Okay, I have a pretty solid math background but it’s been a long time (I’ve spent the last three years learning useful stuff like the parol evidence rule and the rule against perpetuities). But I have a general interest in stuff like game theory, so here goes my simple question.

Given a basic payoff matrix, as follows:

P Q

R S

with player A deciding which row and B deciding which column to select, what’s the formula for an optimal strategy?

If A chooses row 1 x% of the time, and B chooses column 1 y% of the time, then A’s expected payoff is

P(a)= x*y*P+x*(1-y)*Q + (1-x)*y*R + (1-x) * (1-y)*S

Since this is a zero sum game, B’s is simply (-P(a)), and of course both players wish to maximize their profit/ minimize their loss given perfect play from the other.

So the basic question is how do you calculate this? Is there a simple formula, or do you have to do some sort of recursive iteration?

Thanks all.