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Evolutionary Multiplayer Games

Chaitanya S. Gokhale, Arne Traulsen

arXiv:1404.1421v1q-bio.PE

TL;DR

Evolutionary game theory has mostly examined two-player interactions, while many natural and social systems involve nonlinear interactions. This review surveys multiplayer evolutionary games in infinite and finite well-mixed populations, their applications, and future directions. It concludes that multiplayer games can capture nonlinear interactions while retaining the simplicity of evolutionary games.

Abstract

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Evolutionary game theory has become one of the most diverse and far reaching theories in biology. Applications of this theory range from cell dynamics to social evolution. However, many applications make it clear that inherent non-linearities of natural systems need to be taken into account. One way of introducing such non-linearities into evolutionary games is by the inclusion of multiple players. An example is of social dilemmas, where group benefits could e.g.\ increase less than linear with the number of cooperators. Such multiplayer games can be introduced in all the fields where evolutionary game theory is already well established. However, the inclusion of non-linearities can help to advance the analysis of systems which are known to be complex, e.g. in the case of non-Mendelian inheritance. We review the diachronic theory and applications of multiplayer evolutionary games and present the current state of the field. Our aim is a summary of the theoretical results from well-mixed populations in infinite as well as finite populations. We also discuss examples from three fields where the theory has been successfully applied, ecology, social sciences and population genetics. In closing, we probe certain future directions which can be explored using the complexity of multiplayer games while preserving the promise of simplicity of evolutionary games.

INTRODUCTION

Evolutionary game theory models how natural selection changes the frequencies of fixed strategies according to their payoffs. The paper reviews multiplayer games as a way to represent nonlinear interactions, alongside the dynamics used to study them.

INTRODUCTION

  • Natural selection changes strategy frequencies as individuals receive payoffs from interactions, with higher-payoff strategies spreading at the expense of less successful ones.Evolutionary game theory does not rely on the rationality assumption typical of classical game theory.Source: INTRODUCTION · p. 1
  • Evolutionarily stable strategies describe resistance to invasion by a small number of individuals using a different strategy, and the concept was generalized to multiplayer games in the early eighties.A weak ESS can permit initially neutral mutants to invade.Source: INTRODUCTION · p. 1
  • Evolutionary dynamics connect the question of whether a strategy is stable to whether it is attainable.Taylor, Jonker and Zeeman extended evolutionary game theory to include dynamics.Source: INTRODUCTION · p. 1

FROM PAIRWISE CONTESTS TO SOCIAL INTERACTIONS

  • Multiplayer interactions add nonlinear dynamics to evolutionary games, potentially producing dynamics unavailable in conventional two-player games.The review surveys current work on these interactions and their applications.Source: FROM PAIRWISE CONTESTS TO SOCIAL… · p. 2

Replicator dynamics

  • The replicator equation tracks strategy-frequency change by comparing each strategy’s fitness with average population fitness, with fitness assumed equal to payoff here.For two strategies, the population frequencies are x and 1−x; the dynamics have fixed points at x = 0, x = 1, or equal strategy fitness.Source: Replicator dynamics · p. 2

1. Equilibrium points in multiplayer games

With two strategies, adding players makes payoff differences nonlinear in strategy frequency and allows more internal equilibria. The review also describes how random-game analyses characterize equilibrium patterns and connect multiplayer games to population genetics.

1. Equilibrium points in multiplayer games

  • In a three-player, two-strategy game, the payoff difference is quadratic and can allow two internal equilibria in addition to the trivial roots x = 0 and x = 1.The payoff matrix treats AB and BA interactions as equivalent; distinguishing player order would require separate payoff entries.Source: §1 · pp. 2–3
  • For a d-player game with n strategies, the maximum number of distinct internal equilibria is (d−1)^(n−1).For two strategies, the payoff difference can have up to d−1 nontrivial roots.Source: §1 · p. 3
  • For randomly drawn d-player games with two strategies, a given equilibrium has probability 1/2 of being stable or unstable.For d-player games with n strategies, probabilities for given numbers of equilibria have been calculated in several cases, but not in closed form across all cases.Source: §1 · p. 3
  • A study of multiplayer games proves Karlin and Feldman’s conjecture on the maximum number of fixed points in deterministic viability selection among n haplotypes under random matching.Altenberg’s alternative proof establishes the conjecture without that assumption.Source: §1 · p. 4

From infinite to finite populations

Finite populations shift evolutionary questions from deterministic stability to stochastic outcomes such as fixation, extinction, and maintenance of multiple strategies.

