Nash Equilibria in Certain Two-Choice Multi-Player Games Played on the Ladder Graph

Abstract

In this paper, we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with [Formula: see text] players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players [Formula: see text], as [Formula: see text], where [Formula: see text] is the golden ratio and [Formula: see text]. In addition, the value of the scaling factor [Formula: see text] depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is, [Formula: see text] is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.