 Solution

A mixed strategy for Player II is a probability vector $q\ge 0$ with ${\sum }_j{q}_j=1$, specifying the random choice of a column. If Player I chooses row $i$, the expected payment is $(Aq{)}_i$. A mixture of rows cannot exceed the maximum of these row expectations, so the smallest payment Player II can guarantee isAn optimal strategy is a vector attaining that minimum, which exists by compactness of the probability simplex and continuity. The minimax theorem identifies this guaranteed loss with Player I's maximal guaranteed gain, the value of the finite zero-sum game.