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.
The paper proposes the concept of a weak Berge equilibrium. Unlike the Berge equilibrium, the moral basis of this equilibrium is the Hippocratic Oath “First do no harm”. On the other hand, any Berge equilibrium is a weak Berge equilibrium. But, there are weak Berge equilibria, which are not the Berge equilibria. The properties of the weak Berge equilibrium have been investigated. The existence of the weak Berge equilibrium in mixed strategies has been established for finite games. The weak Berge equilibria for finite three-person non-cooperative games are computed.
The efficiency of using the local search algorithm (а cutting plane algorithm ) to find the Nash equilibrium for n person games in a general formulation using Python software environments is studied. The use of multiplication of a multidimensional matrix by a vector in the local search procedure proved to be an effective tool.
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