$3 . \mathrm{I} . 5 \mathrm{H} \quad$

Consider a two-person zero-sum game with a payoff matrix

$\left(\begin{array}{ll} 3 & b \\ 5 & 2 \end{array}\right)$

where $0<b<\infty$. Here, the $(i, j)$ entry of the matrix indicates the payoff to player one if he chooses move $i$ and player two move $j$. Suppose player one chooses moves 1 and 2 with probabilities $p$ and $1-p, 0 \leq p \leq 1$. Write down the maximization problem for the optimal strategy and solve it for each value of $b$.

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