Value of a zero-sum game ($v={\max }_r{\min }_s{r}^{\mathsf{T}}As={\min }_s{\max }_r{r}^{\mathsf{T}}As$)

For a finite matrix zero-sum game, the value is the payoff that the maximizing player can guarantee and the minimizing player can prevent the payoff from exceeding, using mixed strategies. The common value of the two displayed optimizations exists by the minimax theorem. A pair of strategies can be verified directly by checking that every pure column pays the row mixture at least $v$ and every pure row earns at most $v$ against the column mixture.