 Solution

In a matrix game, entry $a_{ij}$ is the row player's payoff, and the column player's payoff is $-a_{ij}$. A mixed strategy is a probability vector over the available rows or columns. With strategies $p,q$, the row payoff is $p^{T}Aq$. Its optimal mixed strategy maximizes its guaranteed payoff, while the column player minimizes the largest row payoff:The equality is the finite minimax theorem. Equivalently the row player maximizes ${\min }_j(p^{T}A{)}_j$, and the column player minimizes ${\max }_i(Aq{)}_i$. Each optimal strategy guarantees its corresponding bound against every opponent strategy; for the column player's own negative payoff this is also a maximin strategy.