I was solving for a stable equilibrium in the following 2 player zero sum game. I need to calculate the equilibrium using maxmin and minmax strategies. In this game they should come out to be identical and coincide with the mixed strategy Nash's equilibrium.
P2
L R
L (0.6,0.4) (0.8,0.2)
P1
R (0.9,0.1) (0.7,0.3)
If I solve it formally, the expected payoff of P1 will be the expression:
E = { 0.6(pq) + 0.8p(1-q) + 0.9(1-p)(q) + 0.7(1-p)(1-q) }
where p is probability of P1 playing L and q is the probability pf P2 playing L. Now, P2 is trying to minimize E keeping in mind that P1 is trying to maximize it
i.e. min max {E}
and P1 is trying to maximize E keeping in mind that P1 will try to minimize E i.e.
max min {E}
and vice versa. Essentially both strategies played simultaneously should give the same equilibrium. How to go about it, I am stuck in the maths part of the problem now.
Setting the derivative with respect to $p$ to $0$ yields $0.6q+0.8(1-q)-0.9q-0.7(1-q)=0$ and thus $q=0.25$, and doing the same for $q$ yields $0.6p-0.8p+0.9(1-p)-0.7(1-p)=0$ and thus $p=0.5$. By the principle of indifference, this is the mixed Nash equilibrium.