Let $A$ be the $m \times n$ payoff matrix of a two-person, zero-sum game. What is Player I's optimization problem?

Write down a sufficient condition that a vector $p \in \mathbb{R}^{m}$ is an optimal mixed strategy for Player I in terms of the optimal mixed strategy of Player II and the value of the game. If $m=n$ and $A$ is an invertible, symmetric matrix such that $A^{-1} e \geqslant 0$, where $e=(1, \ldots, 1)^{T} \in \mathbb{R}^{m}$, show that the value of the game is $\left(e^{T} A^{-1} e\right)^{-1}$

Consider the following game: Players I and II each have three cards labelled 1,2 , and 3. Each player chooses one of her cards, independently of the other, and places it in the same envelope. If the sum of the numbers in the envelope is smaller than or equal to 4, then Player II pays Player I the sum (in $£$ ), and otherwise Player I pays Player II the sum. (For instance, if Player I chooses card 3 and Player II choose card 2, then Player I pays Player II £5.) What is the optimal strategy for each player?