 360522225 V_{3} - 18533025}) \end{matrix}

-->	D_2aa1 : radcan(S_2aa1 − F_2aa);

\begin{matrix} D_2aa1 & - (\frac{(718848 {V_{3}}^{8} - 5784480 {V_{3}}^{7} + 19986966 {V_{3}}^{6} - 38633652 {V_{3}}^{5} + 45461034 {V_{3}}^{4} - 33030072 {V_{3}}^{3} + 14203242 {V_{3}}^{2} - 3173364 V_{3} + 251478) ε - 175616 {V_{3}}^{9} + 1780800 {V_{3}}^{8} - 7839015 {V_{3}}^{7} + 19633248 {V_{3}}^{6} - 30752079 {V_{3}}^{5} + 31087050 {V_{3}}^{4} - 20091981 {V_{3}}^{3} + 7852020 {V_{3}}^{2} - 1607469 V_{3} + 113074}{(933120 {V_{3}}^{7} - 6059448 {V_{3}}^{6} + 16481232 {V_{3}}^{5} - 24196968 {V_{3}}^{4} + 20505312 {V_{3}}^{3} - 9850248 {V_{3}}^{2} + 2391120 V_{3} - 204120) ε - 397440 {V_{3}}^{8} + 3652236 {V_{3}}^{7} - 14287968 {V_{3}}^{6} + 30972348 {V_{3}}^{5} - 40455720 {V_{3}}^{4} + 32284548 {V_{3}}^{3} - 15092784 {V_{3}}^{2} + 3634740 V_{3} - 309960}) \end{matrix}

-->	lim_D_2aa1 : radcan(limit(D_2aa1, ε, 0, plus));

\begin{matrix} lim_D_2aa1 & - (\frac{175616 {V_{3}}^{8} - 1429568 {V_{3}}^{7} + 4979879 {V_{3}}^{6} - 9673490 {V_{3}}^{5} + 11405099 {V_{3}}^{4} - 8276852 {V_{3}}^{3} + 3538277 {V_{3}}^{2} - 775466 V_{3} + 56537}{397440 {V_{3}}^{7} - 2857356 {V_{3}}^{6} + 8573256 {V_{3}}^{5} - 13825836 {V_{3}}^{4} + 12804048 {V_{3}}^{3} - 6676452 {V_{3}}^{2} + 1739880 V_{3} - 154980}) \end{matrix}

We establish that for ε → 0 both prize differences are positive (case III). To do so, we check the denominators of both expressions for roots to prove that the function is continous within the interval. Then, we also inspect the numerators for roots. Lastly, we cestablish that for at least one V_3 both expressions are indeed positive, concluding our proof. We use the function nroots to check for roots, which internally uses Sturm's theorem.

-->	denom_D_1aa1 : denom(radcan(lim_D_1aa1));

\begin{matrix} denom_D_1aa1 & 121881600 {V_{3}}^{11} - 1386866880 {V_{3}}^{10} + 7031773881 {V_{3}}^{9} - 20905814529 {V_{3}}^{8} + 40332058488 {V_{3}}^{7} - 52729366176 {V_{3}}^{6} + 47313846630 {V_{3}}^{5} - 28826345934 {V_{3}}^{4} + 11502251640 {V_{3}}^{3} - 2795407920 {V_{3}}^{2} + 360522225 V_{3} - 18533025 \end{matrix}

-->	nroots(denom_D_1aa1, 0, 0.11);

\begin{matrix} (%o249) & 0 \end{matrix}

-->	num_D_1aa1 : num(radcan(lim_D_1aa1));

\begin{matrix} num_D_1aa1 & - (76041232 {V_{3}}^{12}) + 924699840 {V_{3}}^{11} - 5055385143 {V_{3}}^{10} + 16388299682 {V_{3}}^{9} - 34972224129 {V_{3}}^{8} + 51540970104 {V_{3}}^{7} - 53500204122 {V_{3}}^{6} + 39126863124 {V_{3}}^{5} - 19813986918 {V_{3}}^{4} + 6684652232 {V_{3}}^{3} - 1399964655 {V_{3}}^{2} + 159783306 V_{3} - 7461961 \end{matrix}

-->	nroots(num_D_1aa1, 0, 0.11);

\begin{matrix} (%o251) & 0 \end{matrix}

-->	lim_D_1aa1, V_3 = 0.1;

\begin{matrix} (%o252) & 0.31446945214681643 \end{matrix}

-->	denom_D_2aa1 : denom(radcan(lim_D_2aa1));

\begin{matrix} denom_D_2aa1 & 397440 {V_{3}}^{7} - 2857356 {V_{3}}^{6} + 8573256 {V_{3}}^{5} - 13825836 {V_{3}}^{4} + 12804048 {V_{3}}^{3} - 6676452 {V_{3}}^{2} + 1739880 V_{3} - 154980 \end{matrix}

-->	nroots(denom_D_2aa1, 0, 0.11);

\begin{matrix} (%o254) & 0 \end{matrix}

-->	num_D_2aa1 : num(radcan(lim_D_2aa1));

\begin{matrix} num_D_2aa1 & - (175616 {V_{3}}^{8}) + 1429568 {V_{3}}^{7} - 4979879 {V_{3}}^{6} + 9673490 {V_{3}}^{5} - 11405099 {V_{3}}^{4} + 8276852 {V_{3}}^{3} - 3538277 {V_{3}}^{2} + 775466 V_{3} - 56537 \end{matrix}

-->	nroots(num_D_2aa1, 0, 0.11);

\begin{matrix} (%o256) & 0 \end{matrix}

-->	lim_D_2aa1, V_3 = 0.1;

\begin{matrix} (%o257) & 0.19708992489429206 \end{matrix}

We calculate best responses:

-->	x_1aa1 : x_i3, S_i = S_1aa, F_i = F_1aa1, F_j = F_2aa, S_j = S_2aa1;

\begin{matrix} x_1aa1 & \frac{(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2}} ((- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2}) ((- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2}) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})^(2) - ε \end{matrix}

Auxiliary calculations:

-->	lim_S_1aa : radcan(limit(S_1aa, ε, 0, plus));

\begin{matrix} lim_S_1aa & \frac{770 {V_{3}}^{6} - 2295 {V_{3}}^{5} - 3729 {V_{3}}^{4} + 23758 {V_{3}}^{3} - 37404 {V_{3}}^{2} + 25197 V_{3} - 6289}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129} \end{matrix}

-->	lim_F_1aa1 : radcan(limit(F_1aa1, ε, 0, plus));

\begin{matrix} lim_F_1aa1 & \frac{20112 {V_{3}}^{7} - 98672 {V_{3}}^{6} + 186129 {V_{3}}^{5} - 167988 {V_{3}}^{4} + 72974 {V_{3}}^{3} - 13284 {V_{3}}^{2} + 729 V_{3} + 16}{25600 {V_{3}}^{6} - 127680 {V_{3}}^{5} + 245601 {V_{3}}^{4} - 226660 {V_{3}}^{3} + 100830 {V_{3}}^{2} - 18900 V_{3} + 1225} \end{matrix}

-->	lim_F_2aa : radcan(limit(F_2aa, ε, 0, plus));

\begin{matrix} lim_F_2aa & \frac{371 {V_{3}}^{5} - 1766 {V_{3}}^{4} + 3104 {V_{3}}^{3} - 2383 {V_{3}}^{2} + 655 V_{3} + 17}{621 {V_{3}}^{4} - 2916 {V_{3}}^{3} + 5076 {V_{3}}^{2} - 3888 V_{3} + 1107} \end{matrix}

-->	lim_S_2aa1 : radcan(limit(S_2aa1, ε, 0, plus));

\begin{matrix} lim_S_2aa1 & \frac{896 {V_{3}}^{5} - 932 {V_{3}}^{4} - 4183 {V_{3}}^{3} + 8213 {V_{3}}^{2} - 4505 V_{3} + 479}{5760 {V_{3}}^{4} - 20124 {V_{3}}^{3} + 24084 {V_{3}}^{2} - 10980 V_{3} + 1260} \end{matrix}

-->	lim_x_1aa1 : radcan(limit(x_i3, ε, 0, plus)), S_i = lim_S_1aa, F_i = lim_F_1aa1, F_j = lim_F_2aa, S_j = lim_S_2aa1;

\begin{matrix} lim_x_1aa1 & - (\frac{4061835785581964558336 {V_{3}}^{32} - 131852541675713679392768 {V_{3}}^{31} + 2060079152395765747653632 {V_{3}}^{30} - 20629403399236403805669376 {V_{3}}^{29} + 148747014226611360719074432 {V_{3}}^{28} - 822529694323694016651242752 {V_{3}}^{27} + 3627687728445419744107217244 {V_{3}}^{26} - 13102878372977997619095437784 {V_{3}}^{25} + 39491529179888736219845497284 {V_{3}}^{24} - 100692823212770044089516489696 {V_{3}}^{23} + 219424150815811130350446069432 {V_{3}}^{22} - 411790729753174473174097385040 {V_{3}}^{21} + 669311024075316166613480104200 {V_{3}}^{20} - 946015672198481635208912854944 {V_{3}}^{19} + 1165910467395807925847996107524 {V_{3}}^{18} - 1254862931365846416946663230408 {V_{3}}^{17} + 1180030328767943088952727275740 {V_{3}}^{16} - 968975213036358325977553491648 {V_{3}}^{15} + 693695849041424849630143149456 {V_{3}}^{14} - 431835503076762617067642428640 {V_{3}}^{13} + 232875079121174230845371254128 {V_{3}}^{12} - 108241979512932407214156403776 {V_{3}}^{11} + 43083911421071552416848193956 {V_{3}}^{10} - 14564821947248613481572446184 {V_{3}}^{9} + 4138677867795395527027861308 {V_{3}}^{8} - 975656699567568881737685856 {V_{3}}^{7} + 187651158709319676572365368 {V_{3}}^{6} - 28811144611325162551601104 {V_{3}}^{5} + 3428886527845394583191176 {V_{3}}^{4} - 303328103000642523492640 {V_{3}}^{3} + 18683361995788803424124 {V_{3}}^{2} - 711985533688000290680 V_{3} + 12592115571778643108}{26824306693722779811840 {V_{3}}^{31} - 849008549900752246419456 {V_{3}}^{30} + 12919362841259624287199232 {V_{3}}^{29} - 125850583423035131130739200 {V_{3}}^{28} + 881583278884551343507728768 {V_{3}}^{27} - 4729344709130717852843638704 {V_{3}}^{26} + 20204515470728567161674559368 {V_{3}}^{25} - 70571590138264352380713754083 {V_{3}}^{24} + 205314837590916220185326471256 {V_{3}}^{23} - 504313699181759451179915754330 {V_{3}}^{22} + 1056376574708103105768836157708 {V_{3}}^{21} - 1901035235944211884240312213992 {V_{3}}^{20} + 2954975321926616486866014534912 {V_{3}}^{19} - 3982328577351536273353847260710 {V_{3}}^{18} + 4664046108483816238490402447892 {V_{3}}^{17} - 4752443950666310936116020744243 {V_{3}}^{16} + 4212945093427437159736708624272 {V_{3}}^{15} - 3245404645215062734169487452580 {V_{3}}^{14} + 2167557727467027102752135098008 {V_{3}}^{13} - 1250749903091525967983759584488 {V_{3}}^{12} + 620537715709068807519479876448 {V_{3}}^{11} - 263027684167508926996507725660 {V_{3}}^{10} + 94476736175303254450352172624 {V_{3}}^{9} - 28460040876194079687013055673 {V_{3}}^{8} + 7096062908524681116666471384 {V_{3}}^{7} - 1439970041391140917592050914 {V_{3}}^{6} + 232648086626473240872799452 {V_{3}}^{5} - 29053113750861389839107120 {V_{3}}^{4} + 2688651659149098184807008 {V_{3}}^{3} - 172699330133852812175598 {V_{3}}^{2} + 6841493821011959559060 V_{3} - 125403176085148215345}) \end{matrix}

-->	x_2aa1 : x_j3, S_i = S_1aa, F_i = F_1aa1, F_j = F_2aa, S_j = S_2aa1;

\begin{matrix} x_2aa1 & \frac{{(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2}} ({(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})) ({(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}})}^{2} (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}})^(2) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3}) (ε + \frac{V_{3}}{3} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} + \frac{(- V_{3} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}) (ε + \frac{1}{4})}{- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4}} - V_{3} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - V_{3})}^{3}}{{(- (2 V_{3}) + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} - \frac{{(\frac{1}{3} - \frac{2 V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{4})}^{2}})^(2) - ε \end{matrix}

-->	lim_x_2aa1 : radcan(limit(x_2aa1, ε, 0, plus));

\begin{matrix} lim_x_2aa1 & - (\frac{2345186073920929792 {V_{3}}^{28} - 66699713880047747072 {V_{3}}^{27} + 908620046756621402112 {V_{3}}^{26} - 7892000309065139140608 {V_{3}}^{25} + 49082475010557599323920 {V_{3}}^{24} - 232700137801412612440320 {V_{3}}^{23} + 874212086078505554349735 {V_{3}}^{22} - 2670693256482985123993062 {V_{3}}^{21} + 6755776412526968638656267 {V_{3}}^{20} - 14334965260422844699587372 {V_{3}}^{19} + 25753490135324451952244409 {V_{3}}^{18} - 39432613316858600062141350 {V_{3}}^{17} + 51686073724083616831038381 {V_{3}}^{16} - 58145322243146506162720848 {V_{3}}^{15} + 56194957263754740720239190 {V_{3}}^{14} - 46630846927200150102646764 {V_{3}}^{13} + 33153611534836948614101598 {V_{3}}^{12} - 20121480862438068980358408 {V_{3}}^{11} + 10367612332259188429506594 {V_{3}}^{10} - 4501036803470451499056732 {V_{3}}^{9} + 1630082674005794943904890 {V_{3}}^{8} - 486033322964355204242256 {V_{3}}^{7} + 117276005742814256547987 {V_{3}}^{6} - 22384180829508964493166 {V_{3}}^{5} + 3276653426261537822631 {V_{3}}^{4} - 352067622226845725484 {V_{3}}^{3} + 25958164620770662293 {V_{3}}^{2} - 1165038898623183038 V_{3} + 23851653676615609}{21867641326404440064 {V_{3}}^{27} - 600514191133988990976 {V_{3}}^{26} + 7885022601772979375616 {V_{3}}^{25} - 65886977003870001062016 {V_{3}}^{24} + 393385108322061072194832 {V_{3}}^{23} - 1786332560519517064171848 {V_{3}}^{22} + 6411360729060274710880233 {V_{3}}^{21} - 18659731524084974130263253 {V_{3}}^{20} + 44828882138168433229815666 {V_{3}}^{19} - 90030682943006825451658842 {V_{3}}^{18} + 152506472119262017538156145 {V_{3}}^{17} - 219243182073790516342931637 {V_{3}}^{16} + 268538174063563169821773240 {V_{3}}^{15} - 280801574933075061495681384 {V_{3}}^{14} + 250740756065477755341189954 {V_{3}}^{13} - 190930357249728337582467066 {V_{3}}^{12} + 123595063579021720941982476 {V_{3}}^{11} - 67679000406800709473261916 {V_{3}}^{10} + 31129592398090190625675258 {V_{3}}^{9} - 11912817355308053147448450 {V_{3}}^{8} + 3745243877206674424159656 {V_{3}}^{7} - 951288493759646376267072 {V_{3}}^{6} + 190915426087841658278133 {V_{3}}^{5} - 29368247337645947326737 {V_{3}}^{4} + 3316087373917514353938 {V_{3}}^{3} - 257087167353189163098 {V_{3}}^{2} + 12146063406573328245 V_{3} - 262166918644909161}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_aa1 : radcan(limit(lim_x_1aa1 + lim_x_2aa1 + lim_x_1aa1/(lim_x_1aa1 + lim_x_2aa1) · ex_x_ba + lim_x_2aa1/(lim_x_1aa1 + lim_x_2aa1) · ex_x_ca1, ε, 0, plus));

\begin{matrix} ex_x_aa1 & - (\frac{60365834092544 {V_{3}}^{20} - 1246883827934720 {V_{3}}^{19} + 12114488107229024 {V_{3}}^{18} - 73570332713871192 {V_{3}}^{17} + 312986754724137720 {V_{3}}^{16} - 990701962748525247 {V_{3}}^{15} + 2418603042001746153 {V_{3}}^{14} - 4657953961792526781 {V_{3}}^{13} + 7177577155570965711 {V_{3}}^{12} - 8922092404275694259 {V_{3}}^{11} + 8977918717291198541 {V_{3}}^{10} - 7308471900085434617 {V_{3}}^{9} + 4790577374472306603 {V_{3}}^{8} - 2505692050935512061 {V_{3}}^{7} + 1030993656556530315 {V_{3}}^{6} - 326789396135175255 {V_{3}}^{5} + 77408578028666877 {V_{3}}^{4} - 13105139987363217 {V_{3}}^{3} + 1479655805802719 {V_{3}}^{2} - 98666445137051 V_{3} + 2910554481809}{103252372623360 {V_{3}}^{19} - 2005167248118144 {V_{3}}^{18} + 18255595979736576 {V_{3}}^{17} - 103493334177509544 {V_{3}}^{16} + 409237196110303230 {V_{3}}^{15} - 1198060101044431218 {V_{3}}^{14} + 2689711336114037946 {V_{3}}^{13} - 4732193933335827486 {V_{3}}^{12} + 6609989208629453670 {V_{3}}^{11} - 7380013028677761978 {V_{3}}^{10} + 6597128855554013394 {V_{3}}^{9} - 4707398665963179270 {V_{3}}^{8} + 2660201602462824570 {V_{3}}^{7} - 1174646581140314358 {V_{3}}^{6} + 397042322357841390 {V_{3}}^{5} - 99665500061267514 {V_{3}}^{4} + 17762311974780450 {V_{3}}^{3} - 2095962813040734 {V_{3}}^{2} + 145001418536070 V_{3} - 4408512700410}) \end{matrix}

We establish that for ε → 0 the expected effort for V_3 → 0 is larger than for any V_3 in this interval. To do so, we check the denominator of the difference for roots to prove that the function is continous within the interval. Then, we also inspect the numerator for roots. Lastly, we establish that for at least one V_3 the expressions is indeed positive, concluding our proof. We use the function nroots to check for roots, which internally uses Sturm's theorem.

