Existence of Symmetric Equilibrium for Two-Person Symmetric Equal Row-Sums Games

Abstract

A matrix is said to be an equal row-sums matrix if all row-sums are equal to one another and a two-person symmetric bi-matrix game is said to be a two-person symmetric equal row-sums (TPSERS) game if the pay-off matrix of the row player is an equal row-sums matrix. For TPSERS games, we show that the randomized strategy that assigns the same probability to all pure strategies is a symmetric equilibrium for the game and the probabilistic component of every solution of a certain quadratic programming problem is a symmetric equilibrium for the game. The main result here is motivated by the “Equivalence Theorem” in section II of Mangasarian and Stone (1964) for bi-matrix games. The version of the “Equivalence Theorem” applicable for symmetric games is available in Lahiri (2025). The proofs of both need to appeal to a prior “existence of equilibrium result” argument when it comes to establishing that every solution of the relevant quadratic programming problem yields an equilibrium for the game under consideration and hence neither proof is self-contained. For TPSERS games, we show there is a simple and self-contained proof that does not either implicitly or explicitly use any prior “existence of equilibrium result” argument and this proof is the main contribution of the paper.