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

In this article we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on (circular) ladder graph with $2n$ players. We consider set games generic payoff parameters, only requirement that NE occurs if players choose opposite strategies (anti-coordination game). The results show both, and circular ladder, grows exponentially (half) $n$, as $N_{NE}(2n)\sim C(\varphi)^n$, where $\varphi=1.618..$ is golden ratio $C_{circ}>C_{ladder}$. addition, value scaling factor $C_{ladder}$ depends parameters. However, no longer true (3-degree graph), $C_{circ}$ constant, which might suggest topology indeed plays an important role setting NE.