-->	denom_x_aa1 : denom(radcan(ex_eff_0 − ex_x_aa1));

\begin{matrix} denom_x_aa1 & 972626914652881088563200 {V_{3}}^{19} - 18888472820032067299481280 {V_{3}}^{18} + 171965869081671863893605120 {V_{3}}^{17} - 974896748139587919034161780 {V_{3}}^{16} + 3854973026778738548528301975 {V_{3}}^{15} - 11285605066899279765507874785 {V_{3}}^{14} + 25336808943793774745789892645 {V_{3}}^{13} - 44576788581013137499375558695 {V_{3}}^{12} + 62265430290205110414091207275 {V_{3}}^{11} - 69518976850677741943051288485 {V_{3}}^{10} + 62144287063998197466231301905 {V_{3}}^{9} - 44343219668358476489779129275 {V_{3}}^{8} + 25058830235274350537877771525 {V_{3}}^{7} - 11065052075748421853638622835 {V_{3}}^{6} + 3740098548535950992665316175 {V_{3}}^{5} - 938838937634212539725528805 {V_{3}}^{4} + 167319183609966327876869625 {V_{3}}^{3} - 19743757865122107285616455 {V_{3}}^{2} + 1365898707678911505745275 V_{3} - 41527744080504851312325 \end{matrix}

-->	num_x_aa1 : num(radcan(ex_eff_0 − ex_x_aa1));

\begin{matrix} num_x_aa1 & 568640056127827331809280 {V_{3}}^{20} - 11103379349337001960830720 {V_{3}}^{19} + 101646850766876433966320208 {V_{3}}^{18} - 579491109926439741206883852 {V_{3}}^{17} + 2304664734260178941339462778 {V_{3}}^{16} - 6787211602214095158971203650 {V_{3}}^{15} + 15332100494323816028859457926 {V_{3}}^{14} - 27149781654675733892762739222 {V_{3}}^{13} + 38181905527878914236545248802 {V_{3}}^{12} - 42936803338879078654465813370 {V_{3}}^{11} + 38673800623931694166644720606 {V_{3}}^{10} - 27816641526735107708365916118 {V_{3}}^{9} + 15850813916190421057871758350 {V_{3}}^{8} - 7059216951499845637355579910 {V_{3}}^{7} + 2406572131820616308189370546 {V_{3}}^{6} - 609063872681123288182474530 {V_{3}}^{5} + 109347930502134789217084758 {V_{3}}^{4} - 12982904592852095995472190 {V_{3}}^{3} + 903135583153785879126938 {V_{3}}^{2} - 27644760474732201556710 V_{3} \end{matrix}

-->	nroots(denom_x_aa1, 0, 0.11);

\begin{matrix} (%o269) & 0 \end{matrix}

-->	nroots(num_x_aa1, 0, 0.11);

\begin{matrix} (%o270) & 0 \end{matrix}

-->	ex_eff_0 − ex_x_aa1, V_3 = 0.1;

\begin{matrix} (%o271) & 0.07624391966313893 \end{matrix}

We calculate expected utilities:

-->	EU_1aa1 : radcan(EU_i3), S_i = S_1aa, F_i = F_1aa1, F_j = F_2aa, S_j = S_2aa1;

\begin{matrix} (Config tells to suppress the output of long cells) \end{matrix}

-->	radcan(limit(EU_1aa1, ε, 0, plus));

\begin{matrix} (%o273) & \frac{117597066676244870070272 {V_{3}}^{34} - 3997683304310491073339392 {V_{3}}^{33} + 65477700367749976273913856 {V_{3}}^{32} - 688086937144329891914630144 {V_{3}}^{31} + 5212226708009276750386094080 {V_{3}}^{30} - 30313062420603156580200109008 {V_{3}}^{29} + 140769602295252579065179262504 {V_{3}}^{28} - 535988292070268458057883369047 {V_{3}}^{27} + 1704981580783596561138886186254 {V_{3}}^{26} - 4593709458353799824687473223883 {V_{3}}^{25} + 10590563338383480541080938590344 {V_{3}}^{24} - 21051833373450503192496406190094 {V_{3}}^{23} + 36283043915847621663516738927156 {V_{3}}^{22} - 54434426338267719376852925129958 {V_{3}}^{21} + 71268827388943593231318650892768 {V_{3}}^{20} - 81531380154707898490993932409089 {V_{3}}^{19} + 81499553312060082869887276547754 {V_{3}}^{18} - 71095567686852461497720640301645 {V_{3}}^{17} + 53978686471545598858723024629552 {V_{3}}^{16} - 35513109662540198810492884056996 {V_{3}}^{15} + 20112267875497800735434838401592 {V_{3}}^{14} - 9708300719928954667042611593604 {V_{3}}^{13} + 3933925273406789249255505166440 {V_{3}}^{12} - 1304693950288032630386704558953 {V_{3}}^{11} + 337207119358249305685174981506 {V_{3}}^{10} - 59774038095221640917711612085 {V_{3}}^{9} + 3303346073747346211376647080 {V_{3}}^{8} + 2108364980476759391646242834 {V_{3}}^{7} - 906404127808034703641520556 {V_{3}}^{6} + 209051073401986714581997338 {V_{3}}^{5} - 32491052376779693600421104 {V_{3}}^{4} + 3494615373661911648748129 {V_{3}}^{3} - 250697822670933718596090 {V_{3}}^{2} + 10815931276735497404141 V_{3} - 212322457099174690528}{205653017985207978557440 {V_{3}}^{33} - 7063434554242704671993856 {V_{3}}^{32} + 116961224005753543951958016 {V_{3}}^{31} - 1243457755144611854638850560 {V_{3}}^{30} + 9536281821021501208852920192 {V_{3}}^{29} - 56197655173731044763081060336 {V_{3}}^{28} + 264689380742376052226212448200 {V_{3}}^{27} - 1023243221813203566915609403863 {V_{3}}^{26} + 3308688329154444721831806175374 {V_{3}}^{25} - 9074056735828703825412522494955 {V_{3}}^{24} + 21327339636261007477812131239080 {V_{3}}^{23} - 43298673241690467797857189542414 {V_{3}}^{22} + 76380018865295181119113324678836 {V_{3}}^{21} - 117581490637416081242228062978182 {V_{3}}^{20} + 158443806829971432798241460798976 {V_{3}}^{19} - 187250847087578248508193722192001 {V_{3}}^{18} + 194258384179326266163748028288298 {V_{3}}^{17} - 176899034869922098390110332209389 {V_{3}}^{16} + 141266554855200145477004127636432 {V_{3}}^{15} - 98739139109292783245069277358500 {V_{3}}^{14} + 60229576093043767933592890138296 {V_{3}}^{13} - 31934573712189178691487190997988 {V_{3}}^{12} + 14640909231496449811480084631880 {V_{3}}^{11} - 5765424544629710538526650578409 {V_{3}}^{10} + 1933759388135844679580859123714 {V_{3}}^{9} - 546645629068161245835157893141 {V_{3}}^{8} + 128522542602723849071368956936 {V_{3}}^{7} - 24710391561382643507229039086 {V_{3}}^{6} + 3800567864133079434686660052 {V_{3}}^{5} - 453948717081879895180488390 {V_{3}}^{4} + 40366476950431725000610928 {V_{3}}^{3} - 2502576475146888400271391 {V_{3}}^{2} + 96092081192923177090950 V_{3} - 1713843406497025609715} \end{matrix}

-->	EU_2aa1 : radcan(EU_j3), S_i = S_1aa, F_i = F_1aa1, F_j = F_2aa, S_j = S_2aa1;

