= 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 $v$ are
$$
p\ge0,\quad q\ge0,\quad \mathbf1^Tp=\mathbf1^Tq=1,\qquad
Aq\le v\mathbf1,\quad p^TA\ge v\mathbf1^T.
$$
Indeed $q$ 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,
$$
A=\begin{pmatrix}-1&-1\\0&-3/2\\0&1/2\\1&0\end{pmatrix}.
$$
For example, row $(B,P)$ against call averages a gain of two and a loss of one, giving $1/2$.
Take
$$
\boxed{p=(0,0,2/3,1/3)^T,\qquad q=(1/3,2/3)^T.}
$$
Then $Aq=(-1,-1,1/3,1/3)^T$ and $p^TA=(1/3,1/3)$, so the sufficient inequalities hold with \b[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.
Back to article page
