The Lemke–Howson Algorithm Solving Finite Non-Cooperative Three-Person Games in a Special Setting

Abstract

We present a brief outline of an approximate method (3LP) proposed by E.G. Golshteyn for solving three-person games in mixed strategies. Similar to the 2LP-algorithm (which approximately solves bimatrix games), the solution procedure consists in the search of a global minimum of the so-called Nash function. By making an exhaustive search of the pairs of the initial strategies the algorithm 2LP (3LP) finds an exact solution of the game if the condition of reciprocal complementarity holds. The numerical experiments show that the 2LP-method successfully competes with the Lemke–Howson (LH) algorithm, which efficiently solves the bimatrix games. Unfortunately, the LH-algorithm cannot be applied to solve arbitrary three-person games. However, we have adapted the Lemke–Howson method to the solution of a special setting called the hexamatrix games. We have also conducted a thorough testing of the LH-algorithm to reveal its advantages and minor points as well.