Computing All Nash Equilibria of Multiplayer Games in Electricity Markets by Solving Polynomial Equations

The analysis of the Nash equilibrium (NE) in electricity markets with imperfect competition is subject to two major challenges: the treatment of multiplayer games and the determination of the existence of multiple market equilibria. To resolve these obstacles, a solution method, based on the payoff matrix approach and polynomial equations, is proposed in this paper to calculate all the Nash equilibria of multiplayer games in the electricity markets. First, the proposed method decomposes the game by means of a set of all pure strategies assigned with positive probabilities (support). For each possible support, the NE condition can be characterized by a set of polynomial equations with inequality constraints. Next, the homotopy continuation algorithm is employed to obtain all solutions of the polynomial system. Finally, all Nash equilibria can be found by verifying each solution of all polynomial systems. Two example systems, one involving the Cournot model and the other based on the supply function equilibrium (SFE) model, are applied to test the proposed method, respectively. The results show that the proposed method has the ability to identify all NE under certain conditions, indicating that the proposed algorithm is useful for providing insights into the theoretical analysis of electricity markets.