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Bulletin of the American Mathematical Society

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ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Evolutionary game dynamics
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by Josef Hofbauer and Karl Sigmund PDF
Bull. Amer. Math. Soc. 40 (2003), 479-519 Request permission

Abstract:

Evolutionary game dynamics is the application of population dynamical methods to game theory. It has been introduced by evolutionary biologists, anticipated in part by classical game theorists. In this survey, we present an overview of the many brands of deterministic dynamical systems motivated by evolutionary game theory, including ordinary differential equations (and, in particular, the replicator equation), differential inclusions (the best response dynamics), difference equations (as, for instance, fictitious play) and reaction-diffusion systems. A recurrent theme (the so-called ‘folk theorem of evolutionary game theory’) is the close connection of the dynamical approach with the Nash equilibrium, but we show that a static, equilibrium-based viewpoint is, on principle, unable to always account for the long-term behaviour of players adjusting their behaviour to maximise their payoff.
References
  • Ethan Akin, The geometry of population genetics, Lecture Notes in Biomathematics, vol. 31, Springer-Verlag, Berlin-New York, 1979. MR 559137, DOI 10.1007/978-3-642-93128-4
  • Ethan Akin, The general topology of dynamical systems, Graduate Studies in Mathematics, vol. 1, American Mathematical Society, Providence, RI, 1993. MR 1219737, DOI 10.1090/gsm/001
  • Ethan Akin and Viktor Losert, Evolutionary dynamics of zero-sum games, J. Math. Biol. 20 (1984), no. 3, 231–258. MR 765812, DOI 10.1007/BF00275987
  • [AF00]AF00 C. Alós-Ferrer, Finite population dynamics and mixed equilibria, WP 0008, Department of Economics, University of Vienna, 2000. International Game Theory Review, forthcoming.
  • Carlos Alós-Ferrer and Ana B. Ania, Local equilibria in economic games, Econom. Lett. 70 (2001), no. 2, 165–173. MR 1804650, DOI 10.1016/S0165-1765(00)00371-2
  • [AF02]AF02 C. Alós-Ferrer, A.B. Ania, The evolutionary logic of feeling small, WP 0216, Department of Economics, University of Vienna, 2002.
  • Carlos Alós-Ferrer, Ana B. Ania, and Klaus Reiner Schenk-Hoppé, An evolutionary model of Bertrand oligopoly, Games Econom. Behav. 33 (2000), no. 1, 1–19. MR 1791632, DOI 10.1006/game.1999.0765
  • [ATW02]ATW02 A.B. Ania, T. Tröger and A. Wambach: An evolutionary analysis of insurance markets with adverse selection, Games Econ. Behav. 40 (2002), 153-184.
  • Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
  • [A84]A84 R. Axelrod: The Evolution of Cooperation. New York: Basic Books (1984).
  • Dieter Balkenborg and Karl H. Schlag, Evolutionarily stable sets, Internat. J. Game Theory 29 (2000), no. 4, 571–595 (2001). MR 1831214, DOI 10.1007/s001820100059
  • [Beg02]Beg02 A. Beggs: On the convergence of reinforcement learning, Preprint.
  • Michel Benaïm, Dynamics of stochastic approximation algorithms, Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 1–68. MR 1767993, DOI 10.1007/BFb0096509
  • Michel Benaïm and Morris W. Hirsch, Mixed equilibria and dynamical systems arising from fictitious play in perturbed games, Games Econom. Behav. 29 (1999), no. 1-2, 36–72. Learning in games: a symposium in honor of David Blackwell. MR 1729309, DOI 10.1006/game.1999.0717
  • [BenW00]BenW00 M. Benaim and J. Weibull: Deterministic approximation of stochastic evolution in games, Preprint 2000, Econometrica. (To appear) [Ber98]Ber98 U. Berger: Best response dynamics and Nash dynamics for games, Dissertation, Univ. Vienna (1998).
  • Ulrich Berger, Best response dynamics for role games, Internat. J. Game Theory 30 (2001), no. 4, 527–538 (2002). MR 1907263, DOI 10.1007/s001820200096
  • [Ber02]Ber02 U. Berger: A general model of best response adaptation. Preprint. 2002. [Ber03]Ber03 U. Berger: Fictitious play in $2 \times n$ games. Preprint. 2003. [BerH02]BerH02 U. Berger and J. Hofbauer: Irrational behavior in the Brown–von Neumann–Nash dynamics, Preprint 2002.
