# Fixation Probabilities of Strategies for Bimatrix Games in Finite Populations

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## Abstract

Recent developments in stochastic evolutionary game theory in finite populations yield insights that complement the conventional deterministic evolutionary game theory in infinite populations. However, most studies of stochastic evolutionary game theory have investigated dynamics of symmetric games, although not all social and biological phenomena are described by symmetric games, e.g., social interactions between individuals having conflicting preferences or different roles. In this paper, we describe the stochastic evolutionary dynamics of two-player \(2 \times 2\) bimatrix games in finite populations. The stochastic process is modeled by a frequency-dependent Moran process without mutation. We obtained the fixation probability that the evolutionary dynamics starting from a given initial state converges to a specific absorbing state. Applying the formula to the ultimatum game, we show that evolutionary dynamics favors fairness. Furthermore, we present two novel concepts of stability for bimatrix games, based on our formula for the fixation probability, and demonstrate that one of the two serves as a criterion for equilibrium selection.

## Keywords

Bimatrix games Equilibrium selection Finite population Fixation probability Stability Stochastic evolution## Notes

### Acknowledgments

The authors thank the two anonymous reviewers for their helpful comments. This work was supported by the Grant-in-Aid for JSPS Fellows Grant Number 13J05358 (TS) and JSPS KAKENHI Grant Number 25118006 (HO).

