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arXiv · 2101.09103

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

Victoria Sánchez Muñoz·Michael Mc Gettrick
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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.

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Victoria Sánchez Muñoz, Michael Mc Gettrick. 2021-01-22. Nash Equilibria in certain two-choice multi-player games played on the ladder graph. https://doi.org/10.1142/s0219198920500206

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