  • Lawrence E. Blume, The statistical mechanics of strategic interaction, Games Econom. Behav. 5 (1993), no. 3, 387–424. MR 1227917, DOI 10.1006/game.1993.1023
  • [Bo83]Bo83 I.M. Bomze: Lotka-Volterra equations and replicator dynamics: A two dimensional classification, Biol. Cybernetics 48 (1983), 201-211.
  • I. M. Bomze, Noncooperative two-person games in biology: a classification, Internat. J. Game Theory 15 (1986), no. 1, 31–57. MR 839095, DOI 10.1007/BF01769275
  • Immanuel M. Bomze, Dynamical aspects of evolutionary stability, Monatsh. Math. 110 (1990), no. 3-4, 189–206. MR 1084311, DOI 10.1007/BF01301675
  • Immanuel M. Bomze, Cross entropy minimization in uninvadable states of complex populations, J. Math. Biol. 30 (1991), no. 1, 73–87. MR 1130789, DOI 10.1007/BF00168008
  • [Bo94]Bo94 I.M. Bomze: Lotka-Volterra equation and replicator dynamics: new issues in classification, Biol. Cybernetics 72 (1994), 447-453.
  • Immanuel M. Bomze, Uniform barriers and evolutionarily stable sets, Game theory, experience, rationality, Vienna Circ. Inst. Yearb., vol. 5, Kluwer Acad. Publ., Dordrecht, 1998, pp. 225–243. MR 1747262
  • [Bo02]Bo02 I.M. Bomze: Regularity vs. degeneracy in dynamics, games, and optimization: a unified approach to different aspects, SIAM Review 44 (2002), 394-414.
  • Immanuel M. Bomze and Benedikt M. Pötscher, Game theoretical foundations of evolutionary stability, Lecture Notes in Economics and Mathematical Systems, vol. 324, Springer-Verlag, Berlin, 1989. MR 1116764, DOI 10.1007/978-3-642-45660-2
  • Immanuel M. Bomze and Reinhard Bürger, Stability by mutation in evolutionary games, Games Econom. Behav. 11 (1995), no. 2, 146–172. Evolutionary game theory in biology and economics. MR 1360036, DOI 10.1006/game.1995.1047
  • [Bö00]Bo00 T. Börgers: When Does Learning Lead to Nash Equilibrium?, in K. Inderfurth et al. (editors), Operations Research Proceedings 1999, Springer, Berlin (2000), 176-202.
  • Tilman Börgers and Rajiv Sarin, Learning through reinforcement and replicator dynamics, J. Econom. Theory 77 (1997), no. 1, 1–14. MR 1484291, DOI 10.1006/jeth.1997.2319
  • [BH87]BH87 D.B. Brown and R.I.C. Hansel: Convergence to an evolutionarily stable strategy in the two-policy game, Amer. Naturalist 130 (1987), 929-940. [Br49]Br49 G.W. Brown: Some notes on computation of games solutions, RAND report P-78 (1949).
  • Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
  • Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
  • Antonio Cabrales and Joel Sobel, On the limit points of discrete selection dynamics, J. Econom. Theory 57 (1992), no. 2, 407–419. MR 1180004, DOI 10.1016/0022-0531(92)90043-H
  • [Co92]Co92 S. Cowan: Dynamical systems arising from game theory, Dissertation (1992), Univ. California, Berkeley.
  • R. Cressman, Strong stability and density-dependent evolutionarily stable strategies, J. Theoret. Biol. 145 (1990), no. 3, 319–330. MR 1067064, DOI 10.1016/S0022-5193(05)80112-2
  • Ross Cressman, The stability concept of evolutionary game theory, Lecture Notes in Biomathematics, vol. 94, Springer-Verlag, Berlin, 1992. A dynamic approach. MR 1176310, DOI 10.1007/978-3-642-49981-4
  • R. Cressman, Evolutionary game theory with two groups of individuals, Games Econom. Behav. 11 (1995), no. 2, 237–253. Evolutionary game theory in biology and economics. MR 1360039, DOI 10.1006/game.1995.1050
  • R. Cressman, Local stability of smooth selection dynamics for normal form games, Math. Social Sci. 34 (1997), no. 1, 1–19. MR 1468413, DOI 10.1016/S0165-4896(97)00009-7
  • [Cr03]Cr03 R. Cressman: Evolutionary dynamics and extensive form games, Cambridge, Mass., MIT Press (2003). [CrGH01]CrGH01 R. Cressman, J. Garay, J. Hofbauer: Evolutionary stability concepts for $N$-species frequency-dependent interactions, J. Theor. Biol. 211 (2001), 1-10.
