Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / T / 1 / d / Solution

For the minimizing player, column one weakly dominates column three, since its payoff is no greater in every row. We may therefore choose an optimum with $q_3=0$. Write $q=(x,1-x,0)$, $0\le x\le 1$. The three row payoffs are $6-5x$, $5-3x$, and $1+2x$.

The maximum of the decreasing line $5-3x$ and the increasing line $1+2x$ is minimized where they meet: $x=4/5$, with value $13/5$. At this point the first line is only $2$, so it does not increase the maximum. Hence the candidate isA dual certificate establishes optimality against every possible three-column mixture. Player I's strategy $p_{\ast }=(0,2/5,3/5)$ gives column payoffs $(13/5,13/5,16/5)$, guaranteeing at least $13/5$. The displayed $q_{\ast }$ holds every row payoff to at most $13/5$. These matching guarantees prove the claimed value and optimality.