title: Structure coefficients and strategy selection in multiplayer games.
abstract: Evolutionary processes based on two-player games such as the Prisoner's Dilemma or Snowdrift Game are abundant in evolutionary game theory. These processes, including those based on games with more than two strategies, have been studied extensively under the assumption that selection is weak. However, games involving more than two players have not received the same level of attention. To address this issue, and to relate two-player games to multiplayer games, we introduce a notion of reducibility for multiplayer games that captures what it means to break down a multiplayer game into a sequence of interactions with fewer players. We discuss the role of reducibility in structured populations, and we give examples of games that are irreducible in any population structure. Since the known conditions for strategy selection, otherwise known as [Formula: see text]-rules, have been established only for two-player games with multiple strategies and for multiplayer games with two strategies, we extend these rules to multiplayer games with many strategies to account for irreducible games that cannot be reduced to those simpler types of games. In particular, we show that the number of structure coefficients required for a symmetric game with [Formula: see text]-player interactions and [Formula: see text] strategies grows in [Formula: see text] like [Formula: see text]. Our results also cover a type of ecologically asymmetric game based on payoff values that are derived not only from the strategies of the players, but also from their spatial positions within the population.
authors.affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada. alexmcavoy@gmail.com.
authors.forename: Alex
authors.initials: A
authors.lastname: McAvoy
authors.affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada.
authors.forename: Christoph
authors.initials: C
authors.lastname: Hauert
mesh_terms: D005075:Biological Evolution; D005716:Game Theory; D056084:Genetic Fitness; D006801:Humans; D016014:Linear Models; D008390:Markov Chains; D055641:Mathematical Concepts; D008954:Models, Biological; D011153:Population; D011336:Probability; D013269:Stochastic Processes
keywords: Evolutionary dynamics; Finite populations; Game theory; Stochastic processes; Weak selection
references.citation: Nature. 2006 May 25;441(7092):502-5
references.pmid: 16724065
references.citation: Philos Trans R Soc Lond B Biol Sci. 2010 Jan 12;365(1537):19-30
references.pmid: 20008382
references.citation: J Theor Biol. 2011 Aug 21;283(1):180-91
references.pmid: 21641915
references.citation: Nature. 2007 May 24;447(7143):469-72
references.pmid: 17522682
references.citation: Bull Math Biol. 2008 Jul;70(5):1410-24
references.pmid: 18386099
references.citation: Proc Natl Acad Sci U S A. 2009 May 26;106(21):8601-4
references.pmid: 19433793
references.citation: Evolution. 2010 Dec;64(12):3344-53
references.pmid: 20812974
references.citation: J Theor Biol. 2012 Jan 7;292:116-28
references.pmid: 21907215
references.citation: Theor Popul Biol. 2013 Nov;89:75-87
references.pmid: 23999503
references.citation: J R Soc Interface. 2014 Mar 05;11(94):20140077
references.pmid: 24598208
references.citation: J Theor Biol. 2007 Jun 7;246(3):522-9
references.pmid: 17292423
references.citation: J Theor Biol. 2012 Apr 21;299:106-12
references.pmid: 21704639
references.citation: Proc Biol Sci. 2009 Apr 7;276(1660):1379-84
references.pmid: 19141423
references.citation: Sci Rep. 2014 Jul 02;4:5536
references.pmid: 24985887
references.citation: J Theor Biol. 2014 Apr 7;346:23-33
references.pmid: 24380778
references.citation: Bull Math Biol. 2004 Nov;66(6):1621-44
references.pmid: 15522348
references.citation: J Theor Biol. 2009 Aug 7;259(3):570-81
references.pmid: 19358858
references.citation: J Theor Biol. 2012 Apr 21;299:9-20
references.pmid: 21723299
references.citation: J Theor Biol. 2006 Mar 21;239(2):195-202
references.pmid: 16242728
references.citation: Bull Math Biol. 1997 Sep;59(5):931-52
references.pmid: 9312326
references.citation: Proc Natl Acad Sci U S A. 2012 Jun 26;109(26):10409-13
references.pmid: 22615375
references.citation: PLoS Comput Biol. 2014 Apr 24;10(4):e1003567
references.pmid: 24762474
references.citation: Nature. 2005 Jan 20;433(7023):312-6
references.pmid: 15662424
references.citation: Nature. 2004 Apr 8;428(6983):646-50
references.pmid: 15071593
references.citation: J Evol Biol. 2007 Nov;20(6):2284-95
references.pmid: 17956391
references.citation: Proc Natl Acad Sci U S A. 2011 Feb 8;108(6):2334-7
references.pmid: 21257906
references.citation: Theor Popul Biol. 2004 Sep;66(2):83-91
references.pmid: 15302218
references.citation: PLoS Comput Biol. 2013;9(12):e1003381
references.pmid: 24339769
references.citation: Philos Trans R Soc Lond B Biol Sci. 2014 Mar 31;369(1642):20130359
references.pmid: 24686931