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The game is degenerate in the nondegeneracy of a bimatrix game sense. Against the first player's pure strategy $4$, the second player's choices $1$ and $2$ are both best responses, with payoff four. A mixed strategy of support size one therefore has two pure best responses. The usual Lemke-Howson algorithm path needs additional tie handling or perturbation in such a game; the zero-sum game structure gives a simpler direct linear program.

Add five to every entry of the first player's matrix, obtainingThis does not change either player's best responses or Nash equilibria; it raises the game value by five. Since all entries are positive, its value $v_B$ is positive. If $p$ is a row mixed strategy guaranteeing $v_B$, put $x=p/v_B$. Then $B^{T}x\ge 1$ and $1^{T}x=1/v_B$.

Conversely any feasible $x$ has $s=1^{T}x>0$, and $p=x/s$ guarantees payoff $1/s$ in the shifted game. Maximizing that guaranteed payoff is therefore equivalent to minimizing $s$ under $B^{T}x\ge 1$, $x\ge 0$. These are precisely the displayed constraints. The dual program maximizes $1^{T}y$ subject to $By\le 1$, $y\ge 0$; normalizing an optimal $y$ gives the column strategy. This is positive-payoff linear programming for a matrix game.

Apply positive affine payoff transformations, if necessary, so both payoff matrices have strictly positive entries. Such transformations preserve best responses and Nash equilibria. The Lemke-Howson algorithm uses unnormalized nonnegative strategy vectors and slack variables satisfyingThus its two polytopes are $\{x\ge 0:Q^{T}x\le 1\}$ and $\{y\ge 0:Py\le 1\}$. A label $i\le m$ occurs when $x_i=0$ or $r_i=0$; a label $m+j$ occurs when $y_j=0$ or $s_j=0$. The complementary pivoting path drops one label from the artificial zero pair and resolves each duplicated label until all labels return.

Terminate at a completely labelled pair other than the artificial zero pair. This is equivalent to complementary slacknessA nonzero completely labelled pair has both vectors nonzero: if $x=0$, then $s=1$ forces $y=0$, and the converse is analogous. Normalize to mixed strategiesThe inequalities $Py\le 1$ imply that every row payoff against $q$ is at most $1/(1^{T}y)$, with equality on every row receiving positive probability in $p$. Thus $p$ is a best response to $q$. Likewise $q$ is a best response to $p$ using $Q^{T}x+s=1$. Hence $(p,q)$ is a Nash equilibrium. Nondegeneracy of a bimatrix game ensures the usual complementary pivoting path has a unique continuation after the label choice.