= Solution
A <mixed strategy> for Player II is a probability vector $q\ge0$ with $\sum_jq_j=1$, 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 is
$$
\boxed{v=\min_{q\ge0,\ \mathbf1^\mathsf Tq=1}\max_i(Aq)_i.}
$$
An 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>.
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