Evolutionary Multiplayer Games

Dynamic Games and Applications

Volumn 4, Issue 4, 2014, Pages 468-488

  Gokhale, Chaitanya S ^a,b   Traulsen, Arne ^a  

^a MAX PLANCK INSTITUTE FOR EVOLUTIONARY BIOLOGY   (Germany)

^b MASSEY UNIVERSITY   (New Zealand)

Author keywords

Homogeneous populations; Non linear interactions; Stochastic effects

Indexed keywords

EID: 84911386246     PISSN: 21530785     EISSN: 21530793     Source Type: Journal    
DOI: 10.1007/s13235-014-0106-2     Document Type: Article

References (169)

1. Chakra MA, Traulsen A (2012) Evolutionary dynamics of strategic behavior in a collective-risk dilemma. PLoS Comput Biol 8(e1002):652
2. 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
3. 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
4. Altrock PM, Traulsen A (2009) Fixation times in evolutionary games under weak selection. New J Phys 11:013012
5. Altrock PM, Gokhale CS, Traulsen A (2010) Stochastic slowdown in evolutionary processes. Phys Rev E 82:011925
6. Altrock PM, Traulsen A, Galla T (2012) The mechanics of stochastic slowdown in evolutionary games. J Theor Biol 311:94–106
7. Antal T, Scheuring I (2006) Fixation of strategies for an evolutionary game in finite populations. Bull Math Biol 68:1923–1944
8. Antal T, Nowak MA, Traulsen A (2009a) Strategy abundance in 2×2 games for arbitrary mutation rates. J Theor Biol 257:340–344
9. 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
10. Archetti M (2000) The origin of autumn colours by coevolution. J Theor Biol 205:625–630
11. Archetti M, Scheuring I (2012) Review: evolution of cooperation in one-shot social dilemmas without assortment. J Theor Biol 299:9–20
12. 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
13. Aumann RJ (2006) War and peace. Proc Natl Acad Sci USA 103(46):17,075–17,078
14. Aviles L (1999) Cooperation and non-linear dynamics: an ecological perspective on the evolution of sociality. Evol Ecol Res 1:459–477
15. Axelrod R (1984) The evolution of cooperation. Basic Books, New York
16. Axelrod R, Axelrod DE, Pienta KJ (2006) Evolution of cooperation among tumor cells. Proc Natl Acad Sci USA 103(36):13,474–13,479
17. 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
18. Bach LA, Helvik T, Christiansen FB (2006) The evolution of $$n$$n-player cooperation - threshold games and ESS bifurcations. J Theor Biol 238:426–434
19. Bartholomew GA Jr (1942) The fishing activities of double-crested cormorants on San Francisco Bay. The Condor 44(1):13–21
20. Barton NH, Turelli M (1991) Natural and sexual selection on many loci. Genetics 127(1):229–255
21. 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
22. Beeman RW, Friesen KS, Denell RE (1992) Maternal-effect selfish genes in flour beetles. Science 256:89–92
23. 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
24. Bishop DT, Cannings C (1976) Models of animal conflict. Adv Appl Probab 8(4):616–621
25. 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
26. Broom M, Rychtář J (2013) Game-theoretical models in biology. Chapman and Hall/CRC, London
27. 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
28. Broom M, Cannings C, Vickers GT (1994) Sequential methods for generating patterns of ess’s. J Math Biol 32:597–615
29. Broom M, Cannings C, Vickers G (1997) Multi-player matrix games. Bull Math Biol 59(5):931–952
30. Broom M, Rychtář J, Stadler B (2009) Evolutionary dynamics on small-order graphs. J Interdiscip Math 12:129–140
31. Bukowski M, Miekisz J (2004) Evolutionary and asymptotic stability in symmetric multi-player games. Int J Game Theory 33(1):41–54
32. Cannings C, Vickers GT (1988) Patterns of ESS’s. II. J Theor Biol 132(4):409–420
33. Cannings C, Tyrer JP, Vickers GT (1993) Routes to polymorphism. J Theor Biol 165(2):213–223
34. Christiansen FB (1988) The effect of population subdivision on multiple loci without selection. Mathematical evolutionary theory. Princeton University Press, Princeton, pp 71–85
35. Cressman R (1988) Frequency-dependent viability selection (a single-locus, multi-phenotype model). J Theor Biol 130:147–165
36. Cressman R (2003) Evolutionary dynamics and extensive form games. MIT Press, Cambridge
37. Crow JF, Kimura M (1970) An introduction to population genetics theory. Harper and Row, New York
38. Damore JA, Gore J (2011) A slowly evolving host moves first in symbiotic interactions. Evolution 65(8):2391–2398
