Equilibrium Numbers of Small Multiplayer Games

A Nash equilibrium of a noncooperative game defines a mixed (randomized) strategy for each player that is optimal against the fixed strategies of the other players. It is characterized by constraints (equations and inequalities) on the mixed-strategy probabilities for the players' expected payoffs and a complementarity condition. The constraints are linear for two players but involve polynomials for more than two players. We show that the simplest multiplayer game of three players with two actions each has generically at most nine equilibria. The proof, using the small size of the game, relies on properties of hyperbolas. Another tool is the index of an equilibrium, an orientation that implies that the number of equilibria is odd. We discuss a more general conjecture by Hertling and Vujic about equilibrium numbers for two-action games with many players.

Joint work with Sahar Jahani.