## References

- 1.Antal T, Traulsen A, Ohtsuki H, Tarnita CE, Nowak MA (2009) Mutation-selection equilibrium in games with multiple strategies. J Theor Biol 258:614–622MathSciNetCrossRefGoogle Scholar
- 2.Bornstein G, Budescu D, Zamir S (1997) Cooperation in intergroup, N-person, and two-person games of chicken. J Conflict Resolut 41:384–406CrossRefGoogle Scholar
- 3.Bøe T (1997) Evolutionary game theory and the battle of sexes. Chr. Michelsen Institute working paperGoogle Scholar
- 4.Chiang YS (2008) A path toward fairness: preferential association and the evolution of strategies in the ultimatum game. Ration Soc 20:173–201CrossRefGoogle Scholar
- 5.Cressman R (2003) Evolutionary dynamics and extensive form games. MIT Press, CambridgeMATHGoogle Scholar
- 6.Camerer CF (2003) Behavioral game theory: experiments on strategic interaction. Princeton University Press, PrincetonGoogle Scholar
- 7.Forber P, Smead R (2014) The evolution of fairness through spite. Proc Biol Sci 281:1–8CrossRefGoogle Scholar
- 8.Gale J, Binmore K, Samuelson L (1995) Learning to be imperfect: the ultimatum game. Games Econ Behav 8:56–90MathSciNetCrossRefMATHGoogle Scholar
- 9.Gaunersdorfer A, Hofbauer J, Sigmund K (1991) On the dynamics of asymmetric games. Theor Popul Biol 39:345–357MathSciNetCrossRefMATHGoogle Scholar
- 10.Güth W, Schmittberger R, Schwarze B (1982) An experimental analysis of ultimatum bargaining. J Econ Behav Organ 3:367–388CrossRefGoogle Scholar
- 11.Hammerstein P (1981) The role of asymmetries in animal contests. Anim Behav 29:193–205CrossRefGoogle Scholar
- 12.Harsanyi JC, Selten R (1988) A general theory of equilibrium selection in games. MIT Press, CambridgeMATHGoogle Scholar
- 13.Hauert C, Traulsen A, Brandt H, Nowak MA, Sigmund K (2007) Via freedom to coercion: the emergence of costly punishment. Science 316:1905–1907MathSciNetCrossRefMATHGoogle Scholar
- 14.Hauert C, Traulsen A, Brandt H, Nowak MA, Sigmund K (2008) Public goods with punishment and abstaining in finite and infinite populations. Biol Theory 3:114–122CrossRefGoogle Scholar
- 15.Hofbauer J (1996) Evolutionary dynamics for bimatrix games: a hamiltonian system? J Math Biol 34:675–688MathSciNetCrossRefMATHGoogle Scholar
- 16.Hofbauer J, Schuster P, Sigmund K (1979) A note on evolutionary stable strategies and game dynamics. J Theor Biol 81:609–612MathSciNetCrossRefGoogle Scholar
- 17.Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- 18.Imhof LA, Fudenberg D, Nowak MA (2006) Evolutionary cycles of cooperation and defection. Proc Natl Acad Sci USA 102:10797–10800CrossRefGoogle Scholar
- 19.Kandori M, Mailath GJ, Rob R (1993) Learning, mutation, and long run equilibria in games. Econometrica 61:29–56MathSciNetCrossRefMATHGoogle Scholar
- 20.Kurokawa S, Ihara Y (2009) Emergence of cooperation in public goods games. Proc R Soc B Biol Sci 276:1379–1384CrossRefGoogle Scholar
- 21.Lehmann L, Rousset F (2009) Perturbation expansions of multilocus fixation probabilities for frequency-dependent selection with applications to the Hill-Robertson effect and to the joint evolution of helping and punishment. Theor Popul Biol 76:35–51CrossRefMATHGoogle Scholar
- 22.Maynard-Smith J (1982) Evolution and the theory of games. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
- 23.Maynard-Smith J, Price GR (1973) The logic of animal conflicts. Nature 246:15–18CrossRefGoogle Scholar
- 24.Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, CambridgeMATHGoogle Scholar
- 25.Nowak MA, Page K, Sigmund K (2000) Fairness versus reason in the ultimatum game. Science 289:1773–1775CrossRefGoogle Scholar
- 26.Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650CrossRefGoogle Scholar
- 27.Ohtsuki H (2010) Stochastic evolutionary dynamics of bimatrix games. J Theor Biol 264:136–142MathSciNetCrossRefGoogle Scholar
- 28.Page KM, Nowak MA, Sigmund K (2000) The spatial ultimatum game. Proc R Soc B Biol Sci 267:2177–2182CrossRefGoogle Scholar
- 29.Rand DG, Tarnita CE, Ohtsuki H, Nowak MA (2013) Evolution of fairness in the one-shot anonymous ultimatum game. Proc Natl Acad Sci USA 110:2581–2586MathSciNetCrossRefMATHGoogle Scholar
- 30.Rousset F (2003) A minimal derivation of convergence stability measures. J Theor Biol 221:665–668CrossRefGoogle Scholar
- 31.Rousset F (2004) Genetic structure and selection in subdivided populations. Princeton University Press, PrincetonGoogle Scholar
- 32.Samuelson L (1997) Evolutionary games and equilibrium selection. MIT Press, CambridgeMATHGoogle Scholar
- 33.Samuelson L, Zhang J (1992) Evolutionary stability in asymmetric games. J Econ Theory 57:363–391MathSciNetCrossRefMATHGoogle Scholar
- 34.Seleten R (1975) Reexamination of the perfectness concept for equilibrium points in extensive games. Int J Game Theory 4:25–55MathSciNetCrossRefGoogle Scholar
- 35.Selten R (1978) The chain-store paradox. Theory Decis 9:127–159MathSciNetCrossRefMATHGoogle Scholar
- 36.Sekiguchi T (2013) General conditions for strategy abundance through a self-referential mechanism under weak selection. Phys A 392:2886–2892MathSciNetCrossRefGoogle Scholar
- 37.Shirata Y (2012) The evolution of fairness under an assortative matching rule in the ultimatum game. Int J Game Theory 41:1–21MathSciNetCrossRefMATHGoogle Scholar
- 38.Taylor PD, Jonker LB (1978) Evolutionarily stable strategies and game dynamics. Math Biosci 40:145–156Google Scholar
- 39.Wakeley J (2009) Coalescent theory: an introduction. Roberts and Company Publishers, ColoradoGoogle Scholar
- 40.Weibull J (1995) Evolutionary game theory. MIT Press, CambridgeMATHGoogle Scholar
- 41.Wild G, Taylor PD (2004) Fitness and evolutionary stability in game theoretic models of finite populations. Proc R Soc Lond B Biol Sci 271:2345–2349CrossRefGoogle Scholar
- 42.Zhang Y, Gao X (2015) Stochastic evolutionary selection in heterogeneous populations for asymmetric games. Comput Econ 45:501–515CrossRefGoogle Scholar