Abstract
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 are 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.
Similar content being viewed by others
References
Chakra MA, Traulsen A (2012) Evolutionary dynamics of strategic behavior in a collective-risk dilemma. PLoS Comput Biol 8(e1002):652
Albin PS, Foley DK (2001) The co-evolution of cooperation and complexity in a multi-player, local-interaction prisoners’ dilemma. Complexity 6(3):54–63
Altenberg L (2010) Proof of the Feldman–Karlin conjecture on the maximum number of equilibria in an evolutionary system. Theor Popul Biol 77(4):263–269
Altrock PM, Traulsen A (2009) Fixation times in evolutionary games under weak selection. New J Phys 11:013012
Altrock PM, Gokhale CS, Traulsen A (2010) Stochastic slowdown in evolutionary processes. Phys Rev E 82:011925
Altrock PM, Traulsen A, Galla T (2012) The mechanics of stochastic slowdown in evolutionary games. J Theor Biol 311:94–106
Antal T, Scheuring I (2006) Fixation of strategies for an evolutionary game in finite populations. Bull Math Biol 68:1923–1944
Antal T, Nowak MA, Traulsen A (2009a) Strategy abundance in \(2\times 2\) games for arbitrary mutation rates. J Theor Biol 257:340–344
Antal T, Traulsen A, Ohtsuki H, Tarnita CE, Nowak MA (2009b) Mutation-selection equilibrium in games with multiple strategies. J Theor Biol 258:614–622
Archetti M (2000) The origin of autumn colours by coevolution. J Theor Biol 205:625–630
Archetti M, Scheuring I (2012) Review: evolution of cooperation in one-shot social dilemmas without assortment. J Theor Biol 299:9–20
Archetti M, Scheuring I, Hoffman M, Frederickson ME, Pierce NE, Yu DW (2011) Economic game theory for mutualism and cooperation. Ecol Lett 14(12):1300–1312
Aumann RJ (2006) War and peace. Proc Natl Acad Sci USA 103(46):17,075–17,078
Aviles L (1999) Cooperation and non-linear dynamics: an ecological perspective on the evolution of sociality. Evol Ecol Res 1:459–477
Axelrod R (1984) The evolution of cooperation. Basic Books, New York
Axelrod R, Axelrod DE, Pienta KJ (2006) Evolution of cooperation among tumor cells. Proc Natl Acad Sci USA 103(36):13,474–13,479
Bach LA, Bentzen SM, Alsner J, Christiansen FB (2001) An evolutionary-game model of tumour-cell interactions: possible relevance to gene therapy. Eur J Cancer 37:2116–2120
Bach LA, Helvik T, Christiansen FB (2006) The evolution of \(n\)-player cooperation - threshold games and ESS bifurcations. J Theor Biol 238:426–434
Bartholomew GA Jr (1942) The fishing activities of double-crested cormorants on San Francisco Bay. The Condor 44(1):13–21
Barton NH, Turelli M (1991) Natural and sexual selection on many loci. Genetics 127(1):229–255
Basanta D, Deutsch A (2008) A game theoretical perspective on the somatic evolution of cancer. In: Bellomo N, Angelis E (eds) Selected topics in cancer modeling. Springer, Heidelberg, pp 97–112
Beeman RW, Friesen KS, Denell RE (1992) Maternal-effect selfish genes in flour beetles. Science 256:89–92
Bergstrom CT, Lachmann M (2003) The Red King effect: when the slowest runner wins the coevolutionary race. Proc Natl Acad Sci USA 100:593–598
Bishop DT, Cannings C (1976) Models of animal conflict. Adv Appl Probab 8(4):616–621
Broom M, Rychtář J (2008) An analysis of the fixation probability of a mutant on special classes of non-directed graphs. Proc R Soc A 464:2609–2627