  • Ross Cressman, Andrea Gaunersdorfer, and Jean-François Wen, Evolutionary and dynamic stability in symmetric evolutionary games with two independent decisions, Int. Game Theory Rev. 2 (2000), no. 1, 67–81. MR 1788666, DOI 10.1142/S0219198900000081
  • R. Cressman, J. Hofbauer, and W. G. S. Hines, Evolutionary stability in strategic models of single-locus frequency-dependent viability selection, J. Math. Biol. 34 (1996), no. 7, 707–733. MR 1410024, DOI 10.1007/s002850050027
  • [CrVi97]CrVi97 R. Cressman, G. T. Vickers: Spatial and density effects in evolutionary game theory, J. Theor. Biol. 184 (1997), 359-369.
  • Eddie Dekel and Suzanne Scotchmer, On the evolution of optimizing behavior, J. Econom. Theory 57 (1992), no. 2, 392–406. MR 1180003, DOI 10.1016/0022-0531(92)90042-G
  • Ulf Dieckmann and Richard Law, The dynamical theory of coevolution: a derivation from stochastic ecological processes, J. Math. Biol. 34 (1996), no. 5-6, 579–612. MR 1393842, DOI 10.1007/s002850050022
  • [DR98]DR98 L.A. Dugatkin and H.K. Reeve (eds.): Game Theory and Animal Behaviour, Oxford UP (1998)
  • Rick Durrett, Stochastic spatial models, SIAM Rev. 41 (1999), no. 4, 677–718. MR 1722998, DOI 10.1137/S0036144599354707
  • Glenn Ellison, Learning, local interaction, and coordination, Econometrica 61 (1993), no. 5, 1047–1071. MR 1234793, DOI 10.2307/2951493
  • [ErR98]ErR98 I. Erev and A. Roth: Predicting how people play games: Reinforcement learning in experimental games with unique, mixed strategy equilibrium, Amer. Econ. Review 88 (1998), 79–96.
  • Ilan Eshel, Evolutionarily stable strategies and viability selection in Mendelian populations, Theoret. Population Biol. 22 (1982), no. 2, 204–217. MR 679303, DOI 10.1016/0040-5809(82)90042-9
  • Ilan Eshel, Evolutionary and continuous stability, J. Theoret. Biol. 103 (1983), no. 1, 99–111. MR 714279, DOI 10.1016/0022-5193(83)90201-1
  • [Es96]Es96 I. Eshel: On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution, J. Math. Biol. 34 (1996), 485-510.
  • I. Eshel and E. Akin, Coevolutionary instability and mixed Nash solutions, J. Math. Biol. 18 (1983), no. 2, 123–133. MR 723584, DOI 10.1007/BF00280661
  • [EsMS97]EsMS97 I. Eshel, U. Motro and E. Sansone, Continuous stability and evolutionary convergence, J. Theor. Biology 185 (1997), 333–343. [EsSS98]EsSS98 I. Eshel, L. Samuelson and A. Shaked: Altruists, egoists and hooligans in a local interaction model, Amer. Econ. Review 88 (1998), 157-179. [FaW89]FaW89 J. Farrell and R. Ware: Evolutionary stability in the repeated prisoner’s dilemma, Theor. Popul. Biol. 36 (1989), 161-166. [FG00]FG00 E. Fehr and S. Gächter: Cooperation and punishment in public goods experiments, Amer. Econ. Review 90 (2000), 980-994.
