Author(s)

Yoshihiro Tanaka

Faculty of Economics, Hokkaido University, Sapporo, Japan.

It is well established that Nash equilibrium exists within the framework of mixed strategies in strategic-form non-cooperative games. However, finding the Nash equilibrium generally belongs to the class of problems known as PPAD (Polynomial Parity Argument on Directed graphs), for which no polynomial-time solution methods are known, even for two-player games. This paper demonstrates that in fixed-sum two-player games (including zero-sum games), the Nash equilibrium forms a convex set, and has a unique expected payoff. Furthermore, these equilibria are Pareto optimal. Additionally, it is shown that the Nash equilibrium of fixed-sum two-player games can theoretically be found in polynomial time using the principal-dual interior point method, a solution method of linear programming.

Nash Equilibrium, Fixed-Sum Two-Player Game, Principal-Dual Interior Point Method

Tanaka, Y. (2024) Nash Equilibrium of a Fixed-Sum Two-Player Game. American Journal of Computational Mathematics, 14, 346-357. doi: 10.4236/ajcm.2024.143017.