From infinite to finite populations

  • Finite-population analysis asks about fixation, extinction, and maintenance of multiple strategies, rather than only evolutionary stability in the deterministic regime.The replicator dynamics describes strategy-frequency dynamics in an infinitely large population; finite-population studies can challenge results based on that assumption.Source: From infinite to finite populations · p. 4

Moran process

The review introduces a finite-population Moran process in which fitness affects reproduction while population size remains constant. It also describes selection intensity as a way to tune how strongly payoffs affect fitness.

Moran process

  • In the evolutionary-game Moran process, an individual reproduces according to fitness and a randomly chosen individual dies, keeping population size constant.The offspring inherits the reproducing individual’s strategy.Source: Moran process · p. 4
  • Selection intensity w controls how strongly payoff differences affect fitness: weak selection makes strategy fitnesses nearly equal, while stronger selection gives the game more influence.Fitness is a non-decreasing function of payoff in this setup.Source: Moran process · p. 4

Fixation probability

In finite populations, fixation probabilities track whether a new strategy goes extinct or takes over, and can characterize evolutionary stability. Multiplayer analysis extends two-player fixation criteria but changes the one-third rule’s role.

Fixation probability

  • With very low mutation rates, a single new strategy either goes extinct or reaches fixation, making its fixation probability useful for studying its spread.Source: Fixation probability · p. 4
  • For large finite populations with random matching, selection favors A replacing B when the internal equilibrium x* is below one-third.This one-third rule holds for a wide range of evolutionary processes.Source: Fixation probability · p. 5
  • For two-player games, A has a higher fixation probability than B when Na0 + (N −2)a1 > (N − 2)b0 + Nb1.For large enough populations, this reduces to a0 + a1 > b0 + b1.Source: Fixation probability · p. 5
  • Multiplayer games generalize the one-third rule and risk-dominance condition, but the neutrality condition is no longer explicitly tied to the equilibrium point.The one-third rule is therefore special to the two-player case; under strong selection, finite-population stability becomes equivalent to infinite-population stability.Source: Fixation probability · p. 5

Fixation time

Fixation time measures how long populations take to reach a boundary, either without conditioning or given a particular boundary. Under weak selection, multiplayer games can exhibit stochastic slowdown that intensifies with group size.

Fixation time

  • Conditional fixation time is the expected time to reach a specified boundary given that it is reached, while unconditional fixation time averages time to either boundary.Source: Fixation time · p. 5
  • Under weak selection, fixation-time expressions depend on transition probabilities, which in turn depend on payoff-determined fitnesses.The conditional fixation-time expansion has zeroth-order term N(N−1).Source: Fixation time · p. 5
  • Increasing the number of players can intensify stochastic slowdown, in which a strategy with a frequency-varying fitness advantage can take longer to fix than a neutral mutant.For multiplayer games, the first-order correction to fixation time can be calculated for a given process or in more general terms.Source: Fixation time · pp. 5–6

Mutation selection equilibrium

At intermediate mutation rates, polymorphism makes fixation less useful as a measure of strategy success, so average abundance is used instead. For multiplayer games, abundance results are available in limited forms, and large well-mixed populations with negligible mutation recover risk dominance.

Mutation selection equilibrium

  • At intermediate mutation rates, new mutations can arise while earlier ones remain at intermediate frequency, producing polymorphic populations where fixation is not a useful success measure.Average abundance compares the strategies’ frequencies at mutation-selection equilibrium.Source: Mutation selection equilibrium · p. 6
  • For multiplayer games, average abundance can be written in closed form for three-player games, but no closed form has been obtained for arbitrary player numbers.The underlying results are weak-selection approximations derived using neutral coalescence theory.Source: Mutation selection equilibrium · p. 6
  • For d-player games, capturing process effects in the abundance condition requires d−1 structure parameters rather than a single parameter.In large well-mixed populations with negligible mutation, the condition for higher abundance reduces to risk dominance.Source: Mutation selection equilibrium · p. 6

APPLICATION OF MULTIPLAYER GAMES

The replicator equation is used across biological and social contexts, and results can sometimes transfer between fields.

APPLICATION OF MULTIPLAYER GAMES

Ecology

Multiplayer games address ecological questions about biodiversity and interspecies mutualism by representing nonlinear interactions among multiple individuals. In mutualisms, these interactions can reverse the Red King effect and support richer dynamics.