  • Paul C. Fife, Mathematical aspects of reacting and diffusing systems, Lecture Notes in Biomathematics, vol. 28, Springer-Verlag, Berlin-New York, 1979. MR 527914, DOI 10.1007/978-3-642-93111-6
  • R. A. Fisher, The genetical theory of natural selection, A complete variorum edition, Oxford University Press, Oxford, 1999. Revised reprint of the 1930 original; Edited, with a foreword and notes, by J. H. Bennett. MR 1785121
  • J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
  • Dean P. Foster and H. Peyton Young, On the nonconvergence of fictitious play in coordination games, Games Econom. Behav. 25 (1998), no. 1, 79–96. MR 1653871, DOI 10.1006/game.1997.0626
  • M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260, Springer-Verlag, New York, 1984. Translated from the Russian by Joseph Szücs. MR 722136, DOI 10.1007/978-1-4684-0176-9
  • Daniel Friedman, Evolutionary games in economics, Econometrica 59 (1991), no. 3, 637–666. MR 1106508, DOI 10.2307/2938222
  • Drew Fudenberg and David K. Levine, The theory of learning in games, MIT Press Series on Economic Learning and Social Evolution, vol. 2, MIT Press, Cambridge, MA, 1998. MR 1629477
  • Drew Fudenberg and Jean Tirole, Game theory, MIT Press, Cambridge, MA, 1991. MR 1124618
  • John Gale, Kenneth G. Binmore, and Larry Samuelson, Learning to be imperfect: the ultimatum game, Games Econom. Behav. 8 (1995), no. 1, 56–90. Nobel Symposium on Game Theory (Björkborn, 1993). MR 1315990, DOI 10.1016/S0899-8256(05)80017-X
  • [GH03]GH03 B. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax and discretizations. SIAM J. Math. Anal. 34 (2003), 1007–1093.
  • Andrea Gaunersdorfer, Time averages for heteroclinic attractors, SIAM J. Appl. Math. 52 (1992), no. 5, 1476–1489. MR 1182135, DOI 10.1137/0152085
  • Andrea Gaunersdorfer, Josef Hofbauer, and Karl Sigmund, On the dynamics of asymmetric games, Theoret. Population Biol. 39 (1991), no. 3, 345–357. MR 1115666, DOI 10.1016/0040-5809(91)90028-E
  • Andrea Gaunersdorfer and Josef Hofbauer, Fictitious play, Shapley polygons, and the replicator equation, Games Econom. Behav. 11 (1995), no. 2, 279–303. Evolutionary game theory in biology and economics. MR 1360041, DOI 10.1006/game.1995.1052
  • S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs, and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol. 44 (2002), no. 6, 548–560. MR 1917846, DOI 10.1007/s002850100136
  • [GiM91]GiM91 I. Gilboa, A. Matsui: Social stability and equilibrium, Econometrica 59 (1991), 859-867. [Gi00]Gi00 H. Gintis: Game theory evolving, Princeton UP (2000). [Go95]Go95 Godfray, H.C.J. (1995) Evolutionary theory of parent- offspring conflict, Nature 376, 133-138 [H67]H67 W.D. Hamilton: Extraordinary sex ratios, Science 156 (1967), 477-488. [Ha96]Ha96 P. Hammerstein: Darwinian adaptation, population genetics and the streetcar theory of evolution, J. Math. Biol. 34 (1996), 511-532. [HaS94]HaS94 P. Hammerstein, R. Selten: Game theory and evolutionary biology, in R.J. Aumann, S. Hart (eds.): Handbook of Game Theory II, Amsterdam: North-Holland (1994), 931-993.
  • J. C. Harsanyi, Oddness of the number of equilibrium points: a new proof, Internat. J. Game Theory 2 (1973), 235–250. MR 526058, DOI 10.1007/BF01737572
  • John C. Harsanyi and Reinhard Selten, A general theory of equilibrium selection in games, MIT Press, Cambridge, MA, 1988. With a foreword by Robert Aumann. MR 956053
  • Christopher Harris, On the rate of convergence of continuous-time fictitious play, Games Econom. Behav. 22 (1998), no. 2, 238–259. MR 1610081, DOI 10.1006/game.1997.0582
  • Sergiu Hart and Andreu Mas-Colell, A general class of adaptive strategies, J. Econom. Theory 98 (2001), no. 1, 26–54. MR 1832647, DOI 10.1006/jeth.2000.2746
  • [HB01]HB01 J. Henrich, R. Boyd, S. Bowles, C. Camerer, E. Fehr, H. Gintis and R. McElreath: In search of homo economicus: behavioral experiments in 15 small-scale societies, Amer. Econ. Rev. 91 (2001), 73-78
  • W. G. S. Hines, Evolutionary stable strategies: a review of basic theory, Theoret. Population Biol. 31 (1987), no. 2, 195–272. MR 887490, DOI 10.1016/0040-5809(87)90029-3
  • W. G. S. Hines, ESS modelling of diploid populations. I. Anatomy of one-locus allelic frequency simplices, Adv. in Appl. Probab. 26 (1994), no. 2, 341–360. MR 1272716, DOI 10.2307/1427440
  • Morris W. Hirsch, Systems of differential equations which are competitive or cooperative. III. Competing species, Nonlinearity 1 (1988), no. 1, 51–71. MR 928948, DOI 10.1088/0951-7715/1/1/003
  • Josef Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka equation, Nonlinear Anal. 5 (1981), no. 9, 1003–1007. MR 633014, DOI 10.1016/0362-546X(81)90059-6
  • Josef Hofbauer, A difference equation model for the hypercycle, SIAM J. Appl. Math. 44 (1984), no. 4, 762–772. MR 750948, DOI 10.1137/0144054
  • [Ho95a]Ho95a J. Hofbauer: Stability for the best response dynamics, Preprint. [Ho95b]Ho95b J. Hofbauer: Imitation dynamics for games, Preprint.
