Abstract
William D. Hamilton famously stated that “human life is a many person game and not just a disjoined collection of two person games”. However, most of the theoretical results in evolutionary game theory have been developed for two player games. In spite of a multitude of examples ranging from humans to bacteria, multi-player games have received less attention than pairwise games due to their inherent complexity. Such complexities arise from the fact that group interactions cannot always be considered as a sum of multiple pairwise interactions. Mathematically, multi-player games provide a natural way to introduce non-linear, polynomial fitness functions into evolutionary game theory, whereas pairwise games lead to linear fitness functions. Similarly, studying finite populations is a natural way of introducing intrinsic stochasticity into population dynamics. While these topics have been dealt with individually, few have addressed the combination of finite populations and multi-player games so far. We are investigating the dynamical properties of evolutionary multi-player games in finite populations. Properties of the fixation probability and fixation time, which are relevant for rare mutations, are addressed in well mixed populations. For more frequent mutations, the average abundance is investigated in well mixed as well as in structured populations. While the fixation properties are generalizations of the results from two player scenarios, addressing the average abundance in multi-player games gives rise to novel outcomes not possible in pairwise games.
Highlights
The analysis of stochastic evolutionary game dynamics has rapidly developed in the past decade [1,2,3,4,5,6,7,8,9,10]
When we consider the evolutionary process in a finite population, it is important to investigate the stochastic properties of the system, such as fixation probability, fixation time and average abundance in mutation-selection equilibrium
For a multi-player game, i.e., d > 2, the payoffs are nonlinear, but remain polynomials. Such nonlinearities mimic the interaction pattern among individuals, like the public goods, for example, the invertase produced by the cooperator yeast in the above example is a saturating function of cooperators’ concentration. Dynamical properties of such multi-player games in infinitely large populations have been previously addressed [24,25,26] and we focus on their finite population version
Summary
The analysis of stochastic evolutionary game dynamics has rapidly developed in the past decade [1,2,3,4,5,6,7,8,9,10]. When we consider the evolutionary process in a finite population, it is important to investigate the stochastic properties of the system, such as fixation probability, fixation time and average abundance in mutation-selection equilibrium. Such nonlinearities mimic the interaction pattern among individuals, like the public goods, for example, the invertase produced by the cooperator yeast in the above example is a saturating function of cooperators’ concentration Dynamical properties of such multi-player games in infinitely large populations have been previously addressed [24,25,26] and we focus on their finite population version. For 2 × 2 games, for a given population structure and evolutionary dynamics with mutation, a single parameter condition is obtained to determine which strategy is more abundant than in the neutral case under weak selection [44]. We generalize the so-called σ rule [45]
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