Ecology

  • Ecological multiplayer-game analysis connects to biological markets and longstanding observations of coalition formation and multiparty animal interactions.Examples cited span cormorants and killer whales.Source: Ecology · p. 6
  • The section distinguishes interspecies interactions from multiplayer games, which involve more than two individuals of the same species interacting.In cleaner-wrasse mutualism, interactions may be treated as two-player when within-species interactions are ignored; relaxing that assumption adds complexity.Source: Ecology · p. 6
  • When one species evolves more slowly in a two-species mutualism, it can remain selfish while the other is forced to contribute generously, an outcome termed the Red King effect.The effect resembles extortioner strategies in repeated two-player games and contrasts with the Red Queen pressure for faster evolution.Source: Ecology · p. 7
  • Nonlinear multiplayer interactions can change equilibrium attraction basins enough to reverse the Red King effect.The analysis extends mutualism from two-player to multi-partner interactions.Source: Ecology · p. 7
  • Multiplayer mutualism can incorporate thresholds, asymmetric player numbers, and asymmetric growth rates, yielding rich dynamics.Threshold effects require a certain number of cooperators to generate a public good.Source: Ecology · p. 7

Population genetics

Multiplayer evolutionary games offer a framework for representing genetic interactions whose outcomes depend on multiple alleles and parental genotypes. The section illustrates the approach with Mendelian inheritance and the non-Mendelian Medea system.

Population genetics

  • Medea inheritance makes wild-type homozygote survival depend on whether the mother carries M, distorting the usual Mendelian inheritance pattern.The survival probability is reduced by t for affected ++ offspring; the system is described as a naturally occurring selfish genetic element.Source: Population genetics · pp. 7–8
  • A four-player, two-strategy game represents an allele interacting with the other three allele copies involved in diploid mating, preserving information about parental genotypes.The focal allele is the row player, while columns encode the other three players’ strategy combinations; the arrangement can accommodate mating and offspring-survival effects.Source: Population genetics · pp. 7–8
  • Under usual Mendelian inheritance, the four-player allele game reduces to frequency-dependent allele fitnesses that can be represented by a two-player payoff matrix.The genotype fitnesses are frequency-independent, while the allele fitnesses are not.Source: Population genetics · p. 8
  • For the Medea system, the allele payoffs are πM = νx + ω(1 − x) and π+ = ωx + (1 − xt)(1 − x).The model accounts for maternal effects in matings involving heterozygotes when calculating these payoffs.Source: Population genetics · p. 8
  • The Medea nonlinearity arises from a four-player game; a two-player representation would require frequency-dependent payoff entries.This is presented as an example of multiplayer games handling non-Mendelian inheritance and genetic nonlinearities.Source: Population genetics · pp. 7–8
  • The framework is proposed as a bridge to complex multilocus population genetics, which can be exceedingly complex and daunting.The stated aim is to retain the simplicity of games while accounting for realistic genetic architectures.Source: Population genetics · pp. 7, 9

Social sciences

Social-science applications use multiplayer games to represent cooperation in groups, including public-goods dilemmas with nonlinear benefits. The section distinguishes linear cases that reduce to two-player matrices from nonlinear cases that do not.

Social sciences

  • Social dilemmas include public-goods and stag-hunt games, while threshold settings can make cooperation beneficial above a required number of participants.The section presents these as contexts where choosing an appropriate interaction model matters.Source: Social sciences · p. 9
  • In a linear public goods game, cooperators pay a cost and their combined contribution is multiplied and shared equally among all group members.With k cooperators among d players, the cooperator payoff is bk/d − c and the defector payoff is bk/d.Source: Social sciences · p. 9
  • For the linear public goods game, average fitnesses can be recovered from a simplified two-player matrix because payoffs vary linearly with the number of players.The supplied cooperator average fitness is fC = (b(x(d − 1) + 1) − cd)/d.Source: Social sciences · p. 9
  • Synergistic or discounted public-goods benefits create nonlinear payoff structures that cannot be simplified to meaningful two-player interpretations.The factor ω produces synergy for ω > 1 and discounting for ω < 1; the ω → 1 limit recovers the linear game.Source: Social sciences · p. 9

DISCUSSION

The discussion identifies open directions for evolutionary game theory, including group-size evolution, novel mutations, and multiplayer games beyond well-mixed normal-form settings.

DISCUSSION

  • Evolution of group size remains an open ecological question that evolutionary games may address proximately.The discussion also identifies mutations to entirely novel types and their persistence under some conditions as topics for further study.Source: DISCUSSION · p. 10
  • The review is limited to well-mixed populations and normal-form games; extensive-form multiplayer analysis is more complicated because players’ temporally distinct actions create multiple possible game trees.The discussion points to new methods and Bernstein polynomials as possible aids for analyzing such scenarios.Source: DISCUSSION · p. 10
  • Some finite-population results have also been extended to multiplayer games in structured populations.The passage does not provide further details of those extensions.Source: DISCUSSION · p. 10
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