  • Josef Hofbauer, Evolutionary dynamics for bimatrix games: a Hamiltonian system?, J. Math. Biol. 34 (1996), no. 5-6, 675–688. MR 1393843, DOI 10.1007/s002850050025
  • Josef Hofbauer, Equilibrium selection via travelling waves, Game theory, experience, rationality, Vienna Circ. Inst. Yearb., vol. 5, Kluwer Acad. Publ., Dordrecht, 1998, pp. 245–259. MR 1747263
  • Josef Hofbauer, The spatially dominant equilibrium of a game, Ann. Oper. Res. 89 (1999), 233–251. Nonlinear dynamical systems and adaptive methods (Vienna, 1997). MR 1704055, DOI 10.1023/A:1018979708014
  • [Ho00]Ho00 J. Hofbauer: From Nash and Brown to Maynard Smith: equilibria, dynamics and ESS, Selection 1 (2000), 81-88. [HoH02]HoH02 J. Hofbauer and E. Hopkins: Learning in perturbed asymmetric games. Preprint. 2002. [HoHV97]HoHV97 J. Hofbauer, V. Hutson and G.T. Vickers, Travelling waves for games in economics and biology, Nonlinear Analysis 30 (1997) 1235–1244. [HoOR03]HoOR03 J. Hofbauer, J. Oechssler and F. Riedel: Continuous and global stability in innovative evolutionary dynamics. Preprint. 2003. [HoSa01]HoSa01 J. Hofbauer, W.H. Sandholm: Evolution and learning in games with randomly perturbed payoffs, Preprint, 2001. [HoSa02]HoSa02 J. Hofbauer, W.H. Sandholm: On the global convergence of stochastic fictitious play, Econometrica 70 (2002), 2265-94. [HoSc00]HoSc00 J. Hofbauer, K. Schlag: Sophisticated imitation in cyclic games, J. Evolutionary Economics 10 (2000), 523-543.
  • J. Hofbauer, P. Schuster, and K. Sigmund, A note on evolutionary stable strategies and game dynamics, J. Theoret. Biol. 81 (1979), no. 3, 609–612. MR 558663, DOI 10.1016/0022-5193(79)90058-4
  • Josef Hofbauer, Peter Schuster, and Karl Sigmund, Game dynamics in Mendelian populations, Biol. Cybernet. 43 (1982), no. 1, 51–57. MR 647356, DOI 10.1007/BF00337287
  • Josef Hofbauer and Karl Sigmund, The theory of evolution and dynamical systems, London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge, 1988. Mathematical aspects of selection; Translated from the German. MR 1071180
  • J. Hofbauer and K. Sigmund, Adaptive dynamics and evolutionary stability, Appl. Math. Lett. 3 (1990), no. 4, 75–79. MR 1080408, DOI 10.1016/0893-9659(90)90051-C
  • Josef Hofbauer and Karl Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998. MR 1635735, DOI 10.1017/CBO9781139173179
  • J. Hofbauer and J. W.-H. So, Multiple limit cycles for three-dimensional Lotka-Volterra equations, Appl. Math. Lett. 7 (1994), no. 6, 65–70. MR 1340732, DOI 10.1016/0893-9659(94)90095-7
  • [HoSo02]HoSo02 J. Hofbauer and S. Sorin: Best response dynamics for continuous zero-sum games. Preprint, 2002. [HoSw96]HoSw96 J. Hofbauer, J. Swinkels: A universal Shapley example. Preprint.