\begin{matrix} EU_2aa1 & \frac{(449565190576672072531968 {V_{3}}^{29} - 12216081058998211826417664 {V_{3}}^{28} + 159417325218031265914159104 {V_{3}}^{27} - 1330161716368106623235358720 {V_{3}}^{26} + 7970793415163207546637201408 {V_{3}}^{25} - 36526476723415601883133188096 {V_{3}}^{24} + 133087931291559911366721749760 {V_{3}}^{23} - 395768732656245196319135974656 {V_{3}}^{22} + 978362553692852827145748483840 {V_{3}}^{21} - 2037445551914794187964355881216 {V_{3}}^{20} + 3609258044806036050972271232256 {V_{3}}^{19} - 5477170243902827898168294273792 {V_{3}}^{18} + 7155720663782914358677458415872 {V_{3}}^{17} - 8074424237658291185685959335680 {V_{3}}^{16} + 7882801276830611156004035943936 {V_{3}}^{15} - 6660586168346393487075385963008 {V_{3}}^{14} + 4866111954234985590816427243008 {V_{3}}^{13} - 3066745958005833638411246959104 {V_{3}}^{12} + 1660968618395246299439510137344 {V_{3}}^{11} - 768943136248197806594238432768 {V_{3}}^{10} + 302078103872942148276076319232 {V_{3}}^{9} - 99743339314328638520608151040 {V_{3}}^{8} + 27338421695111177161264770816 {V_{3}}^{7} - 6119298530798799996523968768 {V_{3}}^{6} + 1094595591311874158489586432 {V_{3}}^{5} - 151899288665232056877649152 {V_{3}}^{4} + 15672119755139449738576128 {V_{3}}^{3} - 1125092671403518449522432 {V_{3}}^{2} + 49884670740899491478784 V_{3} - 1023678908822586753792) ε^{6} + (- (712722589054766253342720 {V_{3}}^{30}) + 21533257234637581647347712 {V_{3}}^{29} - 312114584944161919223857152 {V_{3}}^{28} + 2890403038912070184108982272 {V_{3}}^{27} - 19214315162150088351609556992 {V_{3}}^{26} + 97658859842707419683544250368 {V_{3}}^{25} - 394685801559939089991697014528 {V_{3}}^{24} + 1302301621634003748225572241888 {V_{3}}^{23} - 3574376266997538542504856720672 {V_{3}}^{22} + 8272220511308030717721207163488 {V_{3}}^{21} - 16305627823571570521775090171040 {V_{3}}^{20} + 27578038821459794411746901935392 {V_{3}}^{19} - 40237026085043732623902453321696 {V_{3}}^{18} + 50830383557429145257727634211232 {V_{3}}^{17} - 55722459673716583374404756949600 {V_{3}}^{16} + 53059027344055227275939944641216 {V_{3}}^{15} - 43872481252206728941694735852352 {V_{3}}^{14} + 31454569171607782116465022974912 {V_{3}}^{13} - 19500242094965852243761416745536 {V_{3}}^{12} + 10410454878787708725592322160192 {V_{3}}^{11} - 4758863168825405649192044606400 {V_{3}}^{10} + 1848713421561270034390854914880 {V_{3}}^{9} - 604386506593494865379540564928 {V_{3}}^{8} + 164184176518544031130607271264 {V_{3}}^{7} - 36454073074474180683702584736 {V_{3}}^{6} + 6472327870246040896664320992 {V_{3}}^{5} - 891905417067183961519996704 {V_{3}}^{4} + 91403081307656091356566176 {V_{3}}^{3} - 6518229795651562256100960 {V_{3}}^{2} + 287070306174219147817248 V_{3} - 5850354143745379413216) ε^{5} + (390642502147888243802112 {V_{3}}^{31} - 13272256198130242392686592 {V_{3}}^{30} + 215653153682915761927716864 {V_{3}}^{29} - 2233293058521788893536264192 {V_{3}}^{28} + 16571211705392032951854400512 {V_{3}}^{27} - 93886087439008011832506874368 {V_{3}}^{26} + 422586842303691860700926781504 {V_{3}}^{25} - 1552190036652821995330010319792 {V_{3}}^{24} + 4742038340517092414127456243888 {V_{3}}^{23} - 12219247511690205401483554388016 {V_{3}}^{22} + 26835306325212713773625079788880 {V_{3}}^{21} - 50621539591382595368889208215024 {V_{3}}^{20} + 82496009812763079094844765809296 {V_{3}}^{19} - 116623821933392375758176319675536 {V_{3}}^{18} + 143410685603711433308201552171184 {V_{3}}^{17} - 153625434685969409477832228621696 {V_{3}}^{16} + 143411083877931223564033753155552 {V_{3}}^{15} - 116575793917653392659970325711072 {V_{3}}^{14} + 82360283663283555754684740439584 {V_{3}}^{13} - 50415979478980512195609437613024 {V_{3}}^{12} + 26622005754211736739318921114144 {V_{3}}^{11} - 12054654838078527956778061233696 {V_{3}}^{10} + 4644514507620817731151932382368 {V_{3}}^{9} - 1507484910887041530420190059696 {V_{3}}^{8} + 406911983622915087806119886448 {V_{3}}^{7} - 89831305549012141737389920752 {V_{3}}^{6} + 15865542724231150332168621456 {V_{3}}^{5} - 2175428877830959526799288624 {V_{3}}^{4} + 221848695121814247899072592 {V_{3}}^{3} - 15741931609232229825501264 {V_{3}}^{2} + 689676897091078494411120 V_{3} - 13977023871804049424160) ε^{4} + (- (54681299672363794169856 {V_{3}}^{32}) + 2404093046348032589168640 {V_{3}}^{31} - 48249114053778779731329024 {V_{3}}^{30} + 599481098591975208587870208 {V_{3}}^{29} - 5234093958353386843413633024 {V_{3}}^{28} + 34423849323393239404214737920 {V_{3}}^{27} - 178126111005871999345032090240 {V_{3}}^{26} + 746874309544065385468558674480 {V_{3}}^{25} - 2591404819337700315376103194128 {V_{3}}^{24} + 7556060255249643654738728077488 {V_{3}}^{23} - 18730752500828948641199048065104 {V_{3}}^{22} + 39821210417891947358689900720272 {V_{3}}^{21} - 73085856349869807492561512928624 {V_{3}}^{20} + 116364940730389495647958304175120 {V_{3}}^{19} - 161275122070696752033016596676464 {V_{3}}^{18} + 194991014687225926583369336089824 {V_{3}}^{17} - 205882734060657863793531034809696 {V_{3}}^{16} + 189835243549487794779782390380512 {V_{3}}^{15} - 152694009081128524911791690716704 {V_{3}}^{14} + 106910688705627602412347195236896 {V_{3}}^{13} - 64943268533247445636042363661280 {V_{3}}^{12} + 34068745393683787274711281920288 {V_{3}}^{11} - 15340101037600844104958799160416 {V_{3}}^{10} + 5881765414720010305665674000880 {V_{3}}^{9} - 1900992343941502494373294563024 {V_{3}}^{8} + 511193103234850202711735132784 {V_{3}}^{7} - 112459526143758073046399731344 {V_{3}}^{6} + 19795278890033907630287422032 {V_{3}}^{5} - 2705005863017103957357257136 {V_{3}}^{4} + 274852030591228202706625488 {V_{3}}^{3} - 19424479894190154575998128 {V_{3}}^{2} + 847144392685175725816320 V_{3} - 17079535361756468064960) ε^{3} + (- (25319594887747222044672 {V_{3}}^{33}) + 760119444746392307171328 {V_{3}}^{32} - 10481980244813156956446720 {V_{3}}^{31} + 85706207820553940210866176 {V_{3}}^{30} - 432010984598501902795107072 {V_{3}}^{29} + 1039250750289140344441765248 {V_{3}}^{28} + 3119276795424106669832523600 {V_{3}}^{27} - 46912907683392879481312804200 {V_{3}}^{26} + 278100074847751312456982537064 {V_{3}}^{25} - 1150597055738936367191842369992 {V_{3}}^{24} + 3722784336472681599842285512920 {V_{3}}^{23} - 9865128661156794346925621809032 {V_{3}}^{22} + 21940920647594305354162802268792 {V_{3}}^{21} - 41567206838516549501159193785208 {V_{3}}^{20} + 67723536052090170332691214895928 {V_{3}}^{19} - 95487802927389517739869892702304 {V_{3}}^{18} + 116979681991402094739317871444144 {V_{3}}^{17} - 124796459055145143495100737578256 {V_{3}}^{16} + 116028874145702373284151168955632 {V_{3}}^{15} - 93967352309253366477424350409104 {V_{3}}^{14} + 66170499773554589912620067970288 {V_{3}}^{13} - 40392413611236993241278077031792 {V_{3}}^{12} + 21279049218838671280519304350656 {V_{3}}^{11} - 9616312901353788961906744390824 {V_{3}}^{10} + 3698707754757168075938022263880 {V_{3}}^{9} - 1198582382590495125700301877096 {V_{3}}^{8} + 322991174693707282680594547896 {V_{3}}^{7} - 71165295234219774825477184104 {V_{3}}^{6} + 12537481745345272476471980568 {V_{3}}^{5} - 1713387034167290503731994392 {V_{3}}^{4} + 173953384197473103451756344 {V_{3}}^{3} - 12271233785927171188759056 {V_{3}}^{2} + 533610256598715946586400 V_{3} - 10714701734677531103040) ε^{2} + (10142308234722855813120 {V_{3}}^{34} - 370497220226370613739520 {V_{3}}^{33} + 6497872639361215566077952 {V_{3}}^{32} - 72821014434252714838726656 {V_{3}}^{31} + 585471746441127574103498496 {V_{3}}^{30} - 3593492715596282592372335232 {V_{3}}^{29} + 17490309604807557100772097936 {V_{3}}^{28} - 69199153803597340954948645386 {V_{3}}^{27} + 226216237420496487153642129558 {V_{3}}^{26} - 617274348565670264724230689074 {V_{3}}^{25} + 1412565110861890399991170625214 {V_{3}}^{24} - 2707082495935228502258085901206 {V_{3}}^{23} + 4298998715030409074450920527354 {V_{3}}^{22} - 5494502504046812624116445262942 {V_{3}}^{21} + 5199290221403743534052477910594 {V_{3}}^{20} - 2464734727023772444585002947268 {V_{3}}^{19} - 2732239295496581131419427244916 {V_{3}}^{18} + 9082770835217541614325490734540 {V_{3}}^{17} - 14415623359923571253330407523604 {V_{3}}^{16} + 16865640817328532871060892209332 {V_{3}}^{15} - 15907584294403762175714564947116 {V_{3}}^{14} + 12490831433780603082704944396260 {V_{3}}^{13} - 8273460106153705501814009862348 {V_{3}}^{12} + 4643313348846339483406465953006 {V_{3}}^{11} - 2206549418219971273921579607250 {V_{3}}^{10} + 883877244542791526031395447718 {V_{3}}^{9} - 296088474235251915111948030378 {V_{3}}^{8} + 81994364774957568425259220290 {V_{3}}^{7} - 18474594048947663254908806382 {V_{3}}^{6} + 3314415168921466622925987930 {V_{3}}^{5} - 459540618356033368715674278 {V_{3}}^{4} + 47173536569020374604084608 {V_{3}}^{3} - 3354221824225274074876848 {V_{3}}^{2} + 146596277393374427936352 V_{3} - 2950966691061739829856) ε - 1049102166280197521408 {V_{3}}^{35} + 42030653936344485281792 {V_{3}}^{34} - 812078846341980008757248 {V_{3}}^{33} + 10077865771660870231572992 {V_{3}}^{32} - 90263371580216485800145472 {V_{3}}^{31} + 621583837413953736933910688 {V_{3}}^{30} - 3423445168881262036921986020 {V_{3}}^{29} + 15487851946455637715037138137 {V_{3}}^{28} - 58659572625335535571823300609 {V_{3}}^{27} + 188619488008146527201332718025 {V_{3}}^{26} - 520356917444241434308466936079 {V_{3}}^{25} + 1241485601876272612879889192157 {V_{3}}^{24} - 2577003512934931491622720339515 {V_{3}}^{23} + 4674615970448434003237618241475 {V_{3}}^{22} - 7433388077522211632847866936817 {V_{3}}^{21} + 10382161999910863785907863247176 {V_{3}}^{20} - 12747838436895504667907836224066 {V_{3}}^{19} + 13758522865188606290887442228730 {V_{3}}^{18} - 13036825438529592164255285275302 {V_{3}}^{17} + 10819668564179467803152881213290 {V_{3}}^{16} - 7836019104061340259789623967510 {V_{3}}^{15} + 4926023683014604934844398675238 {V_{3}}^{14} - 2667734791375741074630765334374 {V_{3}}^{13} + 1231339647361462645013981928537 {V_{3}}^{12} - 476789548127145176947862490261 {V_{3}}^{11} + 151017707066630242040507498541 {V_{3}}^{10} - 37357344212299389128080433643 {V_{3}}^{9} + 6457994566394362176229042153 {V_{3}}^{8} - 456982148297188367843687839 {V_{3}}^{7} - 137599942409190647200762265 {V_{3}}^{6} + 59450805419377337024798699 {V_{3}}^{5} - 12119611039285586110409738 {V_{3}}^{4} + 1561714391560545857817608 {V_{3}}^{3} - 128818088560398703338464 {V_{3}}^{2} + 6230139396395736549104 V_{3} - 134541691494334256288}{(224782595288336036265984 {V_{3}}^{29} - 6108040529499105913208832 {V_{3}}^{28} + 79708662609015632957079552 {V_{3}}^{27} - 665080858184053311617679360 {V_{3}}^{26} + 3985396707581603773318600704 {V_{3}}^{25} - 18263238361707800941566594048 {V_{3}}^{24} + 66543965645779955683360874880 {V_{3}}^{23} - 197884366328122598159567987328 {V_{3}}^{22} + 489181276846426413572874241920 {V_{3}}^{21} - 1018722775957397093982177940608 {V_{3}}^{20} + 1804629022403018025486135616128 {V_{3}}^{19} - 2738585121951413949084147136896 {V_{3}}^{18} + 3577860331891457179338729207936 {V_{3}}^{17} - 4037212118829145592842979667840 {V_{3}}^{16} + 3941400638415305578002017971968 {V_{3}}^{15} - 3330293084173196743537692981504 {V_{3}}^{14} + 2433055977117492795408213621504 {V_{3}}^{13} - 1533372979002916819205623479552 {V_{3}}^{12} + 830484309197623149719755068672 {V_{3}}^{11} - 384471568124098903297119216384 {V_{3}}^{10} + 151039051936471074138038159616 {V_{3}}^{9} - 49871669657164319260304075520 {V_{3}}^{8} + 13669210847555588580632385408 {V_{3}}^{7} - 3059649265399399998261984384 {V_{3}}^{6} + 547297795655937079244793216 {V_{3}}^{5} - 75949644332616028438824576 {V_{3}}^{4} + 7836059877569724869288064 {V_{3}}^{3} - 562546335701759224761216 {V_{3}}^{2} + 24942335370449745739392 V_{3} - 511839454411293376896) ε^{5} + (- (431542637670449868963840 {V_{3}}^{30}) + 12830024015943117833502720 {V_{3}}^{29} - 183245281875050247665123328 {V_{3}}^{28} + 1674159783905302613905268736 {V_{3}}^{27} - 10991159452631392442411387904 {V_{3}}^{26} + 55222995811168017499389912576 {V_{3}}^{25} - 220808051223963448200391426752 {V_{3}}^{24} + 721367682281776132424725475520 {V_{3}}^{23} - 1961641280484256159370469281472 {V_{3}}^{22} + 4500679738173703597801860644160 {V_{3}}^{21} - 8799664103039582477132197109184 {V_{3}}^{20} + 14769876316166046481445381510208 {V_{3}}^{19} - 21395171892271132532963238760512 {V_{3}}^{18} + 26844963287792436788928851132352 {V_{3}}^{17} - 29239914640288661589711059536896 {V_{3}}^{16} + 27672868426299964586470671223680 {V_{3}}^{15} - 22749222761663361022051583356800 {V_{3}}^{14} + 16220250640156843524064810283136 {V_{3}}^{13} - 10002871427086758267653064780672 {V_{3}}^{12} + 5313369148670539275348011453568 {V_{3}}^{11} - 2417236622722660612961861269632 {V_{3}}^{10} + 934757080336373433916725166464 {V_{3}}^{9} - 304266811513491878508973331136 {V_{3}}^{8} + 82315517095367420765258874816 {V_{3}}^{7} - 18205907073026648217588116928 {V_{3}}^{6} + 3220742892202075018823538240 {V_{3}}^{5} - 442351649449074149270389440 {V_{3}}^{4} + 45195865328341649153344320 {V_{3}}^{3} - 3214420735183001450544960 {V_{3}}^{2} + 141237211781689670980800 V_{3} - 2872659695693978931840) ε^{4} + (333608765748277153628160 {V_{3}}^{31} - 10777738550474453353758720 {V_{3}}^{30} + 167399647301483497058107392 {V_{3}}^{29} - 1664606442325901741813587968 {V_{3}}^{28} + 11905943276340721190669073408 {V_{3}}^{27} - 65238760777244701552399535616 {V_{3}}^{26} + 284825946132323344998026698176 {V_{3}}^{25} - 1017360573746556973700677075776 {V_{3}}^{24} + 3029256344529743440599709525440 {V_{3}}^{23} - 7622853682913726962522522992192 {V_{3}}^{22} + 16377427173256331977889649071424 {V_{3}}^{21} - 30270539731103011555878342815424 {V_{3}}^{20} + 48402755655102946077322955008320 {V_{3}}^{19} - 67223282802302218471981030026432 {V_{3}}^{18} + 81301144516940714611889539753344 {V_{3}}^{17} - 85743232052146872801520788438912 {V_{3}}^{16} + 78874686948033297715897255370112 {V_{3}}^{15} - 63232678708428024336751685089920 {V_{3}}^{14} + 44091826036800179829526236917376 {V_{3}}^{13} - 26657417950122849766293920047488 {V_{3}}^{12} + 13911745154087331275351289971328 {V_{3}}^{11} - 6229480342205719279932227203968 {V_{3}}^{10} + 2374914888680270068813930377408 {V_{3}}^{9} - 763169339205322366925859932736 {V_{3}}^{8} + 204070151641215566090234812608 {V_{3}}^{7} - 44655677356856677479721675584 {V_{3}}^{6} + 7822535688048354195722117184 {V_{3}}^{5} - 1064572871214354157441543104 {V_{3}}^{4} + 107830810943801003430296640 {V_{3}}^{3} - 7605675899755985450151360 {V_{3}}^{2} + 331489178179912859166720 V_{3} - 6688537815578486019840) ε^{3} + (- (129851483490220904939520 {V_{3}}^{32}) + 4532067505595367899136000 {V_{3}}^{31} - 76125581834215228897443840 {V_{3}}^{30} + 819543126038009631338704896 {V_{3}}^{29} - 6353573681376091534767146496 {V_{3}}^{28} + 37783472275138877132928903936 {V_{3}}^{27} - 179270627231246443624578058848 {V_{3}}^{26} + 696914553597364007554531377888 {V_{3}}^{25} - 2262115919472904010625260836320 {V_{3}}^{24} + 6216357389898148502553244751904 {V_{3}}^{23} - 14613200440986204088636285152864 {V_{3}}^{22} + 29616322701825144755816316207264 {V_{3}}^{21} - 52050503871499586569893406697760 {V_{3}}^{20} + 79665560293694204291440329264096 {V_{3}}^{19} - 106496649905548659062265212884992 {V_{3}}^{18} + 124561874977807073810596639781184 {V_{3}}^{17} - 127562405360110194108204567258048 {V_{3}}^{16} + 114344356993617359881314545898816 {V_{3}}^{15} - 89596416711120488355950991665856 {V_{3}}^{14} + 61224609384226069211085130651968 {V_{3}}^{13} - 36358473942041319898276108037184 {V_{3}}^{12} + 18675102383503633886392611009216 {V_{3}}^{11} - 8244991362080922840651490010208 {V_{3}}^{10} + 3103901732909879606770331868000 {V_{3}}^{9} - 986229047440428280431342797664 {V_{3}}^{8} + 261051927652197659435210191008 {V_{3}}^{7} - 56601968412642024665112555744 {V_{3}}^{6} + 9832263103411990502436262944 {V_{3}}^{5} - 1327715619638260599400065696 {V_{3}}^{4} + 133505741610201653173254240 {V_{3}}^{3} - 9351048526753940096397120 {V_{3}}^{2} + 404795433102792027361920 V_{3} - 8112758895290272677120) ε^{2} + (25456486288458862264320 {V_{3}}^{33} - 955077747805705035939840 {V_{3}}^{32} + 17265262077354539234414592 {V_{3}}^{31} - 200285741527252669234068480 {V_{3}}^{30} + 1675317598001527836669906816 {V_{3}}^{29} - 10764176536527242386216019904 {V_{3}}^{28} + 55261855352386468310564305656 {V_{3}}^{27} - 232816974875717520435307314696 {V_{3}}^{26} + 820352039991347545068861010584 {V_{3}}^{25} - 2451661589902718172049895186856 {V_{3}}^{24} + 6280057799803500243017360578056 {V_{3}}^{23} - 13898677629663681778992452245560 {V_{3}}^{22} + 26736777704359656301464212129064 {V_{3}}^{21} - 44907169911601434568596604519128 {V_{3}}^{20} + 66066208161014335866028427443248 {V_{3}}^{19} - 85310595823460099376901851703440 {V_{3}}^{18} + 96795346016897120372459083811568 {V_{3}}^{17} - 96514854313523800066792718294160 {V_{3}}^{16} + 84505274943321555693416464724880 {V_{3}}^{15} - 64860407608776649743291066601200 {V_{3}}^{14} + 43521985192126220255730291848784 {V_{3}}^{13} - 25434693717207344543201885857008 {V_{3}}^{12} + 12881101533853442036627005983576 {V_{3}}^{11} - 5616672303684616599112725602088 {V_{3}}^{10} + 2091384736516636474811467709304 {V_{3}}^{9} - 658104766865869389145104240072 {V_{3}}^{8} + 172707445682442688076510756328 {V_{3}}^{7} - 37160658527990820498337988952 {V_{3}}^{6} + 6410638650962184439628948616 {V_{3}}^{5} - 860212653405108767403750648 {V_{3}}^{4} + 85988620315999187810035200 {V_{3}}^{3} - 5989113577298709397886400 {V_{3}}^{2} + 257845778223955254990720 V_{3} - 5139558236172473642880) ε - 2011823002029208485888 {V_{3}}^{34} + 80788710653567373164544 {V_{3}}^{33} - 1565113618355794922981376 {V_{3}}^{32} + 19482695801853798393368064 {V_{3}}^{31} - 175112282107196587059581376 {V_{3}}^{30} + 1210728767172848252081625312 {V_{3}}^{29} - 6698850317312375141349946236 {V_{3}}^{28} + 30464985377271946412092543164 {V_{3}}^{27} - 116077146686126919110414241756 {V_{3}}^{26} + 375809427279012900044266009956 {V_{3}}^{25} - 1044939318351982202644033450572 {V_{3}}^{24} + 2515637956483870642519731222228 {V_{3}}^{23} - 5276367346053950058045448805940 {V_{3}}^{22} + 9686920836653124676084874463372 {V_{3}}^{21} - 15620284159165620927906636607488 {V_{3}}^{20} + 22175152444401045850950267097080 {V_{3}}^{19} - 27754041008800162358112364444440 {V_{3}}^{18} + 30640238365050250478790773213736 {V_{3}}^{17} - 29827766219987655798294185939160 {V_{3}}^{16} + 25574127210407506345103476203048 {V_{3}}^{15} - 19272585881215391679629785762344 {V_{3}}^{14} + 12727287815073532774621527450744 {V_{3}}^{13} - 7335506139353547903246114586044 {V_{3}}^{12} + 3670631179292243754927267646572 {V_{3}}^{11} - 1584041653937995636859238009420 {V_{3}}^{10} + 584590860354915156959707922676 {V_{3}}^{9} - 182555606995222495083762770460 {V_{3}}^{8} + 47595850283642811890061042756 {V_{3}}^{7} - 10183607096232780002709610020 {V_{3}}^{6} + 1748273497356690068841324252 {V_{3}}^{5} - 233593566728528694761982216 {V_{3}}^{4} + 23260996949830927258241952 {V_{3}}^{3} - 1614353806405371545981568 {V_{3}}^{2} + 69262727396684033923776 V_{3} - 1375851989048483276928} \end{matrix}

-->	radcan(limit(EU_2aa1, ε, 0, plus));

\begin{matrix} (%o275) & \frac{1049102166280197521408 {V_{3}}^{30} - 31539632273542510067712 {V_{3}}^{29} + 454718436955347007223808 {V_{3}}^{28} - 4185167937863283954913792 {V_{3}}^{27} + 27615855968183949354925632 {V_{3}}^{26} - 139084343409950757484096032 {V_{3}}^{25} + 555767317836152650590825636 {V_{3}}^{24} - 1808172129229731140746724721 {V_{3}}^{23} + 4877217295329716507044053495 {V_{3}}^{22} - 11048589594989957475671840571 {V_{3}}^{21} + 21216474926764071723491288985 {V_{3}}^{20} - 34761560673520899975934001835 {V_{3}}^{19} + 48802849768234307802562045413 {V_{3}}^{18} - 58848930634381662016177637625 {V_{3}}^{17} + 60987065386546325408086580175 {V_{3}}^{16} - 54254087673267133748486913066 {V_{3}}^{15} + 41302570939033929370441184646 {V_{3}}^{14} - 26762774609844803620430982414 {V_{3}}^{13} + 14636829784497036189764202042 {V_{3}}^{12} - 6670500587137833864981513750 {V_{3}}^{11} + 2482457355233554327886165418 {V_{3}}^{10} - 728464380323112296543805906 {V_{3}}^{9} + 156636117344458316431957986 {V_{3}}^{8} - 19522481477356486041313653 {V_{3}}^{7} - 879014350734203635111629 {V_{3}}^{6} + 1077229836062895212336025 {V_{3}}^{5} - 270991395391461655261059 {V_{3}}^{4} + 39451361656117043723729 {V_{3}}^{3} - 3554602231641034857087 {V_{3}}^{2} + 184180786489371903387 V_{3} - 4204427859197945509}{2011823002029208485888 {V_{3}}^{29} - 60670480633275288305664 {V_{3}}^{28} + 877935891941873700489216 {V_{3}}^{27} - 8115571816941713177378304 {V_{3}}^{26} + 53824543969280147522905536 {V_{3}}^{25} - 272707566660301395982298976 {V_{3}}^{24} + 1097370819847203521155552188 {V_{3}}^{23} - 3599671300947966546806700132 {V_{3}}^{22} + 9803333734250454494032686228 {V_{3}}^{21} - 22460581570972335139910072124 {V_{3}}^{20} + 43710141475497421375211527380 {V_{3}}^{19} - 72757009839018827227084484652 {V_{3}}^{18} + 104088882595706830552907791836 {V_{3}}^{17} - 128387093921708216453643239316 {V_{3}}^{16} + 136747200687850928593089270360 {V_{3}}^{15} - 125800758660237800467493825640 {V_{3}}^{14} + 99842958441940977923069372616 {V_{3}}^{13} - 68190622393978499428676349144 {V_{3}}^{12} + 39917905028470435721416204968 {V_{3}}^{11} - 19915474178130024518347675608 {V_{3}}^{10} + 8404194294106202490200646264 {V_{3}}^{9} - 2970041069243663439989711112 {V_{3}}^{8} + 867703761514368339438005676 {V_{3}}^{7} - 206060217401430164114836596 {V_{3}}^{6} + 38898535256542638215206884 {V_{3}}^{5} - 5662434251501972324366508 {V_{3}}^{4} + 608713384659468012692676 {V_{3}}^{3} - 45198638527268539795644 {V_{3}}^{2} + 2056971794501963304108 V_{3} - 42995374657765102404} \end{matrix}

-->	EU_3aa1 : radcan(EU_o3), S_i = S_1aa, F_i = F_1aa1, F_j = F_2aa, S_j = S_2aa1, S_o = EU_3ba, F_o = EU_3ca1;

