= Value of a zero-sum game
{title2=$v=\max_r\min_s r^{\mathsf T}As=\min_s\max_r r^{\mathsf T}As$}
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.
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