Glad to help.

Speeding things up: While you are experimenting, lower the precision to speed things up by adding

AccuracyGoal -> 4, PrecisionGoal -> 4

or even

AccuracyGoal -> 2, PrecisionGoal -> 2

to the findroot. Also, if you can give it a better starting point it helps. You can also set limits for the variables to narrow the search space.

I thought of a simple way for you to fix this problem. I will post something when I return to be in front of a computer. I assume that you are OK with a numerical solution and not an analytical one. Correct?

Not at all. Solve, or NSolve might be giving ArgMax a non-numerical variable, but I don't know how to prevent that. Here is the relevant piece of code again (the original had Solve, now it's NSolve:

sol2 = NSolve[{v1n == ArgMax[f1[r, v1, v2n, x1star, x2star], v1], 
    v2n == ArgMax[f2[r, v1n, v2, x1star, x2star], v2]}, {v1n, 
    v2n}][[1, 1]]

v1n and v2n are both real numbers, but perhaps NSolve takes them as something else in intermediate steps. How could this be reformulated?

Hi Michael,

Thanks again for your help. Unfortunately, the article you mentioned is about a non-linear programming algorithm, which has only 1 utility function. In game theory, there are 2 or more (2 in the simple example I submitted) utility functions to be optimized simultaneously, sharing dependent variables. The problem is much simpler (and can easily be solved by known techniques) when the functions are linear, as is the case for many classical (bi-matrix) games, but even this is not true in my case. It appears that for bi-linear games, it is possible to use Reduce and the KT algorithm, as that article does.

Still, I would be surprised if nothing exists out there. I will keep searching.

I'm not sure I believe you that Solve finds a solution with v1n=1.16, even in the newer version. When I try

ArgMax[f2[v1,v2]/.v1->1.16,v2]

The output is v2=-0.96

And when I try

ArgMax[f1[v1,v2]/.v2->-0.96, v1]

The output is v1=0.62.

So whatever you found is not a Nash Equilibrium (there is no Nash equation, look up the concept of Nash equilibrium, which is basically what the Argmax equations say for continuous variables).

I wonder what algorithm Solve uses to find the solution. Just setting the derivatives to zero does not give the correct solution, since the second derivatives have to be negative. I tried it with FindRoot, setting derivatives to zero, and it does not give the solution with negative second derivatives, but with zero or positive ones. I think FindRoot might use a multifunction version of Newton-Raphson, but it does not work in this case. What was the v2n that you say Solve gave you?