39. Darwin C (1859) On the origin of species by means of natural selection. Reprinted in Harvard University Press (1964), Cambridge-London
40. Dawkins R (1982) The extended phenotype. Oxford University Press, Oxford
41. Düsing, C (1884) Jenaische Zeitschrift für Naturwissenschaft, Gustav Fischer, Die Regulierung des Geschlechtsverhältnisses bei der Vermehrung der Menschen, Tiere und Pflanzen 17:593–940.
42. Edwards AW (2000) Carl Düsing (1884) on the regulation of the sex-ratio. Theor Pop Biol 58:255–257
43. 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
44. Ewens WJ (1979) Mathematical population genetics. Springer, Berlin
45. 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
46. 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
47. Fisher RA (1930) The genetical theory of natural selection. Clarendon Press, Oxford
48. Fisher RA (1958) Polymorphism and natural selection. J Ecol 46(2):289–293
49. Fogel G, Andrews P, Fogel D (1998) On the instability of evolutionary stable strategies in small populations. Ecol Model 109:283–294
50. Foster D, Young P (1990) Stochastic evolutionary game dynamics. Theor Popul Biol 38:219–232
51. Frey E (2010) Evolutionary game theory: theoretical concepts and applications to microbial communities. Phys A 389(20):4265–4298
52. Fudenberg D, Imhof LA (2006) Imitation processes with small mutations. J Eco Theory 131:251–262
53. Galla T, Farmer JD (2013) Complex dynamics in learning complicated games. Proc Natl Acad Sci USA 110(4):1232–1236
54. 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).
55. Goel N, Richter-Dyn N (1974) Stochastic models in biology. Academic Press, New York
56. Gokhale CS, Traulsen A (2010) Evolutionary games in the multiverse. Proc Natl Acad Sci USA 107:5500–5504
57. Gokhale CS, Traulsen A (2011) Strategy abundance in evolutionary many-player games with multiple strategies. J Theor Biol 238:180–191
58. Gokhale CS, Traulsen A (2012) Mutualism and evolutionary multiplayer games: revisiting the Red King. Proc R Soc B 279:4611–4616
59. Haigh J, Cannings C (1989) The n-person war of attrition. Acta Appl Math 14:59–74
60. Hamilton W (1967) Extraordinary sex ratios. Science 156:477–488
61. 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
62. Hardin G (1968) The tragedy of the commons. Science 162:1243–1248
63. 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
64. Hauert C, Michor F, Nowak MA, Doebeli M (2006) Synergy and discounting of cooperation in social dilemmas. J Theor Biol 239:195–202
65. Hauert C, Traulsen A, Brandt H, Nowak MA, Sigmund K (2007) Via freedom to coercion: the emergence of costly punishment. Science 316:1905–1907
66. 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
67. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge
68. Huang W, Haubold B, Hauert C, Traulsen A (2012) Emergence of stable polymorphism driven by evolutionary games between mutants. Nat Commun 3:919
69. Imhof LA, Nowak MA (2006) Evolutionary game dynamics in a Wright-Fisher process. J Math Biol 52:667–681
70. Imhof LA, Nowak MA (2010) Stochastic evolutionary dynamics of direct reciprocity. Proc R Soc B 277:463–468
71. Imhof LA, Fudenberg D, Nowak MA (2005) Evolutionary cycles of cooperation and defection. Proc Natl Acad Sci USA 102:10,797–10,800
72. Kamiński D, Miekisz J, Zaborowski M (2005) Stochastic stability in three-player games. Bull Math Biol 67(6):1195–1205
73. Kandori M, Mailath GJ, Rob R (1993) Learning, mutation, and long run equilibria in games. Econometrica 61:29–56
74. Karlin S (1980) The number of stable equilibria for the classical one-locus multiallele selection model. J Math Biol 9(2):189–192
75. 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
76. Karlin S, Feldman MW (1970) Linkage and selection: two locus symmetric viability model. Theor Popul Biol 1(1):39–71
77. Karlin S, Liberman U (1979) Central equilibria in multilocus systems. I. Generalized nonepistatic selection regimes. Genetics 91(4):777–798
78. Karlin S, McGregor J (1972) Polymorphisms for genetic and ecological systems with weak coupling. Theor Popul Biol 3:210–238
79. Karlin S, Taylor HMA (1975) A first course in stochastic processes, 2nd edn. Academic, London
80. Kim Y (1996) Equilibrium selection in $$n$$n-person coordination games. Games Econ Behav 15:203–227
81. Kimura M (1983) The neutral theory of molecular evolution. Cambridge University Press, Cambridge
