Nash Equilibria in certain two-choice multi-player games played on the ladder graph

TL;DR

This paper analytically computes the number of Nash Equilibria in a two-choice anti-coordination game played on ladder and circular ladder graphs, revealing exponential growth related to graph topology and payoff parameters.

Contribution

It provides the first analytical characterization of NE counts in these graph-based games, highlighting the influence of topology and payoff parameters on equilibrium multiplicity.

Findings

Number of NE grows exponentially with number of players

Growth rate involves the golden ratio, 7

Topology affects the scaling factor for NE count

Abstract

In this article 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 $N_{NE}(2n)\sim C(\varphi)^n$, where $\varphi=1.618..$ is the golden ratio and $C_{circ}>C_{ladder}$. In addition, the value of the scaling factor $C_{ladder}$ depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is $C_{circ}$ is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.