Broom M, Rychtář J (2013) Game-theoretical models in biology. Chapman and Hall/CRC, London
Broom M, Cannings C, Vickers GT (1993) On the number of local maxima of a constrained quadratic form. Proc R Soc A: Math Phys Eng Sci 443(1919):573–584
Broom M, Cannings C, Vickers GT (1994) Sequential methods for generating patterns of ess’s. J Math Biol 32:597–615
Broom M, Cannings C, Vickers G (1997) Multi-player matrix games. Bull Math Biol 59(5):931–952
Broom M, Rychtář J, Stadler B (2009) Evolutionary dynamics on small-order graphs. J Interdiscip Math 12:129–140
Bukowski M, Miekisz J (2004) Evolutionary and asymptotic stability in symmetric multi-player games. Int J Game Theory 33(1):41–54
Cannings C, Vickers GT (1988) Patterns of ESS’s. II. J Theor Biol 132(4):409–420
Cannings C, Tyrer JP, Vickers GT (1993) Routes to polymorphism. J Theor Biol 165(2):213–223
Christiansen FB (1988) The effect of population subdivision on multiple loci without selection. Mathematical evolutionary theory. Princeton University Press, Princeton, pp 71–85
Cressman R (1988) Frequency-dependent viability selection (a single-locus, multi-phenotype model). J Theor Biol 130:147–165
Cressman R (2003) Evolutionary dynamics and extensive form games. MIT Press, Cambridge
Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New York
Damore JA, Gore J (2011) A slowly evolving host moves first in symbiotic interactions. Evolution 65(8):2391–2398
Darwin C (1859) On the origin of species by means of natural selection. Reprinted in Harvard University Press (1964), Cambridge-London
Dawkins R (1982) The extended phenotype. Oxford University Press, Oxford
Düsing, C (1884) Jenaische Zeitschrift für Naturwissenschaft, Gustav Fischer, Die Regulierung des Geschlechtsverhältnisses bei der Vermehrung der Menschen, Tiere und Pflanzen 17:593–940.
Edwards AW (2000) Carl Düsing (1884) on the regulation of the sex-ratio. Theor Pop Biol 58:255–257
Eshel I, Motro U (1988) The three brothers’ problem: kin selection with more than one potential helper. 1. The case of immediate help. Am Nat 132:550–566
Ewens WJ (1979) Mathematical population genetics. Springer, Berlin
Feldman MW, Franklin I, Thomson GJ (1974) Selection in complex genetic systems. I. The symmetric equilibria of the three-locus symmetric viability model. Genetics 76(1):135–162
Ficici S, Pollack J (2000) Effects of finite populations on evolutionary stable strategies. In: Whitley D, Goldberg D, Cantu-Paz E, Spector L, Parmee I, Beyer HG (eds) Proceedings GECCO. Morgan-Kaufmann, San Francisco, pp 927–934
Fisher RA (1930) The genetical theory of natural selection. Clarendon Press, Oxford
Fisher RA (1958) Polymorphism and natural selection. J Ecol 46(2):289–293
Fogel G, Andrews P, Fogel D (1998) On the instability of evolutionary stable strategies in small populations. Ecol Model 109:283–294
Foster D, Young P (1990) Stochastic evolutionary game dynamics. Theor Popul Biol 38:219–232
Frey E (2010) Evolutionary game theory: theoretical concepts and applications to microbial communities. Phys A 389(20):4265–4298
Fudenberg D, Imhof LA (2006) Imitation processes with small mutations. J Eco Theory 131:251–262
Galla T, Farmer JD (2013) Complex dynamics in learning complicated games. Proc Natl Acad Sci USA 110(4):1232–1236
Ganzfried S, Sandholm T (2009) Computing equilibria in multiplayer stochastic games of imperfect information. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI).
