= Positive-payoff linear programming for a matrix game
{title2=$\min\mathbf1^Tx:\ B^Tx\ge\mathbf1,\ x\ge0$}
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 $1/(\mathbf1^Tx)$. The dual maximizes $\mathbf1^Ty$ under $By\le\mathbf1$, $y\ge0$ 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>.
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