The matrix $M$ of a linear complementarity problem can be viewed as a payoff matrix of a two-person zero-sum game. Lemke’s algorithm can be successfully applied to reach a complementary solution or infeasibility when the game satisfies the following conditions: (i) The value of $M$ is equal to zero. (ii) For all principal minors of $M^T$ (transpose of $M$) the value is non-negative. (iii) For any optimal mixed strategy $y$ of the maximizer either $y_i>0$ or $(My)_i>0$ for each coordinate $i$.