Goel N, Richter-Dyn N (1974) Stochastic models in biology. Academic Press, New York
Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Natl Acad Sci USA 107:5500–5504
Gokhale CS, Traulsen A (2011) Strategy abundance in evolutionary many-player games with multiple strategies. J Theor Biol 238:180–191
Gokhale CS, Traulsen A (2012) Mutualism and evolutionary multiplayer games: revisiting the Red King. Proc R Soc B 279:4611–4616
Haigh J, Cannings C (1989) The n-person war of attrition. Acta Appl Math 14:59–74
Hamilton W (1967) Extraordinary sex ratios. Science 156:477–488
Han TA, Traulsen A, Gokhale CS (2012) On equilibrium properties of evolutionary multi-player games with random payoff matrices. Theor Popul Biol 81:264–272
Hardin G (1968) The tragedy of the commons. Science 162:1243–1248
Hashimoto K, Aihara K (2009) Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations. J Theor Biol 258:637–645
Hauert C, Michor F, Nowak MA, Doebeli M (2006) Synergy and discounting of cooperation in social dilemmas. J Theor Biol 239:195–202
Hauert C, Traulsen A, Brandt H, Nowak MA, Sigmund K (2007) Via freedom to coercion: the emergence of costly punishment. Science 316:1905–1907
Hilbe C, Nowak MA, Sigmund K (2013) The evolution of extortion in iterated prisoner’s dilemma games. Proc Natl Acad Sci USA 110:6913–6918
Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge
Huang W, Haubold B, Hauert C, Traulsen A (2012) Emergence of stable polymorphism driven by evolutionary games between mutants. Nat Commun 3:919
Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a Wright-Fisher process. J Math Biol 52:667–681
Imhof LA, Nowak MA (2010) Stochastic evolutionary dynamics of direct reciprocity. Proc R Soc B 277:463–468
Imhof LA, Fudenberg D, Nowak MA (2005) Evolutionary cycles of cooperation and defection. Proc Natl Acad Sci USA 102:10,797–10,800
Kamiński D, Miekisz J, Zaborowski M (2005) Stochastic stability in three-player games. Bull Math Biol 67(6):1195–1205
Kandori M, Mailath GJ, Rob R (1993) Learning, mutation, and long run equilibria in games. Econometrica 61:29–56
Karlin S (1980) The number of stable equilibria for the classical one-locus multiallele selection model. J Math Biol 9(2):189–192
Karlin S, Feldman MW (1969) Linkage and selection: new equilibrium properties of the two-locus symmetric viability model. Proc Natl Acad Sci USA 62(1):70–74
Karlin S, Feldman MW (1970) Linkage and selection: two locus symmetric viability model. Theor Popul Biol 1(1):39–71
Karlin S, Liberman U (1979) Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes. Genetics 91(4):777–798
Karlin S, McGregor J (1972) Polymorphisms for genetic and ecological systems with weak coupling. Theor Popul Biol 3:210–238
Karlin S, Taylor HMA (1975) A first course in stochastic processes, 2nd edn. Academic, London
Kim Y (1996) Equilibrium selection in \(n\)-person coordination games. Games Econ Behav 15:203–227
Kimura M (1983) The neutral theory of molecular evolution. Cambridge University Press, Cambridge
Kingman JFC (1982) On the genealogy of large populations. J Appl Probab 19A:27–43
Kurokawa S, Ihara Y (2009) Emergence of cooperation in public goods games. Proc R Soc B 276:1379–1384
Kurokawa SS, Ihara Y (2013) Evolution of social behavior in finite populations: a payoff transformation in general n-player games and its implications. Theor Popul Biol 84:1–8
Lessard S (2011) On the robustness of the extension of the one-third law of evolution to the multi-player game. Dyn Games Appl 1:408–418
Lessard S, Ladret V (2007) The probability of fixation of a single mutant in an exchangeable selection model. J Math Biol 54:721–744
Lewontin RC (1961) Evolution and the theory of games. J Theor Biol 1:382–403
Lotka AJ (1932) The growth of mixed populations: two species competing for a common food supply. J Washington Acad Sci 22:461–469
May RM (1972) Will a large complex system be stable? Nature 238:413–414
Smith JM (1982) Evolution and the theory of games. Cambridge University Press, Cambridge
Smith JM, Price GR (1973) The logic of animal conflict. Nature 246:15–18
Miekisz J (2004a) Stochastic stability in spatial games. J Stat Phys 117(1/2):99–110
Miekisz J (2004b) Stochastic stability in spatial three-player games. Phys A 343:175–184
Miekisz J (2008) Evolutionary game theory and population dynamics. Lect Notes Math 1940:269–316
