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 2n 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 n, as NNE(2n)∼C(φ)n, where φ=1.618.. is the golden ratio and Ccirc>Cladder. In addition, the value of the scaling factor Cladder depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is, Ccirc is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.