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

Against $q$, choosing row $i$ gives Player I expected payoff $(Aq{)}_i={\sum }_j{a}_{ij}{q}_j$. The best response maximizes this number; randomizing among rows cannot exceed the largest row payoff. Player II therefore minimizes that maximum, exactly the printed optimization problem.

The minimax theorem for a finite zero-sum game statesThis common number is the value of a zero-sum game: Player I can guarantee an expected payoff of at least $v$, and Player II can hold it to at most $v$. At optimal mixed strategies their expected transfer is $v$.