82. Kingman JFC (1982) On the genealogy of large populations. J Appl Probab 19A:27–43
83. Kurokawa S, Ihara Y (2009) Emergence of cooperation in public goods games. Proc R Soc B 276:1379–1384
84. 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
85. 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
86. Lessard S, Ladret V (2007) The probability of fixation of a single mutant in an exchangeable selection model. J Math Biol 54:721–744
87. Lewontin RC (1961) Evolution and the theory of games. J Theor Biol 1:382–403
88. Lotka AJ (1932) The growth of mixed populations: two species competing for a common food supply. J Washington Acad Sci 22:461–469
89. May RM (1972) Will a large complex system be stable? Nature 238:413–414
90. Smith JM (1982) Evolution and the theory of games. Cambridge University Press, Cambridge
91. Smith JM, Price GR (1973) The logic of animal conflict. Nature 246:15–18
92. Miekisz J (2004a) Stochastic stability in spatial games. J Stat Phys 117(1/2):99–110
93. Miekisz J (2004b) Stochastic stability in spatial three-player games. Phys A 343:175–184
94. Miekisz J (2008) Evolutionary game theory and population dynamics. Lect Notes Math 1940:269–316
95. Moran PAP (1962) The statistical processes of evolutionary theory. Clarendon Press, Oxford
96. Nash J (1951) Non-cooperative games. Ann Math 54:286–299
97. Nash JF (1950) Equilibrium points in n-person games. Proc Natl Acad Sci USA 36:48–49
98. von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton
99. 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
100. Noë R, Hammerstein P (1995) Biological markets. Trends Ecol Evol 10(8):336–339
101. 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
102. Nowak MA (2006) Evolutionary dynamics. Harvard University Press, Cambridge MA
103. Nowak MA, May RM (1992) Evolutionary games and spatial chaos. Nature 359:826–829
104. Nowak MA, Sigmund K (2004) Evolutionary dynamics of biological games. Science 303:793–799
105. Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650
106. Nowak MA, Tarnita CE, Antal T (2010) Evolutionary dynamics in structured populations. Philos Trans R Soc B 365:19–30
107. Ohtsuki H, Nowak MA (2006) The replicator equation on graphs. J Theor Biol 243:86–97
108. Ohtsuki H, Nowak MA (2008) Evolutionary stability on graphs. J Theor Biol 251:698–707
109. Ohtsuki H, Bordalo P, Nowak MA (2007) The one-third law of evolutionary dynamics. J Theor Biol 249:289–295
110. Osborne MJ (1996) Darwin, Fisher, and a theory of the evolution of the sex ratio. URL http://www.economics.utoronto.ca/osborne/research/sexratio
111. 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
112. Page KM, Nowak MA (2002) Unifying evolutionary dynamics. J Theor Biol 219:93–98
113. Palm G (1984) Evolutionary stable strategies and game dynamics for n-person games. J Math Biol 19(3):329–334
114. Peña J (2012) Group size diversity in public goods games. Evolution 66:623–636
115. Peña J, Lehmann L, Nöldeke G (2014) Gains from switching and evolutionary stability in multi-player games. J Theor Biol 346:23–33
116. Peters LL, Barker JE (1993) Novel inheritance of the murine severe combined anemia and thrombocytopenia (scat) phenotype. Cell 74:135–142
117. Poulin R, Vickery WL (1995) Cleaning symbiosis as an evolutionary game: to cheat or not to cheat? J Theor Biol 175:63–70
118. 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
119. Riechert SE, Hammerstein P (1983) Game theory in the ecological context. Annu Rev Ecol Syst 14:377–409
120. Robson AJ, Vega-Redondo F (1996) Efficient equilibrium selection in evolutionary games with random matching. J Econ Theory 70:65–92
121. Rosas A (2010) Evolutionary game theory meets social science: is there a unifying rule for human cooperation? J Theor Biol 264(2):450–456
122. Rowe GW (1987) A dynamic game theory model of diploid genetic system. J Theor Biol 129:243–255
123. 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
124. Samuelson PA (1985) Modes of thought in economics and biology. Am Econ Rev 75(2):166–172
125. Sandholm WH (2010) Population games and evolutionary dynamics. MIT Press, Cambridge
126. Santos FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc Natl Acad Sci USA 103:3490–3494
127. Schuster P, Sigmund K (1983) Replicator dynamics. J Theor Biol 100:533–538
128. 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
129. 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