Moran PAP (1962) The statistical processes of evolutionary theory. Clarendon Press, Oxford
Nash J (1951) Non-cooperative games. Ann Math 54:286–299
Nash JF (1950) Equilibrium points in n-person games. Proc Natl Acad Sci USA 36:48–49
von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton
Noë R (2001) Biological markets: partner choice as the driving force behind the evolution of mutualisms. In: Noë R, van Hooff JA, Hammerstein P (eds) Economics in nature: social dilemmas, mate choice and biological markets. Cambridge University Press, Cambridge
Noë R, Hammerstein P (1995) Biological markets. Trends Ecol Evol 10(8):336–339
Noë R, van Schaik CP, van Hooff JARAM (1991) The market effect: an explanation for pay-off asymmetries among collaborating animals. Ethology 87:97–118
Nowak MA (2006) Evolutionary dynamics. Harvard University Press, Cambridge MA
Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826–829
Nowak MA, Sigmund K (2004) Evolutionary dynamics of biological games. Science 303:793–799
Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650
Nowak MA, Tarnita CE, Antal T (2010) Evolutionary dynamics in structured populations. Philos Trans R Soc B 365:19–30
Ohtsuki H, Nowak MA (2006) The replicator equation on graphs. J Theor Biol 243:86–97
Ohtsuki H, Nowak MA (2008) Evolutionary stability on graphs. J Theor Biol 251:698–707
Ohtsuki H, Bordalo P, Nowak MA (2007) The one-third law of evolutionary dynamics. J Theor Biol 249:289–295
Osborne MJ (1996) Darwin, Fisher, and a theory of the evolution of the sex ratio. URL http://www.economics.utoronto.ca/osborne/research/sexratio
Pacheco JM, Santos FC, Souza MO, Skyrms B (2009) Evolutionary dynamics of collective action in n-person stag hunt dilemmas. Proc R Soc B 276:315–321
Page KM, Nowak MA (2002) Unifying evolutionary dynamics. J Theor Biol 219:93–98
Palm G (1984) Evolutionary stable strategies and game dynamics for n-person games. J Math Biol 19(3):329–334
Peña J (2012) Group size diversity in public goods games. Evolution 66:623–636
Peña J, Lehmann L, Nöldeke G (2014) Gains from switching and evolutionary stability in multi-player games. J Theor Biol 346:23–33
Peters LL, Barker JE (1993) Novel inheritance of the murine severe combined anemia and thrombocytopenia (scat) phenotype. Cell 74:135–142
Poulin R, Vickery WL (1995) Cleaning symbiosis as an evolutionary game: to cheat or not to cheat? J Theor Biol 175:63–70
Press WH, Dyson FD (2012) Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent. Proc Natl Acad Sci USA 109:10,409–10,413
Riechert SE, Hammerstein P (1983) Game theory in the ecological context. Annu Rev Ecol Syst 14:377–409
Robson AJ, Vega-Redondo F (1996) Efficient equilibrium selection in evolutionary games with random matching. J Econ Theory 70:65–92
Rosas A (2010) Evolutionary game theory meets social science: is there a unifying rule for human cooperation? J Theor Biol 264(2):450–456
Rowe GW (1987) A dynamic game theory model of diploid genetic system. J Theor Biol 129:243–255
Rowe GW (1988) To each genotype a separate strategy—a dynamic game theory model of a general diploid system. J Theor Biol 134:89–101
Samuelson PA (1985) Modes of thought in economics and biology. Am Econ Rev 75(2):166–172
Sandholm WH (2010) Population games and evolutionary dynamics. MIT Press, Cambridge
Santos FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc Natl Acad Sci USA 103:3490–3494
Schuster P, Sigmund K (1983) Replicator dynamics. J Theor Biol 100:533–538
Schuster P, Sigmund K, Hofbauer J, Gottlieb R, Merz P (1981a) Selfregulation in behaviour in animal societies iii. games between two populations with selfinteractions. Biol Cybern 40:17–25
Schuster P, Sigmund K, Hofbauer J, Wolff R (1981b) Selfregulation in behaviour in animal societies ii. games between two populations without selfinteractions. Biol Cybern 40:9–15
Schwalbe U, Walker P (2001) Zermelo and the early history of game theory. Games Econ Behav 34(1):123–137
Shirakihara K, Tanaka S (1978) Two fish species competition model with nonlinear interactions and equilibrium catches. Res Popul Ecol 20(1):123–140
Sigmund K (1987a) Game dynamics, mixed strategies, and gradient systems. Theor Popul Biol 32(1):114–126
Sigmund K (1987b) A maximum principle for frequency dependent selection. Math Biosci 84:189–195
Skyrms B (2003) The Stag-Hunt game and the evolution of social structure. Cambridge University Press, Cambridge
