Solving three-player games by the matrix approach with application to an electric power market

In models of imperfect competition of deregulated electricity markets, the key task is to find the Nash equilibrium (NE). The approaches for finding the NE have had two major bottlenecks: computation of mixed strategy equilibrium and treatment of multiplayer games. This paper proposes a payoff matrix approach that resolves these bottlenecks. The proposed method can efficiently find a mixed strategy equilibrium in a multiplayer game. The formulation of the NE condition for a three-player game is introduced and a basic computation scheme of solving nonlinear equalities and checking inequalities is proposed. In order to relieve the inevitable burden of searching the subspace of payoffs, several techniques are adopted in this paper. Two example application problems arising from electricity markets and involving a Cournot and a Bertrand model, respectively, are investigated for verifying the proposed method. The proposed method outperforms a publicly available game theory software for the application problems.