130. Schwalbe U, Walker P (2001) Zermelo and the early history of game theory. Games Econ Behav 34(1):123–137
131. Shirakihara K, Tanaka S (1978) Two fish species competition model with nonlinear interactions and equilibrium catches. Res Popul Ecol 20(1):123–140
132. Sigmund K (1987a) Game dynamics, mixed strategies, and gradient systems. Theor Popul Biol 32(1):114–126
133. Sigmund K (1987b) A maximum principle for frequency dependent selection. Math Biosci 84:189–195
134. Skyrms B (2003) The Stag-Hunt game and the evolution of social structure. Cambridge University Press, Cambridge
135. 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
136. Souza MO, Pacheco JM, Santos FC (2009) Evolution of cooperation under n-person snowdrift games. J Theor Biol 260:581–588
137. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216
138. Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA (2009) Strategy selection in structured populations. J Theor Biol 259:570–581
139. Tarnita CE, Wage N, Nowak MA (2011) Multiple strategies in structured populations. Proc Natl Acad Sci USA 108:2334–2337
140. Taylor C, Fudenberg D, Sasaki A, Nowak MA (2004) Evolutionary game dynamics in finite populations. Bull Math Biol 66:1621–1644
141. Taylor PD, Jonker L (1978) Evolutionarily stable strategies and game dynamics. Math Biosci 40:145–156
142. Thomas B, Pohley HJ (1981) ESS-theory for finite populations. Biosystems 13(3):211–221
143. 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
144. Traulsen A, Reed FA (2012) From genes to games: cooperation and cyclic dominance in meiotic drive. J Theor Biol 299:120–125
145. Traulsen A, Claussen JC, Hauert C (2005) Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95:238701
146. Traulsen A, Pacheco JM, Imhof LA (2006) Stochasticity and evolutionary stability. Phys Rev E 74(021):905
147. Traulsen A, Pacheco JM, Nowak MA (2007) Pairwise comparison and selection temperature in evolutionary game dynamics. J Theor Biol 246:522–529
148. Traulsen A, Claussen JC, Hauert C (2012) Stochastic differential equations for evolutionary dynamics with demographic noise and mutations. Phys Rev E 85:041901
149. Turner PE, Chao L (1999) Prisoner’s dilemma in an RNA virus. Nature 398:441–443
150. Turner PE, Chao L (2003) Escape from prisoner’s dilemma in RNA phage Phi6. Am Nat 161(3):497–505
151. Van Damme E (1994) Evolutionary game theory. Eur Econ Rev 38:847–858
152. 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
153. van Veelen M, Nowak MA (2012) Multi-player games on the cycle. J Theor Biol 292:116–128
154. van Veelen M, García J, Rand DG, Nowak MA (2012) Direct reciprocity in structured populations. Proc Natl Acad Sci USA 109:9929–9934
155. Vickers GT, Cannings C (1988) Patterns of ESS’s I. J Theor Biol 132(4):387–408
156. Vickery WL, Poulin R (2010) The evolution of host manipulation by parasites: a game theory analysis. Evol Ecol 24(4):773–788
157. 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
158. Wade MJ, Beeman RW (1994) The population dynamics of maternal-effect selfish genes. Genetics 138:1309–1314
159. Wakeley J (2008) Coalescent theory: an introduction. Roberts and Company Publishers, Greenwood Village
160. 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
161. Woelfing B, Traulsen A (2009) Stochastic sampling of interaction partners versus deterministic payoff assignment. J Theor Biol 257:689–695
162. Wooders M, Cartwright E, Selten R (2006) Behavioral conformity in games with many players. Games Econ Behav 57(2):347–360
163. Wu B, Altrock PM, Wang L, Traulsen A (2010) Universality of weak selection. Phys Rev E 82:046106
164. Wu B, Gokhale CS, Wang L, Traulsen A (2012) How small are small mutation rates? J Math Biol 64:803–827
165. Wu B, García J, Hauert C, Traulsen A (2013a) Extrapolating weak selection in evolutionary games. PLoS Comput Biol 9:e1003381
166. Wu B, Traulsen A, Gokhale CS (2013b) Dynamic properties of evolutionary multi-player games in finite populations. Games 4(2):182–199
167. Zeeman EC (1980) Population dynamics from game theory. Lect Notes Math 819:471–497
168. Zermelo E (1913) Über eine Anwendung der Mengenlehre auf die Theorie des Schachspiels. In: Proceedings of the Fifth International Congress of Mathematicians.
169. Zhou Qian H (2011) Fixation, transient landscape, and diffusion dilemma in stochastic evolutionary game dynamics. Phys Rev E 84:031907