Past exam of the computer science course of the University of Oxford / 2019 / Computational game theory / 2 / b / Solution

The row player's unique best response to $L$ is $B$ and to $R$ is $T$. The column player's unique best response to $T$ is $R$ and to $B$ is $L$. Thus the pure Nash equilibria are $(T,R)$ and $(B,L)$.

For a fully mixed equilibrium, let $p=\mathrm{Pr}(T)$ and $q=\mathrm{Pr}(L)$. Row-player indifference requiresColumn-player indifference requiresBoth probabilities lie strictly between zero and one, so these indifference conditions make both supported actions best responses. Hence the remaining equilibrium isIts expected payoffs are $(8/3,18/7)$. There are no other partly mixed equilibria: if either player is pure, the other's best response is unique, forcing both to be pure. If both mix, the displayed linear indifference equations uniquely determine their probabilities. The row player's equilibrium mixture differs from its maxmin mixture in part (a); these are different optimization criteria in this non-zero-sum game.