\begin{matrix} EU_3aa1 & \frac{(361221181683793920 {V_{3}}^{21} - 7072819577356197888 {V_{3}}^{20} + 65463936509426319360 {V_{3}}^{19} - 380778040198193241600 {V_{3}}^{18} + 1560767482349507903616 {V_{3}}^{17} - 4791152930090681379456 {V_{3}}^{16} + 11426432042612303539200 {V_{3}}^{15} - 21674185661240219649024 {V_{3}}^{14} + 33201062980672382221824 {V_{3}}^{13} - 41464305063419672618496 {V_{3}}^{12} + 42437615887242652898304 {V_{3}}^{11} - 35647243029094050330624 {V_{3}}^{10} + 24527805962324704531200 {V_{3}}^{9} - 13749509998802410258176 {V_{3}}^{8} + 6220733173728399553536 {V_{3}}^{7} - 2239589595883612105728 {V_{3}}^{6} + 628594418918313303552 {V_{3}}^{5} - 133551357190636340736 {V_{3}}^{4} + 20565377288677297152 {V_{3}}^{3} - 2144372357943507456 {V_{3}}^{2} + 134155595007250560 V_{3} - 3770568282983040) ε^{5} + (- (464161415714832384 {V_{3}}^{22}) + 10537901223464484864 {V_{3}}^{21} - 112889148213927306240 {V_{3}}^{20} + 759222874520767556352 {V_{3}}^{19} - 3597174277707115843200 {V_{3}}^{18} + 12769805867310952075152 {V_{3}}^{17} - 35261157012471044444304 {V_{3}}^{16} + 77595824868989616031104 {V_{3}}^{15} - 138295705023060020258688 {V_{3}}^{14} + 201734823221153090251200 {V_{3}}^{13} - 242386894377754929938880 {V_{3}}^{12} + 240578813226254424727680 {V_{3}}^{11} - 197221299222009882584448 {V_{3}}^{10} + 133111878422525067525984 {V_{3}}^{9} - 73496004976392976830816 {V_{3}}^{8} + 32861552819538664568448 {V_{3}}^{7} - 11723528916586286725248 {V_{3}}^{6} + 3267669948539760948672 {V_{3}}^{5} - 690578165296287396288 {V_{3}}^{4} + 105904506230179606656 {V_{3}}^{3} - 11005583717210121216 {V_{3}}^{2} + 686443576230238608 V_{3} - 19235236821733008) ε^{4} + (176230742248783872 {V_{3}}^{23} - 4752161303703146496 {V_{3}}^{22} + 59989250976554926080 {V_{3}}^{21} - 472787050557151699200 {V_{3}}^{20} + 2615218631315669694528 {V_{3}}^{19} - 10814028875679277045728 {V_{3}}^{18} + 34745695559628108637536 {V_{3}}^{17} - 88976361957620338576512 {V_{3}}^{16} + 184745114462356405337664 {V_{3}}^{15} - 314659529979925083386688 {V_{3}}^{14} + 442931124195285867357120 {V_{3}}^{13} - 517471563282165699039936 {V_{3}}^{12} + 502433181756464975437248 {V_{3}}^{11} - 404882268156917801899776 {V_{3}}^{10} + 269686443063503259953280 {V_{3}}^{9} - 147424373259794351379648 {V_{3}}^{8} + 65432110543534489224384 {V_{3}}^{7} - 23220334028649609950400 {V_{3}}^{6} + 6448593887563345846848 {V_{3}}^{5} - 1359493575284345797440 {V_{3}}^{4} + 208137876872771061504 {V_{3}}^{3} - 21600965658896455968 {V_{3}}^{2} + 1345474664290389024 V_{3} - 37639351728271296) ε^{3} + (4139288039522304 {V_{3}}^{24} + 114196634440998912 {V_{3}}^{23} - 4251457497912740352 {V_{3}}^{22} + 56196625597465887360 {V_{3}}^{21} - 443417768409834676800 {V_{3}}^{20} + 2426697599585447035800 {V_{3}}^{19} - 9900100925118937679544 {V_{3}}^{18} + 31393862886475467453024 {V_{3}}^{17} - 79462726348339517784768 {V_{3}}^{16} + 163392245600355460931040 {V_{3}}^{15} - 276135006346865052085152 {V_{3}}^{14} + 386416142748570378267360 {V_{3}}^{13} - 449568615130123760987040 {V_{3}}^{12} + 435365324823427675662096 {V_{3}}^{11} - 350400855451304749599504 {V_{3}}^{10} + 233382934218102618389472 {V_{3}}^{9} - 127697499833809350170592 {V_{3}}^{8} + 56774223536794455391200 {V_{3}}^{7} - 20194326232733934183456 {V_{3}}^{6} + 5623158364085853164448 {V_{3}}^{5} - 1188740172051365234976 {V_{3}}^{4} + 182458707517886231448 {V_{3}}^{3} - 18973903973676997752 {V_{3}}^{2} + 1183229221258033344 V_{3} - 33105428354827872) ε^{2} + (- (15197452953403392 {V_{3}}^{25}) + 395883101108030976 {V_{3}}^{24} - 4845551922443612160 {V_{3}}^{23} + 36945899460197565984 {V_{3}}^{22} - 195910405054880002776 {V_{3}}^{21} + 762154827287640720336 {V_{3}}^{20} - 2226579296948501552712 {V_{3}}^{19} + 4866204165686832401280 {V_{3}}^{18} - 7548113035220746933584 {V_{3}}^{17} + 6448717539440554444560 {V_{3}}^{16} + 4194959453361305527536 {V_{3}}^{15} - 27774174226241324216016 {V_{3}}^{14} + 59925181979770435299744 {V_{3}}^{13} - 88113573218277894292560 {V_{3}}^{12} + 99076154788034988864480 {V_{3}}^{11} - 88487108381223504112464 {V_{3}}^{10} + 63640403331763223886192 {V_{3}}^{9} - 36937894250927039755536 {V_{3}}^{8} + 17207911388072010267792 {V_{3}}^{7} - 6356403032013254142864 {V_{3}}^{6} + 1825579466755141317048 {V_{3}}^{5} - 395870798430592580160 {V_{3}}^{4} + 62034420220260903144 {V_{3}}^{3} - 6558218867874319344 {V_{3}}^{2} + 414163023716008800 V_{3} - 11693396406314304) ε + 2333731940663296 {V_{3}}^{26} - 70149455811773440 {V_{3}}^{25} + 1003489331871003328 {V_{3}}^{24} - 9089082106529780560 {V_{3}}^{23} + 58501912106578866136 {V_{3}}^{22} - 284678330329560486007 {V_{3}}^{21} + 1087771042104123626959 {V_{3}}^{20} - 3346101384118407761920 {V_{3}}^{19} + 8427907105514813327320 {V_{3}}^{18} - 17583570809630327821492 {V_{3}}^{17} + 30620841336910442358436 {V_{3}}^{16} - 44709645156821238446368 {V_{3}}^{15} + 54830481483191367806848 {V_{3}}^{14} - 56428876734278475665002 {V_{3}}^{13} + 48556096834486968729898 {V_{3}}^{12} - 34689581547687046576864 {V_{3}}^{11} + 20340167146632010820128 {V_{3}}^{10} - 9608669065910414485636 {V_{3}}^{9} + 3544335598464677515828 {V_{3}}^{8} - 960686528500550749264 {V_{3}}^{7} + 162714524085956056744 {V_{3}}^{6} - 4147021539525265303 {V_{3}}^{5} - 6345159916971844865 {V_{3}}^{4} + 1893287540316890528 {V_{3}}^{3} - 275632018128693752 {V_{3}}^{2} + 21191373923198384 V_{3} - 683162001546128}{(180610590841896960 {V_{3}}^{21} - 3536409788678098944 {V_{3}}^{20} + 32731968254713159680 {V_{3}}^{19} - 190389020099096620800 {V_{3}}^{18} + 780383741174753951808 {V_{3}}^{17} - 2395576465045340689728 {V_{3}}^{16} + 5713216021306151769600 {V_{3}}^{15} - 10837092830620109824512 {V_{3}}^{14} + 16600531490336191110912 {V_{3}}^{13} - 20732152531709836309248 {V_{3}}^{12} + 21218807943621326449152 {V_{3}}^{11} - 17823621514547025165312 {V_{3}}^{10} + 12263902981162352265600 {V_{3}}^{9} - 6874754999401205129088 {V_{3}}^{8} + 3110366586864199776768 {V_{3}}^{7} - 1119794797941806052864 {V_{3}}^{6} + 314297209459156651776 {V_{3}}^{5} - 66775678595318170368 {V_{3}}^{4} + 10282688644338648576 {V_{3}}^{3} - 1072186178971753728 {V_{3}}^{2} + 67077797503625280 V_{3} - 1885284141491520) ε^{4} + (- (288760219092910080 {V_{3}}^{22}) + 6408458288615596032 {V_{3}}^{21} - 67286312751294332928 {V_{3}}^{20} + 444505624312482994176 {V_{3}}^{19} - 2072575903945925506176 {V_{3}}^{18} + 7252006225188885072768 {V_{3}}^{17} - 19764104530744344190464 {V_{3}}^{16} + 42975665781932510551296 {V_{3}}^{15} - 75757011521901520492800 {V_{3}}^{14} + 109393618192914490958592 {V_{3}}^{13} - 130207196466722279759616 {V_{3}}^{12} + 128107572492863033047296 {V_{3}}^{11} - 104161221101949438841344 {V_{3}}^{10} + 69762204537740036172288 {V_{3}}^{9} - 38239642485577587269376 {V_{3}}^{8} + 16981096192912059565824 {V_{3}}^{7} - 6019158146693323140864 {V_{3}}^{6} + 1667574437488226862336 {V_{3}}^{5} - 350431890153758440704 {V_{3}}^{4} + 53460499099458620160 {V_{3}}^{3} - 5529115085445968256 {V_{3}}^{2} + 343377768884388480 V_{3} - 9584764672976640) ε^{3} + (173866867567165440 {V_{3}}^{23} - 4314860273911412736 {V_{3}}^{22} + 50753307561062289408 {V_{3}}^{21} - 376373153482773031296 {V_{3}}^{20} + 1974433844941997993568 {V_{3}}^{19} - 7792886363172130723680 {V_{3}}^{18} + 24027025916638112895360 {V_{3}}^{17} - 59305293219619416598272 {V_{3}}^{16} + 119134844969264362694016 {V_{3}}^{15} - 196940151070857854319744 {V_{3}}^{14} + 269796126109508250480000 {V_{3}}^{13} - 307471175364252069774720 {V_{3}}^{12} + 291802841279173449787968 {V_{3}}^{11} - 230247460216431994478400 {V_{3}}^{10} + 150400464170752096380288 {V_{3}}^{9} - 80738074508757822170496 {V_{3}}^{8} + 35233626313735905184128 {V_{3}}^{7} - 12308084754747938930304 {V_{3}}^{6} + 3368355263005664845440 {V_{3}}^{5} - 700535580971983241472 {V_{3}}^{4} + 105921242714786764896 {V_{3}}^{3} - 10868730776921519712 {V_{3}}^{2} + 670114981493280000 V_{3} - 18575799933559680) ε^{2} + (- (46776336986603520 {V_{3}}^{24}) + 1284314832319776768 {V_{3}}^{23} - 16750915973621600256 {V_{3}}^{22} + 138073094621632472832 {V_{3}}^{21} - 807204400843731070752 {V_{3}}^{20} + 3560650511844386894688 {V_{3}}^{19} - 12307994377203872724480 {V_{3}}^{18} + 34178731730528043689280 {V_{3}}^{17} - 77549764223538683542080 {V_{3}}^{16} + 145440237617695147455936 {V_{3}}^{15} - 227196103903379270457408 {V_{3}}^{14} + 296984379417586423674048 {V_{3}}^{13} - 325507863006739135691136 {V_{3}}^{12} + 299042862657067122993792 {V_{3}}^{11} - 229657532293555859946816 {V_{3}}^{10} + 146673274345546629515328 {V_{3}}^{9} - 77278017898767240407232 {V_{3}}^{8} + 33205134325682214222912 {V_{3}}^{7} - 11451788823252389895360 {V_{3}}^{6} + 3100950008976023671104 {V_{3}}^{5} - 639247855463370147168 {V_{3}}^{4} + 95933041819452516384 {V_{3}}^{3} - 9779397553460021184 {V_{3}}^{2} + 599322980854097280 V_{3} - 16516572628872960) ε + 4749609140674560 {V_{3}}^{25} - 143037860744127744 {V_{3}}^{24} + 2051183368715687424 {V_{3}}^{23} - 18636700970466781488 {V_{3}}^{22} + 120427592944812971652 {V_{3}}^{21} - 588894771665996995572 {V_{3}}^{20} + 2263897496114367743232 {V_{3}}^{19} - 7016400538659814437216 {V_{3}}^{18} + 17836668162848655092016 {V_{3}}^{17} - 37641605584581965089584 {V_{3}}^{16} + 66487442851068275157120 {V_{3}}^{15} - 98812623645472199555808 {V_{3}}^{14} + 123911241480903938165784 {V_{3}}^{13} - 131194228545598809052152 {V_{3}}^{12} + 117112121548114942050048 {V_{3}}^{11} - 87825889427621187070656 {V_{3}}^{10} + 55005998023493687728368 {V_{3}}^{9} - 28523968967489419172592 {V_{3}}^{8} + 12100439234660362063488 {V_{3}}^{7} - 4130970943105388516976 {V_{3}}^{6} + 1109704072987123839108 {V_{3}}^{5} - 227343822529648663476 {V_{3}}^{4} + 33952300466699204352 {V_{3}}^{3} - 3447483344060789088 {V_{3}}^{2} + 210556287642813120 V_{3} - 5783968662937920} \end{matrix}

-->	radcan(limit(EU_3aa1, ε, 0, plus));

\begin{matrix} (%o277) & \frac{2333731940663296 {V_{3}}^{22} - 51479600286467072 {V_{3}}^{21} + 535642963003347648 {V_{3}}^{20} - 3493748573526564176 {V_{3}}^{19} + 16011805486068450136 {V_{3}}^{18} - 54769672255684746807 {V_{3}}^{17} + 144960090632099233239 {V_{3}}^{16} - 303670772194564541472 {V_{3}}^{15} + 510680352828908297808 {V_{3}}^{14} - 694991798011381031140 {V_{3}}^{13} + 767752324585941902996 {V_{3}}^{12} - 687319762222460281632 {V_{3}}^{11} + 495244413722354120480 {V_{3}}^{10} - 283303976228358547882 {V_{3}}^{9} + 125529510829801995162 {V_{3}}^{8} - 41112596893329104928 {V_{3}}^{7} + 8920252704990636912 {V_{3}}^{6} - 796752623190260772 {V_{3}}^{5} - 199761303983136524 {V_{3}}^{4} + 87080877116477104 {V_{3}}^{3} - 14684823455385144 {V_{3}}^{2} + 1239065620006633 V_{3} - 42697625096633}{4749609140674560 {V_{3}}^{21} - 105040987618731264 {V_{3}}^{20} + 1096864848389647872 {V_{3}}^{19} - 7188810987998462256 {V_{3}}^{18} + 33155043329423531268 {V_{3}}^{17} - 114342630368277219156 {V_{3}}^{16} + 305843624071800081408 {V_{3}}^{15} - 649446054897232129536 {V_{3}}^{14} + 1111407880891948588656 {V_{3}}^{13} - 1547159163722767159920 {V_{3}}^{12} + 1760608658019142302336 {V_{3}}^{11} - 1639745385074580389472 {V_{3}}^{10} + 1247051274448153258008 {V_{3}}^{9} - 769905432046825407288 {V_{3}}^{8} + 382056654291811299072 {V_{3}}^{7} - 150138880628696415360 {V_{3}}^{6} + 45733058791780066416 {V_{3}}^{5} - 10471511979296585712 {V_{3}}^{4} + 1723079036001707136 {V_{3}}^{3} - 190051918152031728 {V_{3}}^{2} + 12436771894808580 V_{3} - 361498041433620} \end{matrix}

2.7.1.2 Subcase A.2:

We define the remaining prizes in this subcase:

-->	F_1aa2 : EU_1ca2 ;

\begin{matrix} F_1aa2 & \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} \end{matrix}

-->	S_2aa2 : EU_2ca2;

\begin{matrix} S_2aa2 & - (\frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) \end{matrix}

And prize differences:

-->	D_1aa2 : radcan(S_1aa − F_1aa2);

\begin{matrix} D_1aa2 & - (\frac{(- (1994544 {V_{3}}^{8}) + 15090300 {V_{3}}^{7} - 49155012 {V_{3}}^{6} + 89833212 {V_{3}}^{5} - 100383300 {V_{3}}^{4} + 69835284 {V_{3}}^{3} - 29262060 {V_{3}}^{2} + 6639732 V_{3} - 603612) ε^{3} + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} ((56 3^{\frac{15}{2}} {V_{3}}^{7} - 40 3^{\frac{19}{2}} {V_{3}}^{6} + 968 3^{\frac{15}{2}} {V_{3}}^{5} - 1400 3^{\frac{15}{2}} {V_{3}}^{4} + 1160 3^{\frac{15}{2}} {V_{3}}^{3} - 536 3^{\frac{15}{2}} {V_{3}}^{2} + 40 3^{\frac{17}{2}} V_{3} - 8 3^{\frac{15}{2}}) ε^{2} + (- (1288 3^{\frac{9}{2}} {V_{3}}^{8}) + 11752 3^{\frac{9}{2}} {V_{3}}^{7} - 45592 3^{\frac{9}{2}} {V_{3}}^{6} + 32600 3^{\frac{11}{2}} {V_{3}}^{5} - 4664 3^{\frac{15}{2}} {V_{3}}^{4} + 32776 3^{\frac{11}{2}} {V_{3}}^{3} - 44264 3^{\frac{9}{2}} {V_{3}}^{2} + 9848 3^{\frac{9}{2}} V_{3} - 656 3^{\frac{9}{2}}) ε + 7406 3^{\frac{3}{2}} {V_{3}}^{9} - 87538 3^{\frac{3}{2}} {V_{3}}^{8} + 450106 3^{\frac{3}{2}} {V_{3}}^{7} - 1316758 3^{\frac{3}{2}} {V_{3}}^{6} + 2402650 3^{\frac{3}{2}} {V_{3}}^{5} - 2812270 3^{\frac{3}{2}} {V_{3}}^{4} + 2081614 3^{\frac{3}{2}} {V_{3}}^{3} - 913810 3^{\frac{3}{2}} {V_{3}}^{2} + 202048 3^{\frac{3}{2}} V_{3} - 13448 3^{\frac{3}{2}}) + (3250368 {V_{3}}^{9} - 29836512 {V_{3}}^{8} + 120672828 {V_{3}}^{7} - 281622420 {V_{3}}^{6} + 416664324 {V_{3}}^{5} - 403524828 {V_{3}}^{4} + 254177028 {V_{3}}^{3} - 99409356 {V_{3}}^{2} + 21528828 V_{3} - 1900260) ε^{2} + (- (1683324 {V_{3}}^{10}) + 18815067 {V_{3}}^{9} - 92859453 {V_{3}}^{8} + 266244219 {V_{3}}^{7} - 490452381 {V_{3}}^{6} + 605326041 {V_{3}}^{5} - 505431927 {V_{3}}^{4} + 280621773 {V_{3}}^{3} - 98405199 {V_{3}}^{2} + 19420740 V_{3} - 1595556) ε + 303316 {V_{3}}^{11} - 4129693 {V_{3}}^{10} + 25069360 {V_{3}}^{9} - 89392470 {V_{3}}^{8} + 207524328 {V_{3}}^{7} - 328271388 {V_{3}}^{6} + 359509692 {V_{3}}^{5} - 271025238 {V_{3}}^{4} + 136764732 {V_{3}}^{3} - 43534127 {V_{3}}^{2} + 7760108 V_{3} - 578588}{(1994544 {V_{3}}^{8} - 15090300 {V_{3}}^{7} + 49155012 {V_{3}}^{6} - 89833212 {V_{3}}^{5} + 100383300 {V_{3}}^{4} - 69835284 {V_{3}}^{3} + 29262060 {V_{3}}^{2} - 6639732 V_{3} + 603612) ε^{2} + (- (1699056 {V_{3}}^{9}) + 17434764 {V_{3}}^{8} - 77854284 {V_{3}}^{7} + 198277308 {V_{3}}^{6} - 316657188 {V_{3}}^{5} + 327697164 {V_{3}}^{4} - 218496852 {V_{3}}^{3} + 89652420 {V_{3}}^{2} - 20187468 V_{3} + 1833192) ε + 361836 {V_{3}}^{10} - 4688343 {V_{3}}^{9} + 26872101 {V_{3}}^{8} - 89573931 {V_{3}}^{7} + 191875149 {V_{3}}^{6} - 275155569 {V_{3}}^{5} + 266368311 {V_{3}}^{4} - 170770653 {V_{3}}^{3} + 68663727 {V_{3}}^{2} - 15344496 V_{3} + 1391868}) \end{matrix}

-->	lim_D_1aa2 : radcan(limit(D_1aa2, ε, 0, plus));

\begin{matrix} lim_D_1aa2 & - (\frac{3991 {V_{3}}^{6} - 28134 {V_{3}}^{5} + 80967 {V_{3}}^{4} - 121084 {V_{3}}^{3} + 98289 {V_{3}}^{2} - 40326 V_{3} + 6289}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129}) \end{matrix}

