Evolutionary games in the multiverse

TL;DR

This paper explores the dynamics of multiplayer evolutionary games with multiple strategies, revealing increased complexity and limitations of pairwise assumptions, and providing bounds on internal equilibria.

Contribution

It generalizes evolutionary game theory to multiplayer settings with multiple strategies, establishing bounds on internal equilibria and highlighting the complexity beyond pairwise interactions.

Findings

Maximum one internal equilibrium in two-player, multi-strategy games.

At most (d-1)^(n-1) internal equilibria in d-player, n-strategy games.

Multiplayer games exhibit complex dynamics not reducible to pairwise interactions.

Abstract

Evolutionary game dynamics of two players with two strategies has been studied in great detail. These games have been used to model many biologically relevant scenarios, ranging from social dilemmas in mammals to microbial diversity. Some of these games may in fact take place between a number of individuals and not just between two. Here, we address one-shot games with multiple players. As long as we have only two strategies, many results from two player games can be generalized to multiple players. For games with multiple players and more than two strategies, we show that statements derived for pairwise interactions do no longer hold. For two player games with any number of strategies there can be at most one isolated internal equilibrium. For any number of players $\boldsymbol{d}$ with any number of strategies n, there can be at most (d-1)^(n-1) isolated internal equilibria. Multiplayer games show a great dynamical complexity that cannot be captured based on pairwise interactions. Our results hold for any game and can easily be applied for specific cases, e.g. public goods games or multiplayer stag hunts.