Matrix game

A matrix game is a finite two-player zero-sum game whose entry $A_{ij}$ is the row player's payoff when the players choose row $i$ and column $j$.

When every entry of a payoff matrix $B$ is positive, its game value $v$ is positive. Setting $x=p/v$ converts the row player's probability vector and guaranteed value into the displayed linear program, whose optimum is $1/v$. Conversely normalizing a feasible $x$ gives a mixed strategy guaranteeing $1/(1^{T}x)$. The dual maximizes $1^{T}y$ under $By\le 1$, $y\ge 0$ and recovers the column strategy by normalization. Adding a constant to every payoff preserves Nash equilibria and lets this positive-payoff formulation handle general finite zero-sum games.

A payoff matrix records the row player's gains when row $i$ meets column $j$. In a zero-sum game the other player's payoff is the negative. For mixed strategies $p,q$, the expected row payoff is $p^{T}Aq$.

The row player solves

If the column player has an optimal strategy at value $v$ and a row probability vector $p$ satisfies $p^{T}A\ge v{e}^T$, then $p$ is optimal: it guarantees $v$, while the optimal column strategy prevents every row strategy from exceeding $v$.

If a symmetric invertible payoff matrix satisfies $A^{-1}e\ge 0$, both players may useand the game value is $(e^T{A}^{-1}e{)}^{-1}$.

Forboth players optimally choose cards $1,2,3$ with probabilities $41/50,2/25,1/10$, and the row player's expected payoff is $57/25$.