I have to calculate a threshold value for a probability, lambda*. My code for the simplified version is working but it breaks down for the more complex version.
Here are defined probabilities and the working calculation for one alternative
r = \[Epsilon]*\[Lambda] + (1 - \[Eta])*(1 - \[Lambda]);
s = \[Eta]*(1 - \[Lambda]) + (1 - \[Epsilon])*\[Lambda];
\[Lambda][r] = (\[Epsilon]*\[Lambda])/(\[Epsilon]*\[Lambda] + (1 - \[Eta])*(1 -\[Lambda]));
\[Lambda][s] = (\[Eta]*(1 - \[Lambda]))/(\[Eta]*(1 - \[Lambda]) + (1 - \[Epsilon])*\[Lambda]);
Simplify[r*(y - m - c + \[Alpha]*\[Lambda][r]) + s*(y - m + \[Alpha]*(1 - \[Lambda][s])) ==
y + \[Alpha]*(1 - \[Lambda])];
Solve[r*(y - m - c + \[Alpha]*\[Lambda][r]) + s*(y - m + Subscript[\[Alpha], i]*(1 - \[Lambda][s])) ==
y + \[Alpha]*(1 - \[Lambda]), {\[Lambda]}]
It does not work for two alternatives (1,2):
\[Mu] = 1 - c1 - c2;
\[Xi] = c1 - y;
\[Beta] = 1 - c1;
\[Gamma] = c1 - y;
(*Further probabilities *)
r1 = \[Epsilon]1*\[Lambda]1 + (1 - \[Eta]1)*(1 - \[Lambda]1);
r2 = \[Epsilon]2*\[Lambda]2 + (1 - \[Eta]2)*(1 - \[Lambda]2);
n1 = \[Eta]1*(1 - \[Lambda]1) + (1 - \[Epsilon]1)*\[Lambda]1;
n2 = \[Eta]2*(1 - \[Lambda]2) + (1 - \[Epsilon]2)*\[Lambda]2;
\[Lambda][r1] = (\[Epsilon]1*\[Lambda]1)/(\[Epsilon]1*\[Lambda]1 + (1 - \
\[Eta]1)*(1 - \[Lambda]1));
\[Lambda][r2] = (\[Epsilon]2*\[Lambda]2)/(\[Epsilon]2*\[Lambda]2 + (1 - \
\[Eta]2)*(1 - \[Lambda]2));
\[Lambda][s1] = (\[Eta]1*(1 - \[Lambda]1))/(\[Eta]1*(1 - \[Lambda]1) + (1 - \
\[Epsilon]1)*\[Lambda]1);
\[Lambda][s2] = (\[Eta]2*(1 - \[Lambda]2))/(\[Eta]2*(1 - \[Lambda]2) + (1 - \
\[Epsilon]2)*\[Lambda]2);
(* Expected Payoff *)
EMS1 = \[Mu]*\[Xi]*(\[Lambda]2*\[Alpha]2 + (1 - \[Lambda]1) \[Alpha]1 \
- c2 + y) + \[Mu]*(1 - \[Xi])*(\[Lambda]1*\[Alpha]1 + \[Lambda]2*\
\[Alpha]2) + (1 - \[Mu])*\[Xi]*(\[Lambda]1*\[Alpha]1 + (1 - \
\[Lambda]2)*\[Alpha]2) + (1 - \[Mu])*(1 - \
\[Xi])*(\[Lambda]1*\[Alpha]1 + \[Lambda]2*\[Alpha]2 + 2*c1 + c2 - y);
(* Solving *)
Simplify[r1*r2*EMS1 + r1*n2*(\[Alpha]1*\[Lambda][r1] - c1 + \[Alpha]2*(1 - \[Lambda]2[n2])) + n1*r2*(\[Alpha]2*\[Lambda][r2] - c2 + \[Alpha]1*(1 - \[Lambda]1[n1])) + n1*n2*( \[Alpha]1*(1 - \[Lambda]1[n1]) + \[Alpha]2*(1 - \[Lambda]2[
n2])) - s1 - s2 == r1*(y^p - s1 - c1 + \[Alpha]1*\[Lambda][r1]) + n1*(y - s1 + \[Alpha]1*(1 - \[Lambda][
n1])) + \[Alpha]2*(1 - \[Lambda]2)]
Solve[ r1*r2*EMS1 + r1*n2*(\[Alpha]1*\[Lambda][r1] - c1 + \[Alpha]2*(1 - \[Lambda]2[n2])) + n1*r2*(\[Alpha]2*\[Lambda][r2] - c2 + \[Alpha]1*(1 - \[Lambda]1[n1])) + n1*n2*( \[Alpha]1*(1 - \[Lambda]1[n1]) + \[Alpha]2*(1 - \[Lambda]2[n2])) - s1 - s2 == r1*(y^p - s1 - c1 + \[Alpha]1*\[Lambda][r1]) + n1*(y - s1 + \[Alpha]1*(1 - \[Lambda][
n1])) + \[Alpha]2*(1 - \[Lambda]2), {\[Lambda]2}]