-->	D_2aa2: radcan(S_2aa2 − F_2aa);

\begin{matrix} D_2aa2 & - (\frac{\sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} ((128 3^{\frac{9}{2}} {V_{3}}^{6} - 20 3^{\frac{15}{2}} {V_{3}}^{5} + 244 3^{\frac{11}{2}} {V_{3}}^{4} - 8 3^{\frac{13}{2}} {V_{3}}^{3} - 232 3^{\frac{11}{2}} {V_{3}}^{2} + 68 3^{\frac{13}{2}} V_{3} - 164 3^{\frac{9}{2}}) ε - 1472 3^{\frac{3}{2}} {V_{3}}^{7} + 10178 3^{\frac{3}{2}} {V_{3}}^{6} - 8770 3^{\frac{5}{2}} {V_{3}}^{5} + 27356 3^{\frac{3}{2}} {V_{3}}^{4} + 2608 3^{\frac{3}{2}} {V_{3}}^{3} - 10270 3^{\frac{5}{2}} {V_{3}}^{2} + 25174 3^{\frac{3}{2}} V_{3} - 6724 3^{\frac{3}{2}}) + (410400 {V_{3}}^{8} - 3166560 {V_{3}}^{7} + 10518390 {V_{3}}^{6} - 19613556 {V_{3}}^{5} + 22389642 {V_{3}}^{4} - 15938424 {V_{3}}^{3} + 6844554 {V_{3}}^{2} - 1592244 V_{3} + 147798) ε - 112784 {V_{3}}^{9} + 1099008 {V_{3}}^{8} - 4615479 {V_{3}}^{7} + 10931808 {V_{3}}^{6} - 16024959 {V_{3}}^{5} + 14988042 {V_{3}}^{4} - 8869821 {V_{3}}^{3} + 3166452 {V_{3}}^{2} - 610029 V_{3} + 47794}{(443232 {V_{3}}^{7} - 2910168 {V_{3}}^{6} + 8013168 {V_{3}}^{5} - 11949768 {V_{3}}^{4} + 10357632 {V_{3}}^{3} - 5161320 {V_{3}}^{2} + 1341360 V_{3} - 134136) ε - 188784 {V_{3}}^{8} + 1748412 {V_{3}}^{7} - 6902064 {V_{3}}^{6} + 15128748 {V_{3}}^{5} - 20055384 {V_{3}}^{4} + 16355412 {V_{3}}^{3} - 7922016 {V_{3}}^{2} + 2039364 V_{3} - 203688}) \end{matrix}

-->	lim_D_2aa2 : radcan(limit(D_2aa2, ε, 0, plus));

\begin{matrix} lim_D_2aa2 & - (\frac{1484 {V_{3}}^{5} - 7685 {V_{3}}^{4} + 15332 {V_{3}}^{3} - 14608 {V_{3}}^{2} + 6508 V_{3} - 1039}{2484 {V_{3}}^{4} - 11664 {V_{3}}^{3} + 20304 {V_{3}}^{2} - 15552 V_{3} + 4428}) \end{matrix}

-->	denom_D_1aa2 : denom(radcan(lim_D_1aa2));

\begin{matrix} denom_D_1aa2 & 4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129 \end{matrix}

-->	nroots(denom_D_1aa2, 0.1, 0.25);

\begin{matrix} (%o285) & 0 \end{matrix}

-->	num_D_1aa2 : num(radcan(lim_D_1aa2));

\begin{matrix} num_D_1aa2 & - (3991 {V_{3}}^{6}) + 28134 {V_{3}}^{5} - 80967 {V_{3}}^{4} + 121084 {V_{3}}^{3} - 98289 {V_{3}}^{2} + 40326 V_{3} - 6289 \end{matrix}

-->	nroots(num_D_1aa2, 0.1, 0.25);

\begin{matrix} (%o287) & 0 \end{matrix}

-->	lim_D_1aa2, V_3 = 0.2;

\begin{matrix} (%o288) & 0.2083878643664485 \end{matrix}

-->	denom_D_2aa2 : denom(radcan(lim_D_2aa2));

\begin{matrix} denom_D_2aa2 & 2484 {V_{3}}^{4} - 11664 {V_{3}}^{3} + 20304 {V_{3}}^{2} - 15552 V_{3} + 4428 \end{matrix}

-->	nroots(denom_D_2aa2, 0.1, 0.25);

\begin{matrix} (%o290) & 0 \end{matrix}

-->	num_D_2aa2 : num(radcan(lim_D_2aa2));

\begin{matrix} num_D_2aa2 & - (1484 {V_{3}}^{5}) + 7685 {V_{3}}^{4} - 15332 {V_{3}}^{3} + 14608 {V_{3}}^{2} - 6508 V_{3} + 1039 \end{matrix}

-->	nroots(num_D_2aa2, 0.1, 0.25);

\begin{matrix} (%o292) & 0 \end{matrix}

-->	lim_D_2aa2, V_3 = 0.2;

\begin{matrix} (%o293) & 0.10335365853658535 \end{matrix}

We calculate best responses:

-->	x_1aa2 : x_i3, S_i = S_1aa, F_i = F_1aa2, F_j = F_2aa, S_j = S_2aa2;

\begin{matrix} x_1aa2 & \frac{(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} ((- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}) ((- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) - ε \end{matrix}

-->	x_1aa2 : radcan(x_1aa2);

\begin{matrix} (Config tells to suppress the output of long cells) \end{matrix}

-->	lim_x_1aa2 : radcan(limit(x_1aa2, ε, 0, plus));

\begin{matrix} lim_x_1aa2 & - (\frac{94549088816 {V_{3}}^{17} - 1822650540308 {V_{3}}^{16} + 16414767591488 {V_{3}}^{15} - 91681488787640 {V_{3}}^{14} + 355485031830032 {V_{3}}^{13} - 1014804696674984 {V_{3}}^{12} + 2206957802709536 {V_{3}}^{11} - 3731560650616184 {V_{3}}^{10} + 4961236283070416 {V_{3}}^{9} - 5210339293545904 {V_{3}}^{8} + 4316585771388160 {V_{3}}^{7} - 2800745071374952 {V_{3}}^{6} + 1403321966056048 {V_{3}}^{5} - 530558358698200 {V_{3}}^{4} + 145854243847840 {V_{3}}^{3} - 27411442518376 {V_{3}}^{2} + 3137613293440 V_{3} - 164376121276}{464227623744 {V_{3}}^{16} - 8657203230192 {V_{3}}^{15} + 75220161876117 {V_{3}}^{14} - 404056836319830 {V_{3}}^{13} + 1501213562366349 {V_{3}}^{12} - 4088472648179508 {V_{3}}^{11} + 8437920096277329 {V_{3}}^{10} - 13451827614587778 {V_{3}}^{9} + 16726862938208409 {V_{3}}^{8} - 16260433692473064 {V_{3}}^{7} + 12301361643028767 {V_{3}}^{6} - 7155530861800410 {V_{3}}^{5} + 3131767053308775 {V_{3}}^{4} - 994850187949428 {V_{3}}^{3} + 215765341116795 {V_{3}}^{2} - 28460576692686 V_{3} + 1714597428147}) \end{matrix}

-->	x_2aa2 : x_j3, S_i = S_1aa, F_i = F_1aa2, F_j = F_2aa, S_j = S_2aa2;

\begin{matrix} x_2aa2 & \frac{{(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} ({(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) ({(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}^{2} (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})^(2) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) - ε \end{matrix}

-->	x_2aa2 : radcan(x_2aa2);

\begin{matrix} (Config tells to suppress the output of long cells) \end{matrix}

-->	lim_x_2aa2 : radcan(limit(x_2aa2, ε, 0, plus));

\begin{matrix} lim_x_2aa2 & - (\frac{8789203696 {V_{3}}^{16} - 152989308584 {V_{3}}^{15} + 1237338040863 {V_{3}}^{14} - 6168787681942 {V_{3}}^{13} + 21205537754975 {V_{3}}^{12} - 53255269062924 {V_{3}}^{11} + 100985870442547 {V_{3}}^{10} - 147342454956770 {V_{3}}^{9} + 166968713121507 {V_{3}}^{8} - 147239491429328 {V_{3}}^{7} + 100544332916989 {V_{3}}^{6} - 52511586620442 {V_{3}}^{5} + 20521224706013 {V_{3}}^{4} - 5788162070524 {V_{3}}^{3} + 1108728671025 {V_{3}}^{2} - 128582835182 V_{3} + 6789107569}{60551429184 {V_{3}}^{15} - 1026526258800 {V_{3}}^{14} + 8065428638121 {V_{3}}^{13} - 38942906191893 {V_{3}}^{12} + 129159497304270 {V_{3}}^{11} - 311500462821966 {V_{3}}^{10} + 563939571692763 {V_{3}}^{9} - 779716117910583 {V_{3}}^{8} + 829223909066484 {V_{3}}^{7} - 677462899808196 {V_{3}}^{6} + 421065475942239 {V_{3}}^{5} - 195158222344899 {V_{3}}^{4} + 65147802013566 {V_{3}}^{3} - 14747864469246 {V_{3}}^{2} + 2018222067357 V_{3} - 125458348401}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_aa2 : radcan(limit(lim_x_1aa2 + lim_x_2aa2 + lim_x_1aa2/(lim_x_1aa2 + lim_x_2aa2) · ex_x_ba + lim_x_2aa2/(lim_x_1aa2 + lim_x_2aa2) · ex_x_ca2, ε, 0, plus));

\begin{matrix} ex_x_aa2 & - (\frac{16059784 {V_{3}}^{11} - 209491354 {V_{3}}^{10} + 1231926781 {V_{3}}^{9} - 4308630531 {V_{3}}^{8} + 9951213930 {V_{3}}^{7} - 15920886786 {V_{3}}^{6} + 17981414760 {V_{3}}^{5} - 14311275672 {V_{3}}^{4} + 7846741038 {V_{3}}^{3} - 2812788644 {V_{3}}^{2} + 590163515 V_{3} - 54446693}{33957936 {V_{3}}^{10} - 396361602 {V_{3}}^{9} + 2063547954 {V_{3}}^{8} - 6305128884 {V_{3}}^{7} + 12507295644 {V_{3}}^{6} - 16805409120 {V_{3}}^{5} + 15458076936 {V_{3}}^{4} - 9583796556 {V_{3}}^{3} + 3816862452 {V_{3}}^{2} - 876178206 V_{3} + 87133446}) \end{matrix}

-->	denom_x_aa2 : denom(radcan(ex_eff_0 − ex_x_aa2));

\begin{matrix} denom_x_aa2 & 319880325076303320 {V_{3}}^{10} - 3733686231563789865 {V_{3}}^{9} + 19438413169047159105 {V_{3}}^{8} - 59393676843666516330 {V_{3}}^{7} + 117817460885377500030 {V_{3}}^{6} - 158305255429713764400 {V_{3}}^{5} + 145613522427929270820 {V_{3}}^{4} - 90278394947161576470 {V_{3}}^{3} + 35954458537094132490 {V_{3}}^{2} - 8253510147379165095 V_{3} + 820788254960446395 \end{matrix}

-->	num_x_aa2 : num(radcan(ex_eff_0 − ex_x_aa2));

\begin{matrix} num_x_aa2 & 151281542157780580 {V_{3}}^{11} - 1762198444267743737 {V_{3}}^{10} + 9139600048922251244 {V_{3}}^{9} - 27753383913130961643 {V_{3}}^{8} + 54526991160667372308 {V_{3}}^{7} - 72188603004094792398 {V_{3}}^{6} + 64868026726388866140 {V_{3}}^{5} - 38674926181319493522 {V_{3}}^{4} + 14312597230854460632 {V_{3}}^{3} - 2758607573773361129 {V_{3}}^{2} + 110211424913329672 V_{3} + 29012188429355213 \end{matrix}

-->	nroots(denom_x_aa2, 0.1, 0.25);

\begin{matrix} (%o303) & 0 \end{matrix}

-->	nroots(num_x_aa2, 0.1, 0.25);

\begin{matrix} (%o304) & 0 \end{matrix}

-->	ex_eff_0 − ex_x_aa2, V_3 = 0.2;

\begin{matrix} (%o305) & 0.13432975411028236 \end{matrix}

We calculate expected utilities:

-->	EU_1aa2 : EU_i3, S_i = S_1aa, F_i = F_1aa2, F_j = F_2aa, S_j = S_2aa2;

\begin{matrix} EU_1aa2 & \frac{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3} / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) + ε + \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} \end{matrix}

-->	radcan(limit(EU_1aa2, ε, 0, plus));

\begin{matrix} (%o307) & \frac{2541974908368 {V_{3}}^{19} - 53439035167520 {V_{3}}^{18} + 526904668136337 {V_{3}}^{17} - 3235472104371504 {V_{3}}^{16} + 13853825778482868 {V_{3}}^{15} - 43879968918207816 {V_{3}}^{14} + 106399685330137788 {V_{3}}^{13} - 201581853692771232 {V_{3}}^{12} + 301730160714952092 {V_{3}}^{11} - 358147679784758280 {V_{3}}^{10} + 335981955148295390 {V_{3}}^{9} - 246136831524972768 {V_{3}}^{8} + 137283037667294700 {V_{3}}^{7} - 55272457682536920 {V_{3}}^{6} + 13963928544043212 {V_{3}}^{5} - 911380015481952 {V_{3}}^{4} - 799492788176652 {V_{3}}^{3} + 329660711197512 {V_{3}}^{2} - 57104823251703 V_{3} + 3979832249104}{3559078448704 {V_{3}}^{18} - 75965928988848 {V_{3}}^{17} + 761947885332033 {V_{3}}^{16} - 4770634201371072 {V_{3}}^{15} + 20887820807725428 {V_{3}}^{14} - 67892147354651784 {V_{3}}^{13} + 169702407486176124 {V_{3}}^{12} - 333390153226691040 {V_{3}}^{11} + 521561961210202044 {V_{3}}^{10} - 654193469764633576 {V_{3}}^{9} + 658959862396512126 {V_{3}}^{8} - 531313137693529536 {V_{3}}^{7} + 340009794340635564 {V_{3}}^{6} - 170142625653932568 {V_{3}}^{5} + 65015254561403532 {V_{3}}^{4} - 18273634039699872 {V_{3}}^{3} + 3550123493860836 {V_{3}}^{2} - 424396228315080 V_{3} + 23432831518009} \end{matrix}

-->	EU_2aa2 : EU_j3, S_i = S_1aa, F_i = F_1aa2, F_j = F_2aa, S_j = S_2aa2;

\begin{matrix} EU_2aa2 & \frac{{(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}^{3}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} {(- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}^{3} (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})^(3) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) + \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + ε \end{matrix}

-->	radcan(limit(EU_2aa2, ε, 0, plus));

\begin{matrix} (%o309) & \frac{2751802545856 {V_{3}}^{18} - 54123190490928 {V_{3}}^{17} + 497920480408488 {V_{3}}^{16} - 2844480076191537 {V_{3}}^{15} + 11295551767385919 {V_{3}}^{14} - 33067944139724247 {V_{3}}^{13} + 73843521361247421 {V_{3}}^{12} - 128350774910726193 {V_{3}}^{11} + 175571786449714503 {V_{3}}^{10} - 189759708618091663 {V_{3}}^{9} + 161657893796691309 {V_{3}}^{8} - 107549258360459307 {V_{3}}^{7} + 54878320441102485 {V_{3}}^{6} - 20812019838387261 {V_{3}}^{5} + 5542995672798639 {V_{3}}^{4} - 919447470829683 {V_{3}}^{3} + 62600202452517 {V_{3}}^{2} + 5359082091987 V_{3} - 944451546497}{5570731484928 {V_{3}}^{17} - 109457170247232 {V_{3}}^{16} + 1006528381275708 {V_{3}}^{15} - 5751323978351364 {V_{3}}^{14} + 22863244784234148 {V_{3}}^{13} - 67076234526550284 {V_{3}}^{12} + 150316712933482044 {V_{3}}^{11} - 262676972530381284 {V_{3}}^{10} + 362144286633554244 {V_{3}}^{9} - 395847559568177676 {V_{3}}^{8} + 342741544026021972 {V_{3}}^{7} - 233482710057950124 {V_{3}}^{6} + 123447574981310220 {V_{3}}^{5} - 49519406895098436 {V_{3}}^{4} + 14527386348794676 {V_{3}}^{3} - 2930711013713772 {V_{3}}^{2} + 362102089449996 V_{3} - 20575169137764} \end{matrix}

-->	EU_3aa2 : EU_o3, S_i = S_1aa, F_i = F_1aa2, F_j = F_2aa, S_j = S_2aa2, S_o = EU_3ba, F_o = EU_3ca2;

\begin{matrix} EU_3aa2 & \frac{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{V_{3}}{2})}{\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}} ((\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{V_{3}}{2})) / (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) - \frac{((4 3^{\frac{5}{2}} - 4 3^{\frac{5}{2}} V_{3}) ε + 4 \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} - 3^{\frac{3}{2}} V_{3} + 3^{\frac{3}{2}}) (- (\frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828})}{(4 3^{\frac{3}{2}} V_{3} - 4 3^{\frac{3}{2}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} - \frac{(- (2736 {V_{3}}^{4}) + 9756 {V_{3}}^{3} - 11988 {V_{3}}^{2} + 5796 V_{3} - 828) ε + \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} (64 \sqrt{3} {V_{3}}^{3} - 26 3^{\frac{3}{2}} {V_{3}}^{2} - 20 3^{\frac{3}{2}} V_{3} + 82 \sqrt{3}) \sqrt{ε} - 684 {V_{3}}^{4} + 2439 {V_{3}}^{3} - 2997 {V_{3}}^{2} + 1449 V_{3} - 207}{2736 {V_{3}}^{4} - 9756 {V_{3}}^{3} + 11988 {V_{3}}^{2} - 5796 V_{3} + 828} + ε - \frac{(14 \sqrt{3} {V_{3}}^{2} - 20 \sqrt{3} V_{3} + 2 \sqrt{3}) \sqrt{- (76 {V_{3}}^{3}) + 195 {V_{3}}^{2} - 138 V_{3} + 23} \sqrt{ε} + 228 {V_{3}}^{4} - 585 {V_{3}}^{3} + 414 {V_{3}}^{2} - 69 V_{3}}{228 {V_{3}}^{3} - 585 {V_{3}}^{2} + 414 V_{3} - 69} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})} \end{matrix}

Auxiliary calculations:

-->	lim_S_1aa : radcan(limit(S_1aa, ε, 0, plus));

\begin{matrix} lim_S_1aa & \frac{770 {V_{3}}^{6} - 2295 {V_{3}}^{5} - 3729 {V_{3}}^{4} + 23758 {V_{3}}^{3} - 37404 {V_{3}}^{2} + 25197 V_{3} - 6289}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129} \end{matrix}

