 Solution

In a two-player zero-sum game, entry $a_{ij}$ is player I's gain, and player II's loss, when pure strategies $i,j$ are chosen. Player I maximizes the expected payoff, player II minimizes it. A mixed strategy is a probability vector. Sufficient optimality conditions for $p,q$ and value $v$ areIndeed $q$ keeps every opposing payoff at most $v$, while $p$ guarantees at least $v$ against every opposing mixed strategy; their joint payoff must equal $v$.

A pure strategy for the informed player specifies one action after each possible outcome. Order these contingent choices as $(P,P),(P,B),(B,P),(B,B)$, with the first action for a head. The columns are fold and call. Averaging over the fair coin gives the payoff matrix, in currency units,For example, row $(B,P)$ against call averages a gain of two and a loss of one, giving $1/2$.

TakeThen $Aq=(-1,-1,1/3,1/3{)}^T$ and $p^{T}A=(1/3,1/3)$, so the sufficient inequalities hold with game value $v=1/3$ to player I. In behavioral terms, I always bets on a head and bets with probability $1/3$ on a tail; II calls with probability $2/3$. The four rows describe complete contingent plans, not four coin outcomes.