Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 3 / 5H (Optimization)

Player A maximizes the payoff, while B minimizes it. The fourth row is strictly worse for A than the first in every column, so remove $A_4$ by dominated strategy elimination. On the remaining rows, column $B_3$ is strictly better for B than $B_2$, so remove $B_2$. Now $A_3$ is strictly worse than $A_2$ in the two remaining columns. The reduced zero-sum game has payoff matrixwith rows $A_1,A_2$ and columns $B_1,B_3$.

If A uses mixed strategy $(p,1-p)$, the expected payoffs against these columns are $-2+6p$ and $3-8p$. The first increases and the second decreases, so their minimum is maximized where they are equal: $14p=5$. ThusFor an optimality certificate, B uses mixed strategy $(q,0,1-q)$, making A's first two row payoffs $-5+9q$ and $3-5q$. Equality gives $q=4/7$. Against this B strategy the four row payoffs are $(1/7,1/7,-6/7,-6/7)$, while against the displayed A strategy the three column payoffs are $(1/7,13/7,1/7)$. Therefore A guarantees $1/7$ and B holds A to $1/7$, verifying the value of a zero-sum game and the original, unreduced mixed strategy.