-->	lim_F_1aa2 : radcan(limit(F_1aa2, ε, 0, plus));

\begin{matrix} lim_F_1aa2 & V_{3} \end{matrix}

-->	lim_S_2aa2 : radcan(limit(S_2aa2, ε, 0, plus));

\begin{matrix} lim_S_2aa2 & \frac{1}{4} \end{matrix}

-->	lim_F_2aa : radcan(limit(F_2aa, ε, 0, plus));

\begin{matrix} lim_F_2aa & \frac{371 {V_{3}}^{5} - 1766 {V_{3}}^{4} + 3104 {V_{3}}^{3} - 2383 {V_{3}}^{2} + 655 V_{3} + 17}{621 {V_{3}}^{4} - 2916 {V_{3}}^{3} + 5076 {V_{3}}^{2} - 3888 V_{3} + 1107} \end{matrix}

-->	lim_S_o : radcan(limit(EU_3ba, ε, 0, plus));

\begin{matrix} lim_S_o & \frac{6476 {V_{3}}^{7} - 46735 {V_{3}}^{6} + 137823 {V_{3}}^{5} - 209993 {V_{3}}^{4} + 168562 {V_{3}}^{3} - 58737 {V_{3}}^{2} - 2305 V_{3} + 4913}{14283 {V_{3}}^{6} - 105570 {V_{3}}^{5} + 323001 {V_{3}}^{4} - 523692 {V_{3}}^{3} + 474633 {V_{3}}^{2} - 228042 V_{3} + 45387} \end{matrix}

-->	lim_F_o : radcan(limit(EU_3ca2, ε, 0, plus));

\begin{matrix} lim_F_o & \frac{1}{4} \end{matrix}

-->	radcan(limit(EU_o3, ε, 0, plus)), S_i = lim_S_1aa, F_i = lim_F_1aa2, F_j = lim_F_2aa, S_j = lim_S_2aa2, S_o = lim_S_o, F_o = lim_F_o;

\begin{matrix} (%o317) & \frac{413531456 {V_{3}}^{13} - 5736940252 {V_{3}}^{12} + 35909598729 {V_{3}}^{11} - 133488104491 {V_{3}}^{10} + 325776300275 {V_{3}}^{9} - 542625600357 {V_{3}}^{8} + 615706244994 {V_{3}}^{7} - 448163351718 {V_{3}}^{6} + 158609764206 {V_{3}}^{5} + 42099783266 {V_{3}}^{4} - 81777413323 {V_{3}}^{3} + 43144286097 {V_{3}}^{2} - 11006945377 V_{3} + 1138845983}{1562065056 {V_{3}}^{12} - 22443417756 {V_{3}}^{11} + 146856595284 {V_{3}}^{10} - 578417526324 {V_{3}}^{9} + 1526382513468 {V_{3}}^{8} - 2840974047864 {V_{3}}^{7} + 3820540512744 {V_{3}}^{6} - 3735699729480 {V_{3}}^{5} + 2631528754488 {V_{3}}^{4} - 1299466459116 {V_{3}}^{3} + 425636957124 {V_{3}}^{2} - 82651160196 V_{3} + 7144942572} \end{matrix}

2.7.1.3 Subcase A.3:

We define the remaining prizes in this subcase:

-->	F_1aa3 : EU_1ca3;

\begin{matrix} F_1aa3 & \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} \end{matrix}

-->	S_2aa3 : EU_2ca3;

\begin{matrix} S_2aa3 & \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

And prize differences:

-->	D_1aa3 : radcan(S_1aa − F_1aa3);

\begin{matrix} D_1aa3 & \frac{(52488 {V_{3}}^{5} - 262440 {V_{3}}^{4} + 524880 {V_{3}}^{3} - 524880 {V_{3}}^{2} + 262440 V_{3} - 52488) ε^{3} + (- (65124 {V_{3}}^{6}) + 454896 {V_{3}}^{5} - 1326780 {V_{3}}^{4} + 2064528 {V_{3}}^{3} - 1805004 {V_{3}}^{2} + 839808 V_{3} - 162324) ε^{2} + (26910 {V_{3}}^{7} - 240210 {V_{3}}^{6} + 910224 {V_{3}}^{5} - 1902060 {V_{3}}^{4} + 2372238 {V_{3}}^{3} - 1769418 {V_{3}}^{2} + 732204 V_{3} - 129888) ε - 4279 {V_{3}}^{8} + 48130 {V_{3}}^{7} - 233947 {V_{3}}^{6} + 641584 {V_{3}}^{5} - 1085053 {V_{3}}^{4} + 1157434 {V_{3}}^{3} - 758893 {V_{3}}^{2} + 278620 V_{3} - 43588}{(52488 {V_{3}}^{5} - 262440 {V_{3}}^{4} + 524880 {V_{3}}^{3} - 524880 {V_{3}}^{2} + 262440 V_{3} - 52488) ε^{2} + (- (44712 {V_{3}}^{6}) + 344088 {V_{3}}^{5} - 1084752 {V_{3}}^{4} + 1796256 {V_{3}}^{3} - 1650456 {V_{3}}^{2} + 798984 V_{3} - 159408) ε + 9522 {V_{3}}^{7} - 98946 {V_{3}}^{6} + 435996 {V_{3}}^{5} - 1055988 {V_{3}}^{4} + 1518282 {V_{3}}^{3} - 1295946 {V_{3}}^{2} + 608112 V_{3} - 121032} \end{matrix}

-->	lim_D_1aa3 : radcan(limit(D_1aa3, ε, 0, plus));

\begin{matrix} lim_D_1aa3 & - (\frac{4279 {V_{3}}^{6} - 31014 {V_{3}}^{5} + 92775 {V_{3}}^{4} - 146428 {V_{3}}^{3} + 128241 {V_{3}}^{2} - 58758 V_{3} + 10897}{9522 {V_{3}}^{5} - 60858 {V_{3}}^{4} + 154476 {V_{3}}^{3} - 194652 {V_{3}}^{2} + 121770 V_{3} - 30258}) \end{matrix}

-->	D_2aa3 : radcan(S_2aa3 − F_2aa);

\begin{matrix} D_2aa3 & - (\frac{(3024 {V_{3}}^{5} - 16686 {V_{3}}^{4} + 36720 {V_{3}}^{3} - 40500 {V_{3}}^{2} + 22464 V_{3} - 5022) ε - 744 {V_{3}}^{6} + 5651 {V_{3}}^{5} - 17534 {V_{3}}^{4} + 28556 {V_{3}}^{3} - 25876 {V_{3}}^{2} + 12445 V_{3} - 2506}{(3888 {V_{3}}^{4} - 15552 {V_{3}}^{3} + 23328 {V_{3}}^{2} - 15552 V_{3} + 3888) ε - 1656 {V_{3}}^{5} + 11088 {V_{3}}^{4} - 29088 {V_{3}}^{3} + 37440 {V_{3}}^{2} - 23688 V_{3} + 5904}) \end{matrix}

-->	lim_D_2aa3 : radcan(limit(D_2aa3, ε, 0, plus));

\begin{matrix} lim_D_2aa3 & - (\frac{744 {V_{3}}^{5} - 4163 {V_{3}}^{4} + 9208 {V_{3}}^{3} - 10140 {V_{3}}^{2} + 5596 V_{3} - 1253}{1656 {V_{3}}^{4} - 7776 {V_{3}}^{3} + 13536 {V_{3}}^{2} - 10368 V_{3} + 2952}) \end{matrix}

-->	denom_D_1aa3 : denom(radcan(lim_D_1aa3));

\begin{matrix} denom_D_1aa3 & 9522 {V_{3}}^{5} - 60858 {V_{3}}^{4} + 154476 {V_{3}}^{3} - 194652 {V_{3}}^{2} + 121770 V_{3} - 30258 \end{matrix}

-->	nroots(denom_D_1aa3, 0.24, 0.5);

\begin{matrix} (%o325) & 0 \end{matrix}

-->	num_D_1aa3 : num(radcan(lim_D_1aa3));

\begin{matrix} num_D_1aa3 & - (4279 {V_{3}}^{6}) + 31014 {V_{3}}^{5} - 92775 {V_{3}}^{4} + 146428 {V_{3}}^{3} - 128241 {V_{3}}^{2} + 58758 V_{3} - 10897 \end{matrix}

-->	nroots(num_D_1aa3, 0.24, 0.5);

\begin{matrix} (%o327) & 0 \end{matrix}

-->	lim_D_1aa3, V_3 = 0.3;

\begin{matrix} (%o328) & 0.20371641396446144 \end{matrix}

-->	denom_D_2aa3 : denom(radcan(lim_D_2aa3));

\begin{matrix} denom_D_2aa3 & 1656 {V_{3}}^{4} - 7776 {V_{3}}^{3} + 13536 {V_{3}}^{2} - 10368 V_{3} + 2952 \end{matrix}

-->	nroots(denom_D_2aa3, 0.24, 0.5);

\begin{matrix} (%o330) & 0 \end{matrix}

-->	num_D_2aa3 : num(radcan(lim_D_2aa3));

\begin{matrix} num_D_2aa3 & - (744 {V_{3}}^{5}) + 4163 {V_{3}}^{4} - 9208 {V_{3}}^{3} + 10140 {V_{3}}^{2} - 5596 V_{3} + 1253 \end{matrix}

-->	nroots(num_D_2aa3, 0.24, 0.5);

\begin{matrix} (%o332) & 0 \end{matrix}

-->	lim_D_2aa3, V_3 = 0.3;

\begin{matrix} (%o333) & 0.312864449689425 \end{matrix}

We calculate best responses:

-->	x_1aa3 : x_i3, S_i = S_1aa, F_i = F_1aa3, F_j = F_2aa, S_j = S_2aa3;

\begin{matrix} x_1aa3 & \frac{(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} ((\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}) ((\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214}) {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) - ε \end{matrix}

Auxiliary calculations:

-->	lim_S_1aa : radcan(limit(S_1aa, ε, 0, plus));

\begin{matrix} lim_S_1aa & \frac{770 {V_{3}}^{6} - 2295 {V_{3}}^{5} - 3729 {V_{3}}^{4} + 23758 {V_{3}}^{3} - 37404 {V_{3}}^{2} + 25197 V_{3} - 6289}{4761 {V_{3}}^{5} - 30429 {V_{3}}^{4} + 77238 {V_{3}}^{3} - 97326 {V_{3}}^{2} + 60885 V_{3} - 15129} \end{matrix}

-->	lim_F_1aa3 : radcan(limit(F_1aa3, ε, 0, plus));

\begin{matrix} lim_F_1aa3 & \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} \end{matrix}

-->	lim_F_2aa : radcan(limit(F_2aa, ε, 0, plus));

\begin{matrix} lim_F_2aa & \frac{371 {V_{3}}^{5} - 1766 {V_{3}}^{4} + 3104 {V_{3}}^{3} - 2383 {V_{3}}^{2} + 655 V_{3} + 17}{621 {V_{3}}^{4} - 2916 {V_{3}}^{3} + 5076 {V_{3}}^{2} - 3888 V_{3} + 1107} \end{matrix}

-->	lim_S_2aa3 : radcan(limit(S_2aa3, ε, 0, plus));

\begin{matrix} lim_S_2aa3 & \frac{32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} \end{matrix}

-->	lim_x_1aa3 : radcan(limit(x_i3, ε, 0, plus)), S_i = lim_S_1aa, F_i = lim_F_1aa3, F_j = lim_F_2aa, S_j = lim_S_2aa3;

\begin{matrix} lim_x_1aa3 & - (\frac{27245043408 {V_{3}}^{17} - 547389440422 {V_{3}}^{16} + 5159747663816 {V_{3}}^{15} - 30305928359580 {V_{3}}^{14} + 124225385384848 {V_{3}}^{13} - 377116432593332 {V_{3}}^{12} + 877926350054520 {V_{3}}^{11} - 1600818033200476 {V_{3}}^{10} + 2314434846384416 {V_{3}}^{9} - 2668084854391536 {V_{3}}^{8} + 2452189568970296 {V_{3}}^{7} - 1786366709104180 {V_{3}}^{6} + 1018645867368912 {V_{3}}^{5} - 445055115023276 {V_{3}}^{4} + 143841174961288 {V_{3}}^{3} - 32396252454612 {V_{3}}^{2} + 4538102725040 V_{3} - 297573990154}{242512088688 {V_{3}}^{16} - 4664643524280 {V_{3}}^{15} + 41935834856655 {V_{3}}^{14} - 233916006171906 {V_{3}}^{13} + 906135745921407 {V_{3}}^{12} - 2584961850651300 {V_{3}}^{11} + 5617721132113347 {V_{3}}^{10} - 9487567125464166 {V_{3}}^{9} + 12584604041887491 {V_{3}}^{8} - 13154328530976816 {V_{3}}^{7} + 10799591882123853 {V_{3}}^{6} - 6890891488384014 {V_{3}}^{5} + 3350071536131133 {V_{3}}^{4} - 1199652814301940 {V_{3}}^{3} + 298433611947681 {V_{3}}^{2} - 46081328862474 V_{3} + 3327491271249}) \end{matrix}

-->	x_2aa3 : x_j3, S_i = S_1aa, F_i = F_1aa3, F_j = F_2aa, S_j = S_2aa3;

\begin{matrix} x_2aa3 & \frac{{(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} ({(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) ({(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) - ε \end{matrix}

-->	lim_x_2aa3 : radcan(limit(x_j3, ε, 0, plus)), S_i = lim_S_1aa, F_i = lim_F_1aa3, F_j = lim_F_2aa, S_j = lim_S_2aa3;

\begin{matrix} lim_x_2aa3 & - (\frac{2368580544 {V_{3}}^{16} - 43673819280 {V_{3}}^{15} + 376258130383 {V_{3}}^{14} - 2010796614742 {V_{3}}^{13} + 7462396396183 {V_{3}}^{12} - 20395990069364 {V_{3}}^{11} + 42476138640227 {V_{3}}^{10} - 68767458642914 {V_{3}}^{9} + 87482498979003 {V_{3}}^{8} - 87755777557512 {V_{3}}^{7} + 69194718889677 {V_{3}}^{6} - 42441622508954 {V_{3}}^{5} + 19854933864581 {V_{3}}^{4} - 6849517794932 {V_{3}}^{3} + 1643480635009 {V_{3}}^{2} - 245065496494 V_{3} + 17108388073}{21088007712 {V_{3}}^{15} - 369863249904 {V_{3}}^{14} + 3017601171810 {V_{3}}^{13} - 15193392006906 {V_{3}}^{12} + 52799632683228 {V_{3}}^{11} - 134160386717484 {V_{3}}^{10} + 257505958653606 {V_{3}}^{9} - 380201357531070 {V_{3}}^{8} + 435396254912136 {V_{3}}^{7} - 386740942713144 {V_{3}}^{6} + 264289388112558 {V_{3}}^{5} - 136462870730358 {V_{3}}^{4} + 51537528310140 {V_{3}}^{3} - 13441054160556 {V_{3}}^{2} + 2164731905610 V_{3} - 162316647378}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_aa3 : radcan(limit(lim_x_1aa3 + lim_x_2aa3 + lim_x_1aa3/(lim_x_1aa3 + lim_x_2aa3) · ex_x_ba + lim_x_2aa3/(lim_x_1aa3 + lim_x_2aa3) · ex_x_ca3, ε, 0, plus));

\begin{matrix} ex_x_aa3 & - (\frac{10953872 {V_{3}}^{11} - 148307070 {V_{3}}^{10} + 908329429 {V_{3}}^{9} - 3322578765 {V_{3}}^{8} + 8066625930 {V_{3}}^{7} - 13650396198 {V_{3}}^{6} + 16430679456 {V_{3}}^{5} - 14068540296 {V_{3}}^{4} + 8397526710 {V_{3}}^{3} - 3327617260 {V_{3}}^{2} + 787653915 V_{3} - 84329531}{28340784 {V_{3}}^{10} - 339102252 {V_{3}}^{9} + 1817362764 {V_{3}}^{8} - 5745823128 {V_{3}}^{7} + 11869423560 {V_{3}}^{6} - 16741348704 {V_{3}}^{5} + 16329102768 {V_{3}}^{4} - 10876271976 {V_{3}}^{3} + 4734824328 {V_{3}}^{2} - 1216670580 V_{3} + 140162436}) \end{matrix}

-->	num_x_aa3 : num(radcan(ex_eff_0 − ex_x_aa3));

\begin{matrix} num_x_aa3 & 206368734319083280 {V_{3}}^{11} - 2441564981907752566 {V_{3}}^{10} + 12894898480993753983 {V_{3}}^{9} - 39991868999068721511 {V_{3}}^{8} + 80505519679399231422 {V_{3}}^{7} - 109535645578906158210 {V_{3}}^{6} + 101317314342680239536 {V_{3}}^{5} - 61942715067622705872 {V_{3}}^{4} + 22925730803191343874 {V_{3}}^{3} - 3798627283653155672 {V_{3}}^{2} - 294032998713661855 V_{3} + 154626486529693671 \end{matrix}

-->	denom_x_aa3 : denom(radcan(ex_eff_0 − ex_x_aa3));

\begin{matrix} denom_x_aa3 & 533934641895626160 {V_{3}}^{10} - 6388617883246291980 {V_{3}}^{9} + 34238747121137698860 {V_{3}}^{8} - 108250146299562021720 {V_{3}}^{7} + 223617540644468699400 {V_{3}}^{6} - 315403625570839716960 {V_{3}}^{5} + 307636995465931990320 {V_{3}}^{4} - 204906765552604131240 {V_{3}}^{3} + 89203133265804459720 {V_{3}}^{2} - 22921827795492311700 V_{3} + 2640631962505999140 \end{matrix}

-->	nroots(num_x_aa3, 0.24, 0.5);

\begin{matrix} (%o345) & 0 \end{matrix}

-->	nroots(denom_x_aa3, 0.24, 0.5);

\begin{matrix} (%o346) & 0 \end{matrix}

-->	ex_eff_0 − ex_x_aa3, V_3 = 0.3;

\begin{matrix} (%o347) & 0.1843444291073149 \end{matrix}

We calculate expected utilities:

-->	EU_1aa3 : EU_i3, S_i = S_1aa, F_i = F_1aa3, F_j = F_2aa, S_j = S_2aa3;

\begin{matrix} EU_1aa3 & \frac{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3} / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) + ε + \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} \end{matrix}

