Exponent optimization with constrains

Questions
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1

Hello,
I’m trying to find an optimization for each player (each one has a utilitiy function) with constraints. the problem has an exponent factor.
I don’t know how to construct the problem in mathematica and to find a feasible solution for each player (or potentialy an equilibrium).

Thanks!

2

Can you provide any additional constraint information?

In[1]:= s = 95;
u = 0.1;
x = 100;
k = 0.01;
Subscript[c, A] = (Subscript[p, A]*(s*(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B]*(s*(1 + u) - x)) - k;
Subscript[Z, A]=(1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^\[Beta] (v-Subscript[c,A]) ;
Subscript[Z, B]=(1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^\[Alpha] (Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z, A]] + Norm[Subscript[Z, B]], Subscript[c, A] <= v <= Subscript[c, B]},
   {v, \[Alpha], \[Beta], Subscript[p, A], Subscript[p, B]}]

During evaluation of In[61]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[68]= {1.945590892669753*10^514, {v -> -48.2786, \[Alpha] -> 56.0364, \[Beta] -> -102.072,
   Subscript[p, A] -> -52.0972, Subscript[p, B] -> -10.7259}}
3

0<=Pa<=1, 0<=Pb<=1

4

In[1]:= s = 95; u = 0.1; x = 100; k = 0.01;
Subscript[c, A] = (Subscript[p, A](s(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B](s(1 + u) - x)) - k;
Subscript[Z, A] = (1-(v-Subscript[c,A])/(Subscript[c,B]-Subscript[c,A]))^[Beta](v-Subscript[c,A]);
Subscript[Z, B] = (1-(v-Subscript[c,B])/(Subscript[c,B]-Subscript[c,A]))^[Alpha](Subscript[c,B]-v);
Maximize[{Norm[Subscript[Z,A]]+Norm[Subscript[Z,B]], Subscript[c,A]<=v<=Subscript[c,B] &&
0<=Subscript[p,A]<=1 && 0<=Subscript[p,B]<= 1}, {v, [Alpha], [Beta], Subscript[p,A], Subscript[p,B]}]

During evaluation of In[1]:=NMaximize::cvdiv:Failed to converge to a solution. The function may be unbounded.

Out[9]= {3.97046*10^15, {v -> 0.101697, \[Alpha] -> 53.8285, \[Beta] -> -34.2923, Subscript[p, A] -> 0.0203771, 
  Subscript[p, B] -> 0.0799841}}

You may want to look at the values of each part of ZA and ZB to determine whether more constraints are needed or whether there is something about the way that I have done this which is incorrect.

5

I think that Za and Zb should be maximized seperately. Is this what you’ve wrote?

6

From the initial post I was guessing that you needed to maximize both of the items.

In[1]:= s = 95; u = 0.1; x = 100; k = 0.01;
Subscript[c, A] = (Subscript[p, A](s(1 + u) - x)) + k;
Subscript[c, B] = (Subscript[p, B](s(1 + u) - x)) - k;
Subscript[Z, A] = (1 - (v - Subscript[c, A])/(Subscript[c, B] - Subscript[c, A]))^[Beta] (v - Subscript[c, A]);
Subscript[Z, B] = (1 - (v - Subscript[c, B])/(Subscript[c, B] - Subscript[c, A]))^[Alpha] (Subscript[c, B] - v);
Maximize[{Subscript[Z, A], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 &&
0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[1]:=NMaximize::nrnum:The function value -0.113773+0.104822 I is not a real number at
{v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {1.24375,0.162408,-0.0367399,0.0548333,0.081051}. >>

Out[6]= {0.863531, {v → 0.754547, [Alpha] → -0.523666, [Beta] → -0.734744,
Subscript[p, A] → 0.0000800015, Subscript[p, B] → 0.906934}}

In[7]:= Maximize[{Subscript[Z, B], Subscript[c, A] <= v <= Subscript[c, B] && 0 <= Subscript[p, A] <= 1 &&
0 <= Subscript[p, B] <= 1}, {v, [Alpha], [Beta], Subscript[p, A],Subscript[p, B]}]

During evaluation of In[7]:=NMaximize::nrnum:The function value 1.613510^8-4.6536810^7 I is not a real number at
{v,[Alpha],[Beta],Subscript[p, A],Subscript[p, B]} = {0.377249,18.8636,-19.7831,0.162586,0.150344}. >>

Out[7]= {2.99746*10^7, {v → 0.454771, [Alpha] → 16.1621, [Beta] → -15.78, Subscript[p, A] → 0.187606,
Subscript[p, B] → 0.2869}}