 Solution

Every payoff is positive, so every pair of mixed strategies has positive expected payoff and the game value satisfies $v>0$. Player 1 maximizes $v$ subject to $A^{T}p\ge ve$, $e^{T}p=1$, $p\ge 0$. Setting $x=p/v$ turns these intoMaximizing $v$ is therefore equivalent to the stated minimization of $e^{T}x$.