Smith TG, Siniff DB, Reichle R, Stone S (1981) Coordinated behavior of killer whales, Orcinus orca, hunting a crabeater seal, Lobodon carcinophagus. Can J Zool 59(6):1185–1189
Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under n-person snowdrift games. J Theor Biol 260:581–588
Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216
Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA (2009) Strategy selection in structured populations. J Theor Biol 259:570–581
Tarnita CE, Wage N, Nowak MA (2011) Multiple strategies in structured populations. Proc Natl Acad Sci USA 108:2334–2337
Taylor C, Fudenberg D, Sasaki A, Nowak MA (2004) Evolutionary game dynamics in finite populations. Bull Math Biol 66:1621–1644
Taylor PD, Jonker L (1978) Evolutionarily stable strategies and game dynamics. Math Biosci 40:145–156
Thomas B, Pohley HJ (1981) ESS-theory for finite populations. Biosystems 13(3):211–221
Traulsen A, Hauert C (2009) Stochastic evolutionary game dynamics. In: Schuster HG (ed) Reviews of nonlinear dynamics and complexity. Wiley-VCH, Weinheim, pp 25–61
Traulsen A, Reed FA (2012) From genes to games: cooperation and cyclic dominance in meiotic drive. J Theor Biol 299:120–125
Traulsen A, Claussen JC, Hauert C (2005) Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95:238701
Traulsen A, Pacheco JM, Imhof LA (2006) Stochasticity and evolutionary stability. Phys Rev E 74(021):905
Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246:522–529
Traulsen A, Claussen JC, Hauert C (2012) Stochastic differential equations for evolutionary dynamics with demographic noise and mutations. Phys Rev E 85:041901
Turner PE, Chao L (1999) Prisoner’s dilemma in an RNA virus. Nature 398:441–443
Turner PE, Chao L (2003) Escape from prisoner’s dilemma in RNA phage Phi6. Am Nat 161(3):497–505
Van Damme E (1994) Evolutionary game theory. Eur Econ Rev 38:847–858
Van Segbroeck S, Santos FC, Lenaerts T, Pacheco JM (2009) Reacting differently to adverse ties promotes cooperation in social networks. Phys Rev Lett 102:058105
van Veelen M, Nowak MA (2012) Multi-player games on the cycle. J Theor Biol 292:116–128
van Veelen M, García J, Rand DG, Nowak MA (2012) Direct reciprocity in structured populations. Proc Natl Acad Sci USA 109:9929–9934
Vickers GT, Cannings C (1988) Patterns of ESS’s I. J Theor Biol 132(4):387–408
Vickery WL, Poulin R (2010) The evolution of host manipulation by parasites: a game theory analysis. Evol Ecol 24(4):773–788
Volterra V (1928) Variations and fluctuations of the number of individuals in animal species living together. J Cons Int Explor Mer 3(1):3–51
Wade MJ, Beeman RW (1994) The population dynamics of maternal-effect selfish genes. Genetics 138:1309–1314
Wakeley J (2008) Coalescent theory: an introduction. Roberts and Company Publishers, Greenwood Village
Weichenhan D, Traut W, Kunze B, Winking H (1996) Distortion of mendelian recovery ratio for a mouse hsr is caused by maternal and zygotic effects. Genet Res 68:125–129
Woelfing B, Traulsen A (2009) Stochastic sampling of interaction partners versus deterministic payoff assignment. J Theor Biol 257:689–695
Wooders M, Cartwright E, Selten R (2006) Behavioral conformity in games with many players. Games Econ Behav 57(2):347–360
Wu B, Altrock PM, Wang L, Traulsen A (2010) Universality of weak selection. Phys Rev E 82:046106
Wu B, Gokhale CS, Wang L, Traulsen A (2012) How small are small mutation rates? J Math Biol 64:803–827
Wu B, García J, Hauert C, Traulsen A (2013a) Extrapolating weak selection in evolutionary games. PLoS Comput Biol 9:e1003381
Wu B, Traulsen A, Gokhale CS (2013b) Dynamic properties of evolutionary multi-player games in finite populations. Games 4(2):182–199
Zeeman EC (1980) Population dynamics from game theory. Lect Notes Math 819:471–497
Zermelo E (1913) Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. In: Proceedings of the Fifth International Congress of Mathematicians.
Zhou Qian H (2011) Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics. Phys Rev E 84:031907
Acknowledgments
We appreciate generous funding from the Max Planck Society.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gokhale, C.S., Traulsen, A. Evolutionary Multiplayer Games. Dyn Games Appl 4, 468–488 (2014). https://doi.org/10.1007/s13235-014-0106-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13235-014-0106-2