-->	radcan(limit(EU_1aa3, ε, 0, plus));

\begin{matrix} (%o349) & \frac{5563719316672 {V_{3}}^{20} - 124698886964472 {V_{3}}^{19} + 1317737510094763 {V_{3}}^{18} - 8723464654432440 {V_{3}}^{17} + 40538122621301187 {V_{3}}^{16} - 140403544639845960 {V_{3}}^{15} + 375501201232120656 {V_{3}}^{14} - 792491347119618504 {V_{3}}^{13} + 1336794097191221832 {V_{3}}^{12} - 1812893521033206616 {V_{3}}^{11} + 1975685573518197150 {V_{3}}^{10} - 1716907474388850712 {V_{3}}^{9} + 1168094716013293350 {V_{3}}^{8} - 598414436599004568 {V_{3}}^{7} + 209866525429338024 {V_{3}}^{6} - 33894224169477528 {V_{3}}^{5} - 10484652847159152 {V_{3}}^{4} + 8965052752627248 {V_{3}}^{3} - 2883256033559105 {V_{3}}^{2} + 487892316025392 V_{3} - 35861931415169}{11155556079648 {V_{3}}^{19} - 255800657193840 {V_{3}}^{18} + 2771995292803458 {V_{3}}^{17} - 18870030766813122 {V_{3}}^{16} + 90469248111070632 {V_{3}}^{15} - 324576757666257336 {V_{3}}^{14} + 903703762853841096 {V_{3}}^{13} - 1998245952630343800 {V_{3}}^{12} + 3560995622224488888 {V_{3}}^{11} - 5159573760224719656 {V_{3}}^{10} + 6103426013760640716 {V_{3}}^{9} - 5894640837242718012 {V_{3}}^{8} + 4628925710944927128 {V_{3}}^{7} - 2929786371541444680 {V_{3}}^{6} + 1472836763071425672 {V_{3}}^{5} - 574687800867732120 {V_{3}}^{4} + 167835745339079400 {V_{3}}^{3} - 34529983441491816 {V_{3}}^{2} + 4464132168600162 V_{3} - 272854284242418} \end{matrix}

-->	EU_2aa3 : EU_j3, S_i = S_1aa, F_i = F_1aa3, F_j = F_2aa, S_j = S_2aa3;

\begin{matrix} EU_2aa3 & \frac{{(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}^{3}}{{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2}} {(\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}^{3} / {(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{2} (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})^(2) + \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + ε \end{matrix}

-->	radcan(limit(EU_2aa3, ε, 0, plus));

\begin{matrix} (%o351) & \frac{2823602706304 {V_{3}}^{19} - 59566173588992 {V_{3}}^{18} + 591018994555440 {V_{3}}^{17} - 3663690110217039 {V_{3}}^{16} + 15893982732548484 {V_{3}}^{15} - 51215138656330074 {V_{3}}^{14} + 126936201440022288 {V_{3}}^{13} - 247153593015125178 {V_{3}}^{12} + 382611106146887052 {V_{3}}^{11} - 473264116300186762 {V_{3}}^{10} + 466972444311538184 {V_{3}}^{9} - 364176436572121296 {V_{3}}^{8} + 219969074649701724 {V_{3}}^{7} - 98763369972994782 {V_{3}}^{6} + 29943229654085376 {V_{3}}^{5} - 4230361449831414 {V_{3}}^{4} - 875845247989740 {V_{3}}^{3} + 601890035908290 {V_{3}}^{2} - 130801158912584 V_{3} + 11147089346767}{5820290128512 {V_{3}}^{18} - 123592024839744 {V_{3}}^{17} + 1236183215853672 {V_{3}}^{16} - 7738855665827904 {V_{3}}^{15} + 33981686234924976 {V_{3}}^{14} - 111147584367984480 {V_{3}}^{13} + 280650733904096496 {V_{3}}^{12} - 559391309768211840 {V_{3}}^{11} + 892259026198300080 {V_{3}}^{10} - 1147466489765183136 {V_{3}}^{9} + 1192628471663159424 {V_{3}}^{8} - 999465690806604864 {V_{3}}^{7} + 670354713480552336 {V_{3}}^{6} - 354976496998757280 {V_{3}}^{5} + 145147458640384656 {V_{3}}^{4} - 44222432809434624 {V_{3}}^{3} + 9454170262653072 {V_{3}}^{2} - 1265671473719328 V_{3} + 79859790509976} \end{matrix}

-->	EU_3aa3 : EU_o3, S_i = S_1aa, F_i = F_1aa3, F_j = F_2aa, S_j = S_2aa3, S_o = EU_3ba, F_o = EU_3ca3;

\begin{matrix} EU_3aa3 & \frac{((216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 76 {V_{3}}^{3} - 141 {V_{3}}^{2} + 30 V_{3} + 31) (\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})}{(216 {V_{3}}^{2} - 432 V_{3} + 216) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})} (((216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 76 {V_{3}}^{3} - 141 {V_{3}}^{2} + 30 V_{3} + 31) (\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})) (((216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 76 {V_{3}}^{3} - 141 {V_{3}}^{2} + 30 V_{3} + 31) (\frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214})) / ((216 {V_{3}}^{2} - 432 V_{3} + 216) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) ((216 {V_{3}}^{2} - 432 V_{3} + 216) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})) + \frac{(\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}) (\frac{{(ε - \frac{V_{3}}{6} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{1}{3})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} + ε + \frac{V_{3}}{2})}{\frac{{(1 - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}})}^{3}}{{(ε - \frac{V_{3}}{6} - \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}} + \frac{{(\frac{2}{3} - \frac{V_{3}}{3})}^{3}}{{(1 - V_{3})}^{2}} + \frac{4}{3})}^{2}} - \frac{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε^{2} + (729 {V_{3}}^{5} - 2430 {V_{3}}^{4} + 1944 {V_{3}}^{3} + 1458 {V_{3}}^{2} - 2673 V_{3} + 972) ε - 371 {V_{3}}^{6} + 2508 {V_{3}}^{5} - 6636 {V_{3}}^{4} + 8591 {V_{3}}^{3} - 5421 {V_{3}}^{2} + 1293 V_{3} + 34}{(1458 {V_{3}}^{4} - 5832 {V_{3}}^{3} + 8748 {V_{3}}^{2} - 5832 V_{3} + 1458) ε - 621 {V_{3}}^{5} + 4158 {V_{3}}^{4} - 10908 {V_{3}}^{3} + 14040 {V_{3}}^{2} - 8883 V_{3} + 2214} + \frac{(216 {V_{3}}^{2} - 432 V_{3} + 216) ε + 32 {V_{3}}^{3} + 15 {V_{3}}^{2} - 138 V_{3} + 95}{216 {V_{3}}^{2} - 432 V_{3} + 216} + ε - \frac{11 {V_{3}}^{2} - 8 V_{3} - 1}{18 V_{3} - 18} + \frac{(\frac{1}{3} - \frac{2 V_{3}}{3}) (\frac{V_{3}}{3} + \frac{1}{3})}{1 - V_{3}}} \end{matrix}

-->	radcan(limit(EU_3aa3, ε, 0, plus));

\begin{matrix} (%o353) & \frac{1574716576 {V_{3}}^{14} - 24402619156 {V_{3}}^{13} + 173363468569 {V_{3}}^{12} - 746793508592 {V_{3}}^{11} + 2172689906438 {V_{3}}^{10} - 4496716472444 {V_{3}}^{9} + 6782457756507 {V_{3}}^{8} - 7492186281408 {V_{3}}^{7} + 5972567542980 {V_{3}}^{6} - 3280384081100 {V_{3}}^{5} + 1087800952079 {V_{3}}^{4} - 101515446800 {V_{3}}^{3} - 77330579882 {V_{3}}^{2} + 33264927644 V_{3} - 4390281155}{3911028192 {V_{3}}^{13} - 61249910616 {V_{3}}^{12} + 441252814464 {V_{3}}^{11} - 1936315625904 {V_{3}}^{10} + 5774899588704 {V_{3}}^{9} - 12361702069800 {V_{3}}^{8} + 19540280270592 {V_{3}}^{7} - 23094799639200 {V_{3}}^{6} + 20410061756832 {V_{3}}^{5} - 13321156309416 {V_{3}}^{4} + 6241528752768 {V_{3}}^{3} - 1988152045488 {V_{3}}^{2} + 385921348128 V_{3} - 34479959256} \end{matrix}

2.7.2 Case B:

We define the prizes:

-->	S_1ab : EU_1bb;

\begin{matrix} S_1ab & \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	F_1ab : EU_1cb;

\begin{matrix} F_1ab & \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} \end{matrix}

-->	S_2ab : EU_2cb;

\begin{matrix} S_2ab & \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} \end{matrix}

-->	F_2ab : EU_2bb;

\begin{matrix} F_2ab & \frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} \end{matrix}

-->	S_1ab − F_1ab;

\begin{matrix} (%o358) & \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

-->	D_1ab : radcan(limit(S_1ab − F_1ab, ε, 0, plus));

\begin{matrix} D_1ab & - (\frac{{V_{3}}^{3} - 8 {V_{3}}^{2} + 16 V_{3} - 8}{2 {V_{3}}^{2} - 16 V_{3} + 32}) \end{matrix}

-->	D_1abzp : solve(num(D_1ab)=0, V_3);

\begin{matrix} D_1abzp & [V_{3} = 3 - \sqrt{5}, V_{3} = \sqrt{5} + 3, V_{3} = 2] \end{matrix}

-->	allroots(num(D_1ab));

\begin{matrix} (%o361) & [V_{3} = 0.7639320225002104, V_{3} = 2.0, V_{3} = 5.23606797749979] \end{matrix}

-->	S_2ab − F_2ab;

\begin{matrix} (%o362) & - (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} \end{matrix}

-->	D_2ab: radcan(limit(S_2ab − F_2ab, ε, 0, plus));

\begin{matrix} D_2ab & - (\frac{8 {V_{3}}^{2} - 29 V_{3} + 20}{8 V_{3} - 32}) \end{matrix}

-->	D_2abzp : solve(num(D_2ab)=0, V_3);

\begin{matrix} D_2abzp & [V_{3} = - (\frac{\sqrt{201} - 29}{16}), V_{3} = \frac{\sqrt{201} + 29}{16}] \end{matrix}

-->	allroots(num(D_2ab));

\begin{matrix} (%o365) & [V_{3} = 0.926409570077636, V_{3} = 2.6985904299223638] \end{matrix}

From this subcases B.1, B.2, B.3 follow:

2.7.2.1 Subcase B.1:

S_1ab > F_1ab and S_2ab > F_2ab, hence case III.

We calculate best responses:

-->	x_1ab1 : x_i3, S_i = S_1ab, F_i = F_1ab, F_j = F_2ab, S_j = S_2ab;

\begin{matrix} x_1ab1 & \frac{{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2} (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}})}{{(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2}} - ε \end{matrix}

-->	lim_x_1ab1 : radcan(limit(x_1ab1, ε, 0, plus));

\begin{matrix} lim_x_1ab1 & \frac{16 {V_{3}}^{8} - 294 {V_{3}}^{7} + 2104 {V_{3}}^{6} - 7360 {V_{3}}^{5} + 12640 {V_{3}}^{4} - 7808 {V_{3}}^{3} - 4608 {V_{3}}^{2} + 7808 V_{3} - 2560}{144 {V_{3}}^{7} - 2808 {V_{3}}^{6} + 22377 {V_{3}}^{5} - 93684 {V_{3}}^{4} + 220384 {V_{3}}^{3} - 288128 {V_{3}}^{2} + 191744 V_{3} - 50176} \end{matrix}

-->	x_2ab1 : x_j3, S_i = S_1ab, F_i = F_1ab, F_j = F_2ab, S_j = S_2ab;

\begin{matrix} x_2ab1 & \frac{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) {(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}})}^{2}}{{(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2}} - ε \end{matrix}

-->	radcan(x_2ab1);

\begin{matrix} (Config tells to suppress the output of long cells) \end{matrix}

-->	lim_x_2ab1 : radcan(limit(x_2ab1, ε, 0, plus));

\begin{matrix} lim_x_2ab1 & - (\frac{64 {V_{3}}^{7} - 976 {V_{3}}^{6} + 5897 {V_{3}}^{5} - 18384 {V_{3}}^{4} + 31968 {V_{3}}^{3} - 31048 {V_{3}}^{2} + 15680 V_{3} - 3200}{288 {V_{3}}^{6} - 4464 {V_{3}}^{5} + 26898 {V_{3}}^{4} - 79776 {V_{3}}^{3} + 121664 {V_{3}}^{2} - 89600 V_{3} + 25088}) \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ab1 : radcan(limit(x_1ab1 + x_2ab1 + lim_x_1ab1/(lim_x_1ab1 + lim_x_2ab1) · ex_x_bb + lim_x_2ab1/(lim_x_1ab1 + lim_x_2ab1) · ex_x_cb, ε, 0, plus));

\begin{matrix} ex_x_ab1 & - (\frac{2048 {V_{3}}^{13} - 56832 {V_{3}}^{12} + 753824 {V_{3}}^{11} - 6598752 {V_{3}}^{10} + 43281338 {V_{3}}^{9} - 219181553 {V_{3}}^{8} + 841408160 {V_{3}}^{7} - 2380572304 {V_{3}}^{6} + 4841209984 {V_{3}}^{5} - 6899580160 {V_{3}}^{4} + 6658985984 {V_{3}}^{3} - 4113412096 {V_{3}}^{2} + 1458601984 V_{3} - 224788480}{18432 {V_{3}}^{12} - 582912 {V_{3}}^{11} + 8361024 {V_{3}}^{10} - 71790192 {V_{3}}^{9} + 409699652 {V_{3}}^{8} - 1630250624 {V_{3}}^{7} + 4614131328 {V_{3}}^{6} - 9306512384 {V_{3}}^{5} + 13199216640 {V_{3}}^{4} - 12767985664 {V_{3}}^{3} + 7960494080 {V_{3}}^{2} - 2864447488 V_{3} + 449576960}) \end{matrix}

-->	denom_x_ab1 : denom(radcan(ex_eff_0 − ex_x_ab1));

\begin{matrix} denom_x_ab1 & 3125296388229120 {V_{3}}^{12} - 98837498277745920 {V_{3}}^{11} + 1417680018939723840 {V_{3}}^{10} - 12172614353725860720 {V_{3}}^{9} + 69467927661367586820 {V_{3}}^{8} - 276422330029051059840 {V_{3}}^{7} + 782363713878756132480 {V_{3}}^{6} - 1577995309284113341440 {V_{3}}^{5} + 2238035161265500262400 {V_{3}}^{4} - 2164916421476800266240 {V_{3}}^{3} + 1349766894354562252800 {V_{3}}^{2} - 485690505019443118080 V_{3} + 76229451460450713600 \end{matrix}

-->	num_x_ab1 : num(radcan(ex_eff_0 − ex_x_ab1));

\begin{matrix} num_x_ab1 & 347255154247680 {V_{3}}^{13} - 7572971291576832 {V_{3}}^{12} + 62563290145139232 {V_{3}}^{11} - 182902543197818304 {V_{3}}^{10} - 697805261991948798 {V_{3}}^{9} + 8699557624157695913 {V_{3}}^{8} - 39829796175634982816 {V_{3}}^{7} + 112880668459131444912 {V_{3}}^{6} - 220945216517171858816 {V_{3}}^{5} + 307698168656271932160 {V_{3}}^{4} - 300218985219695374336 {V_{3}}^{3} + 193670086720072847360 {V_{3}}^{2} - 73340965492219543552 V_{3} + 12212898725392547840 \end{matrix}

-->	nroots(denom_x_ab1, 0.49, 0.77);

\begin{matrix} (%o374) & 0 \end{matrix}

-->	nroots(num_x_ab1, 0.49, 0.77);

\begin{matrix} (%o375) & 0 \end{matrix}

-->	ex_eff_0 − ex_x_ab1, V_3 = 0.6;

\begin{matrix} (%o376) & 0.2556010612386184 \end{matrix}

We calculate expected utilities:

-->	EU_1ab1 : EU_i3, S_i = S_1ab, F_i = F_1ab, F_j = F_2ab, S_j = S_2ab;

\begin{matrix} EU_1ab1 & \frac{{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2}} + ε + \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} \end{matrix}

-->	radcan(limit(EU_1ab1, ε, 0, plus));

\begin{matrix} (%o378) & \frac{128 {V_{3}}^{9} - 3000 {V_{3}}^{8} + 29769 {V_{3}}^{7} - 162328 {V_{3}}^{6} + 527536 {V_{3}}^{5} - 1034496 {V_{3}}^{4} + 1177344 {V_{3}}^{3} - 694272 {V_{3}}^{2} + 151552 V_{3} + 8192}{288 {V_{3}}^{8} - 6768 {V_{3}}^{7} + 67218 {V_{3}}^{6} - 366384 {V_{3}}^{5} + 1190240 {V_{3}}^{4} - 2339328 {V_{3}}^{3} + 2688512 {V_{3}}^{2} - 1634304 V_{3} + 401408} \end{matrix}

-->	EU_2ab1 : EU_j3, S_i = S_1ab, F_i = F_1ab, F_j = F_2ab, S_j = S_2ab;

\begin{matrix} EU_2ab1 & \frac{{(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}})}^{3}}{{(- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{2}} + \frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + ε \end{matrix}

-->	radcan(limit(EU_2ab1, ε, 0, plus));

\begin{matrix} (%o380) & \frac{640 {V_{3}}^{8} - 11648 {V_{3}}^{7} + 84400 {V_{3}}^{6} - 308371 {V_{3}}^{5} + 581692 {V_{3}}^{4} - 467424 {V_{3}}^{3} - 71808 {V_{3}}^{2} + 319744 V_{3} - 128000}{1152 {V_{3}}^{7} - 22464 {V_{3}}^{6} + 179016 {V_{3}}^{5} - 749472 {V_{3}}^{4} + 1763072 {V_{3}}^{3} - 2305024 {V_{3}}^{2} + 1533952 V_{3} - 401408} \end{matrix}

-->	EU_3ab1 : EU_o3, S_i = S_1ab, F_i = F_1ab, F_j = F_2ab, S_j = S_2ab, S_o = EU_3bb, F_o = EU_3cb;

\begin{matrix} EU_3ab1 & \frac{((4 \sqrt{3} V_{3} - 8 \sqrt{3}) ε + \sqrt{2 - V_{3}} (8 V_{3} - 4) \sqrt{ε} + 4 \sqrt{3} {V_{3}}^{2} - 7 \sqrt{3} V_{3} - 2 \sqrt{3}) (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}})}{(8 \sqrt{3} V_{3} - 16 \sqrt{3}) (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})} (((4 \sqrt{3} V_{3} - 8 \sqrt{3}) ε + \sqrt{2 - V_{3}} (8 V_{3} - 4) \sqrt{ε} + 4 \sqrt{3} {V_{3}}^{2} - 7 \sqrt{3} V_{3} - 2 \sqrt{3}) (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}})) / ((8 \sqrt{3} V_{3} - 16 \sqrt{3}) (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})) ((8 \sqrt{3} V_{3} - 16 \sqrt{3}) (- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})) + \frac{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) (\frac{{(2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{V_{3}}{2})}{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}} \end{matrix}

Auxiliary calculations:

-->	lim_S_1ab : radcan(limit(S_1ab, ε, 0, plus));

\begin{matrix} lim_S_1ab & \frac{4}{{V_{3}}^{2} - 8 V_{3} + 16} \end{matrix}

-->	lim_F_1ab : radcan(limit(F_1ab, ε, 0, plus));

\begin{matrix} lim_F_1ab & \frac{V_{3}}{2} \end{matrix}

-->	lim_F_2ab : radcan(limit(F_2ab, ε, 0, plus));

\begin{matrix} lim_F_2ab & \frac{{V_{3}}^{2} - 3 V_{3}}{V_{3} - 4} \end{matrix}

-->	lim_S_2ab : radcan(limit(S_2ab, ε, 0, plus));

\begin{matrix} lim_S_2ab & \frac{5}{8} \end{matrix}

-->	lim_S_o : radcan(limit(EU_3bb, ε, 0, plus));

\begin{matrix} lim_S_o & - (\frac{{V_{3}}^{2} - 2 V_{3} - 4}{{V_{3}}^{2} - 8 V_{3} + 16}) \end{matrix}

-->	lim_F_o : radcan(limit(EU_3cb, ε, 0, plus));

\begin{matrix} lim_F_o & \frac{4 V_{3} + 1}{8} \end{matrix}

-->	radcan(limit(EU_o3, ε, 0, plus)), S_i = lim_S_1ab, F_i = lim_F_1ab, F_j = lim_F_2ab, S_j = lim_S_2ab, S_o = lim_S_o, F_o = lim_F_o;

\begin{matrix} (%o388) & \frac{32 {V_{3}}^{6} - 524 {V_{3}}^{5} + 3203 {V_{3}}^{4} - 8720 {V_{3}}^{3} + 9376 {V_{3}}^{2} - 768 V_{3} - 2304}{96 {V_{3}}^{5} - 1512 {V_{3}}^{4} + 9088 {V_{3}}^{3} - 25600 {V_{3}}^{2} + 32768 V_{3} - 14336} \end{matrix}

2.7.2.2 Subcase B.2:

S_1ab ≤ F_1ab and S_2ab > F_2ab, hence case I.

We calculate best responses:

-->	assume[S_2ab − F_2ab >= 0, S_1ab − F_1ab < 0];

\begin{matrix} (%o389) & {assume}_{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} \geq 0, \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} < 0} \end{matrix}

-->	x_1ab2 : 0;

\begin{matrix} x_1ab2 & 0 \end{matrix}

-->	radcan(x_1ab2);

\begin{matrix} (%o391) & 0 \end{matrix}

-->	radcan(limit(x_1ab2, ε, 0, plus));

\begin{matrix} (%o392) & 0 \end{matrix}

-->	x_2ab2 : BR_i(x_1ab2, S_2ab, F_2ab);

\begin{matrix} x_2ab2 & \sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - 2 ε \end{matrix}

-->	radcan(x_2ab2);

\begin{matrix} (%o394) & \frac{\sqrt{ε} \sqrt{(128 3^{\frac{3}{2}} V_{3} - 256 3^{\frac{3}{2}}) ε^{3} + \sqrt{2 - V_{3}} \sqrt{ε} ((3072 - 1536 V_{3}) ε^{2} + (1312 {V_{3}}^{2} - 6448 V_{3} + 6352) ε - 192 {V_{3}}^{3} + 1500 {V_{3}}^{2} - 3420 V_{3} + 2160) + (- (1120 \sqrt{3} {V_{3}}^{2}) + 4720 \sqrt{3} V_{3} - 4672 \sqrt{3}) ε^{2} + (392 \sqrt{3} {V_{3}}^{3} - 2828 \sqrt{3} {V_{3}}^{2} + 6104 \sqrt{3} V_{3} - 436 3^{\frac{5}{2}}) ε - 8 3^{\frac{3}{2}} {V_{3}}^{4} + 77 3^{\frac{3}{2}} {V_{3}}^{3} - 86 3^{\frac{5}{2}} {V_{3}}^{2} + 352 3^{\frac{3}{2}} V_{3} - 160 3^{\frac{3}{2}}} - 2^{\frac{5}{2}} ε \sqrt{(16 3^{\frac{3}{2}} V_{3} - 32 3^{\frac{3}{2}}) ε^{2} + \sqrt{2 - V_{3}} \sqrt{ε} ((240 - 48 V_{3}) ε + 12 {V_{3}}^{2} - 108 V_{3} + 240) + (- (28 \sqrt{3} {V_{3}}^{2}) + 184 \sqrt{3} V_{3} - 292 \sqrt{3}) ε + 3^{\frac{3}{2}} {V_{3}}^{3} - 10 3^{\frac{3}{2}} {V_{3}}^{2} + 32 3^{\frac{3}{2}} V_{3} - 32 3^{\frac{3}{2}}}}{2^{\frac{3}{2}} \sqrt{(16 3^{\frac{3}{2}} V_{3} - 32 3^{\frac{3}{2}}) ε^{2} + \sqrt{2 - V_{3}} \sqrt{ε} ((240 - 48 V_{3}) ε + 12 {V_{3}}^{2} - 108 V_{3} + 240) + (- (28 \sqrt{3} {V_{3}}^{2}) + 184 \sqrt{3} V_{3} - 292 \sqrt{3}) ε + 3^{\frac{3}{2}} {V_{3}}^{3} - 10 3^{\frac{3}{2}} {V_{3}}^{2} + 32 3^{\frac{3}{2}} V_{3} - 32 3^{\frac{3}{2}}}} \end{matrix}

-->	radcan(limit(x_2ab2, ε, 0, plus));

\begin{matrix} (%o395) & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ab2 : radcan(limit(x_1ab2 + x_2ab2 + ex_x_cb, ε, 0, plus));

\begin{matrix} ex_x_ab2 & \frac{1}{4} \end{matrix}

Trivially, expected effort is higher when V_3 → 0 than within this interval:

-->	ex_eff_0 − ex_x_ab2;

\begin{matrix} (%o397) & \frac{278219431801}{678232723140} \end{matrix}

We calculate expected utilities:

-->	EU_1ab2 : EU_i2(x_2ab2, S_1ab, F_1ab);

\begin{matrix} EU_1ab2 & - (2 \sqrt{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) (\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - ε)}) (2 \sqrt{(\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) (\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - ε)} sqrt((\frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}}) (\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - ε))) + \sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} + \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} \end{matrix}

Auxiliary calculations:

-->	lim_x_2ab2: radcan(limit(x_2ab2, ε, 0, plus));

\begin{matrix} lim_x_2ab2 & 0 \end{matrix}

-->	lim_S_1ab : radcan(limit(S_1ab, ε, 0, plus));

\begin{matrix} lim_S_1ab & \frac{4}{{V_{3}}^{2} - 8 V_{3} + 16} \end{matrix}

-->	lim_F_1ab : radcan(limit(F_1ab, ε, 0, plus));

\begin{matrix} lim_F_1ab & \frac{V_{3}}{2} \end{matrix}

-->	radcan(limit(EU_i2(lim_x_2ab2, lim_S_1ab, lim_F_1ab), ε, 0, plus));

\begin{matrix} (%o402) & \frac{4}{{V_{3}}^{2} - 8 V_{3} + 16} \end{matrix}

-->	EU_2ab2 : EU_i(x_2ab2, x_1ab2, S_2ab, F_2ab);

\begin{matrix} EU_2ab2 & - (\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}}) + \frac{(\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}) (1 - \frac{\sqrt{ε}}{\sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}}})}{8 \sqrt{3} \sqrt{2 - V_{3}}} + \frac{\sqrt{ε} (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)})}{\sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}}} + 2 ε \end{matrix}

-->	radcan(limit(EU_2ab2, ε, 0, plus));

\begin{matrix} (%o404) & \frac{5}{8} \end{matrix}

-->	EU_3ab2 : EU_o(x_1ab2, x_2ab2, EU_3bb, EU_3cb);

\begin{matrix} EU_3ab2 & (\frac{{(2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε + \frac{V_{3}}{2}) (1 - \frac{\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - ε}{\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}}}) + \frac{((4 \sqrt{3} V_{3} - 8 \sqrt{3}) ε + \sqrt{2 - V_{3}} (8 V_{3} - 4) \sqrt{ε} + 4 \sqrt{3} {V_{3}}^{2} - 7 \sqrt{3} V_{3} - 2 \sqrt{3}) (\sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}} - ε)}{(8 \sqrt{3} V_{3} - 16 \sqrt{3}) \sqrt{ε} \sqrt{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}}}} \end{matrix}

-->	radcan(limit(EU_3ab2, ε, 0, plus));

\begin{matrix} (%o406) & \frac{4 V_{3} + 1}{8} \end{matrix}

-->	forget[S_2ab − F_2ab >= 0, S_1ab − F_1ab < 0];

\begin{matrix} (%o407) & {forget}_{- (\frac{((\frac{1}{3} - \frac{2 V_{3}}{3}) \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - (\frac{V_{3}}{3} + \frac{1}{3}) V_{3} + V_{3}) (2 ε - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 1)}{(\frac{2}{3} - \frac{V_{3}}{3}) (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}) - \frac{V_{3} (1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}{2 (2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)} + \frac{\sqrt{2 - V_{3}} (8 \sqrt{3} ε + 5 \sqrt{3}) + (8 V_{3} - 16) \sqrt{ε}}{8 \sqrt{3} \sqrt{2 - V_{3}}} \geq 0, \frac{{(1 - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}})}^{3}}{{(2 ε - \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} - 2 \sqrt{\frac{2}{3} - \frac{V_{3}}{3}} \sqrt{ε} - \frac{V_{3}}{2} + 2)}^{2}} + ε - \frac{(\sqrt{3} V_{3} + \sqrt{3}) \sqrt{ε} + 3 \sqrt{2 - V_{3}} V_{3}}{6 \sqrt{2 - V_{3}}} + \frac{(\frac{V_{3}}{3} + \frac{1}{3}) \sqrt{ε}}{\sqrt{\frac{2}{3} - \frac{V_{3}}{3}}} < 0} \end{matrix}

2.7.2.3 Subcase B.3:

S_1ab ≤ F_1ab and S_2ab ≤ F_2ab, hence case II.

We calculate best responses:

-->	x_1ab3 : 0;

\begin{matrix} x_1ab3 & 0 \end{matrix}

-->	radcan(x_1ab3);

\begin{matrix} (%o409) & 0 \end{matrix}

-->	radcan(limit(x_1ab3, ε, 0, plus));

\begin{matrix} (%o410) & 0 \end{matrix}

-->	x_2ab3 : 0;

\begin{matrix} x_2ab3 & 0 \end{matrix}

-->	radcan(x_2ab3);

\begin{matrix} (%o412) & 0 \end{matrix}

-->	radcan(limit(x_2ab3, ε, 0, plus));

\begin{matrix} (%o413) & 0 \end{matrix}

We calculate expected effort in this and all subsequent nodes:

-->	ex_x_ab3 : radcan(limit(x_1ab3 + x_2ab3 + 1/2 · ex_x_bb + 1/2 · ex_x_cb, ε, 0, plus));

\begin{matrix} ex_x_ab3 & \frac{5 V_{3} - 12}{8 V_{3} - 32} \end{matrix}

We establish that for ε → 0 the expected effort for V_3 → 0 is larger than for any V_3 in this interval. To do so, we check the denominator of the difference for roots to prove that the function is continous within the interval. Then, we also inspect the numerator for roots. Lastly, we establish
that for at least one V_3 the expressions is indeed positive, concluding our proof. We use the function nroots to check for roots, which internally uses Sturm's theorem.

-->	denom_x_ab3 : denom(radcan(ex_eff_0 − ex_x_ab3));

\begin{matrix} denom_x_ab3 & 1356465446280 V_{3} - 5425861785120 \end{matrix}

-->	num_x_ab3 : num(radcan(ex_eff_0 − ex_x_ab3));

\begin{matrix} num_x_ab3 & 47764321247 V_{3} - 1547522731268 \end{matrix}

-->	nroots(denom_x_ab3, 0.92, 1);

\begin{matrix} (%o417) & 0 \end{matrix}

-->	nroots(num_x_ab3, 0.92, 1);

\begin{matrix} (%o418) & 0 \end{matrix}

-->	ex_eff_0 − ex_x_ab3, V_3 = 0.95;

\begin{matrix} (%o419) & 0.363081191540701 \end{matrix}

We calculate expected utilities:

-->	EU_1ab3 : radcan(1/2 · S_1ab + 1/2 · F_1ab);

\begin{matrix} EU_1ab3 & \frac{\sqrt{2 - V_{3}} ((288 V_{3} - 576) ε^{3} + (- (168 {V_{3}}^{2}) + 1392 V_{3} - 1032) ε^{2} + (- (30 {V_{3}}^{3}) + 12 {V_{3}}^{2} - 48 V_{3} + 72) ε + 9 {V_{3}}^{4} - 90 {V_{3}}^{3} + 288 {V_{3}}^{2} - 216 V_{3} - 144) + \sqrt{ε} ((80 3^{\frac{3}{2}} {V_{3}}^{2} - 272 3^{\frac{3}{2}} V_{3} + 224 3^{\frac{3}{2}}) ε^{2} + (- (52 \sqrt{3} {V_{3}}^{3}) + 140 3^{\frac{3}{2}} {V_{3}}^{2} - 244 3^{\frac{3}{2}} V_{3} + 92 \sqrt{3}) ε - 3^{\frac{3}{2}} {V_{3}}^{4} + 17 3^{\frac{3}{2}} {V_{3}}^{3} - 110 3^{\frac{3}{2}} {V_{3}}^{2} + 184 3^{\frac{3}{2}} V_{3} - 16 3^{\frac{5}{2}})}{\sqrt{2 - V_{3}} ((576 V_{3} - 1152) ε^{2} + (- (336 {V_{3}}^{2}) + 2208 V_{3} - 3504) ε + 36 {V_{3}}^{3} - 360 {V_{3}}^{2} + 1152 V_{3} - 1152) + \sqrt{ε} ((64 3^{\frac{3}{2}} {V_{3}}^{2} - 448 3^{\frac{3}{2}} V_{3} + 640 3^{\frac{3}{2}}) ε - 16 3^{\frac{3}{2}} {V_{3}}^{3} + 176 3^{\frac{3}{2}} {V_{3}}^{2} - 608 3^{\frac{3}{2}} V_{3} + 640 3^{\frac{3}{2}})} \end{matrix}

-->	radcan(limit(EU_1ab3, ε, 0, plus));

\begin{matrix} (%o421) & \frac{{V_{3}}^{3} - 8 {V_{3}}^{2} + 16 V_{3} + 8}{4 {V_{3}}^{2} - 32 V_{3} + 64} \end{matrix}

-->	EU_2ab3 : radcan(1/2 · S_2ab + 1/2 · F_2ab);

\begin{matrix} EU_2ab3 & - (\frac{(128 3^{\frac{3}{2}} V_{3} - 256 3^{\frac{3}{2}}) ε^{3} + \sqrt{2 - V_{3}} \sqrt{ε} (2304 ε^{2} + (- (672 {V_{3}}^{2}) + 1296 V_{3} + 4560) ε + 144 {V_{3}}^{3} - 900 {V_{3}}^{2} + 804 V_{3} + 1776) + (416 \sqrt{3} {V_{3}}^{2} + 496 \sqrt{3} V_{3} - 3520 \sqrt{3}) ε^{2} + (- (280 \sqrt{3} {V_{3}}^{3}) + 1364 \sqrt{3} {V_{3}}^{2} - 296 \sqrt{3} V_{3} - 3092 \sqrt{3}) ε + 8 3^{\frac{3}{2}} {V_{3}}^{4} - 67 3^{\frac{3}{2}} {V_{3}}^{3} + 158 3^{\frac{3}{2}} {V_{3}}^{2} - 32 3^{\frac{3}{2}} V_{3} - 160 3^{\frac{3}{2}}}{(512 3^{\frac{3}{2}} - 256 3^{\frac{3}{2}} V_{3}) ε^{2} + \sqrt{2 - V_{3}} \sqrt{ε} ((768 V_{3} - 3840) ε - 192 {V_{3}}^{2} + 1728 V_{3} - 3840) + (448 \sqrt{3} {V_{3}}^{2} - 2944 \sqrt{3} V_{3} + 4672 \sqrt{3}) ε - 16 3^{\frac{3}{2}} {V_{3}}^{3} + 160 3^{\frac{3}{2}} {V_{3}}^{2} - 512 3^{\frac{3}{2}} V_{3} + 512 3^{\frac{3}{2}}}) \end{matrix}

-->	radcan(limit(EU_2ab3, ε, 0, plus));

\begin{matrix} (%o423) & \frac{8 {V_{3}}^{2} - 19 V_{3} - 20}{16 V_{3} - 64} \end{matrix}

-->	EU_3ab3 : radcan(1/2 · EU_3bb + 1/2 · EU_3cb);

\begin{matrix} EU_3ab3 & \frac{(- (4032 {V_{3}}^{2}) + 16128 V_{3} - 16128) ε^{3} + \sqrt{2 - V_{3}} \sqrt{ε} ((832 3^{\frac{3}{2}} {V_{3}}^{2} - 4096 3^{\frac{3}{2}} V_{3} + 4864 3^{\frac{3}{2}}) ε^{2} + (- (944 \sqrt{3} {V_{3}}^{3}) + 1936 3^{\frac{3}{2}} {V_{3}}^{2} - 5264 3^{\frac{3}{2}} V_{3} + 16336 \sqrt{3}) ε + 8 3^{\frac{3}{2}} {V_{3}}^{4} + 16 3^{\frac{5}{2}} {V_{3}}^{3} - 316 3^{\frac{3}{2}} {V_{3}}^{2} - 8 3^{\frac{5}{2}} V_{3} + 800 3^{\frac{3}{2}}) + (3504 {V_{3}}^{3} - 25488 {V_{3}}^{2} + 62928 V_{3} - 51936) ε^{2} + (- (348 {V_{3}}^{4}) + 2580 {V_{3}}^{3} - 10512 {V_{3}}^{2} + 24924 V_{3} - 22872) ε - 36 {V_{3}}^{5} + 495 {V_{3}}^{4} - 2196 {V_{3}}^{3} + 3564 {V_{3}}^{2} - 864 V_{3} - 1728}{(- (2304 {V_{3}}^{2}) + 9216 V_{3} - 9216) ε^{2} + \sqrt{2 - V_{3}} \sqrt{ε} ((256 3^{\frac{3}{2}} {V_{3}}^{2} - 1792 3^{\frac{3}{2}} V_{3} + 2560 3^{\frac{3}{2}}) ε - 64 3^{\frac{3}{2}} {V_{3}}^{3} + 704 3^{\frac{3}{2}} {V_{3}}^{2} - 2432 3^{\frac{3}{2}} V_{3} + 2560 3^{\frac{3}{2}}) + (1344 {V_{3}}^{3} - 11520 {V_{3}}^{2} + 31680 V_{3} - 28032) ε - 144 {V_{3}}^{4} + 1728 {V_{3}}^{3} - 7488 {V_{3}}^{2} + 13824 V_{3} - 9216} \end{matrix}

-->	radcan(limit(EU_3ab3, ε, 0, plus));

\begin{matrix} (%o425) & \frac{4 {V_{3}}^{3} - 39 {V_{3}}^{2} + 72 V_{3} + 48}{16 {V_{3}}^{2} - 128 V_{3} + 256} \end{matrix}

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The source of this Maxima session can be downloaded here.