  • Josef Hofbauer and Jörgen W. Weibull, Evolutionary selection against dominated strategies, J. Econom. Theory 71 (1996), no. 2, 558–573. MR 1424183, DOI 10.1006/jeth.1996.0133
  • Ed Hopkins, Learning, matching, and aggregation, Games Econom. Behav. 26 (1999), no. 1, 79–110. MR 1679706, DOI 10.1006/game.1998.0647
  • Ed Hopkins, A note on best response dynamics, Games Econom. Behav. 29 (1999), no. 1-2, 138–150. Learning in games: a symposium in honor of David Blackwell. MR 1729314, DOI 10.1006/game.1997.0636
  • [Hop02]Hop02 E. Hopkins: Two Competing Models of How People Learn in Games, Econometrica 70 (2002), 2141-2166. [HopP02]HopP02 E. Hopkins and M. Posch: Attainability of boundary points under reinforcement learning, Preprint.
  • V. C. L. Hutson and G. T. Vickers, Travelling waves and dominance of ESSs, J. Math. Biol. 30 (1992), no. 5, 457–471. MR 1161112, DOI 10.1007/BF00160531
  • [JJS99]JJS99 H.J. Jacobsen, M. Jensen and B. Sloth: On the structural difference between the evolutionary approach of Young and that of KMR. Preprint. [KR95]KR95 J.H. Kagel and A.E. Roth: Handbook of Experimental Economics, Princeton UP (1995). [Ka97]Ka97 M. Kandori: Evolutionary Game Theory in Economics, in D. M. Kreps and K. F. Wallis (eds.), Advances in Economics and Econometrics: Theory and Applications, I, Cambridge UP (1997).
  • Michihiro Kandori, George J. Mailath, and Rafael Rob, Learning, mutation, and long run equilibria in games, Econometrica 61 (1993), no. 1, 29–56. MR 1201702, DOI 10.2307/2951777
  • Michihiro Kandori and Rafael Rob, Evolution of equilibria in the long run: a general theory and applications, J. Econom. Theory 65 (1995), no. 2, 383–414. MR 1332651, DOI 10.1006/jeth.1995.1014
  • [K02]K02 B. Kerr, M.A. Riley, M.W. Feldman and B.J.M. Bohannan: Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors, Nature 418 (2002), 171-174.
  • Michael Kosfeld, Stochastic strategy adjustment in coordination games, Econom. Theory 20 (2002), no. 2, 321–339. MR 1925047, DOI 10.1007/s001990100223
  • [Kr92]Kr92 V. Krishna: Learning in games with strategic complementarities, Preprint.
  • Vijay Krishna and Tomas Sjöström, On the convergence of fictitious play, Math. Oper. Res. 23 (1998), no. 2, 479–511. MR 1626686, DOI 10.1287/moor.23.2.479
  • Jean-François Laslier, Richard Topol, and Bernard Walliser, A behavioral learning process in games, Games Econom. Behav. 37 (2001), no. 2, 340–366. MR 1866182, DOI 10.1006/game.2000.0841
  • [Le94]Le94 R.J. Leonard: Reading Cournot, reading Nash: The creation and stabilisation of the Nash equilibrium, The Economic Journal 104 (1994), 492–511. [Le84]Le84 S. Lessard: Evolutionary dynamics in frequency dependent two-phenotypes models, Theor. Pop. Biol. 25 (1984), 210-234.
  • Sabin Lessard, Evolutionary stability: one concept, several meanings, Theoret. Population Biol. 37 (1990), no. 1, 159–170. MR 1042080, DOI 10.1016/0040-5809(90)90033-R
  • V. Losert and E. Akin, Dynamics of games and genes: discrete versus continuous time, J. Math. Biol. 17 (1983), no. 2, 241–251. MR 714271, DOI 10.1007/BF00305762
  • [Lu02]Lu02 Z. Lu, Y. Luo, Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle, Computer Math. Appl., to appear. [Mai98]Mai98 G. Mailath: Do people play Nash equilibrium? Lessons from evolutionary game theory, J. Economic Literature 36 (1998), 1347-1374.
  • Akihiko Matsui, Best response dynamics and socially stable strategies, J. Econom. Theory 57 (1992), no. 2, 343–362. MR 1180001, DOI 10.1016/0022-0531(92)90040-O
  • [MS74]MS74 J. Maynard Smith: The theory of games and the evolution of animal conflicts, J. Theor. Biol. 47 (1974), 209-221. MR56:2475 [MS81]MS81 J. Maynard Smith: Will a sexual population evolve to an ESS? Amer. Naturalist 177 (1981), 1015-1018. [MS82]MS82 J. Maynard Smith: Evolution and the Theory of Games. Cambridge Univ. Press (1982).
  • J. Maynard Smith, Can a mixed strategy be stable in a finite population?, J. Theoret. Biol. 130 (1988), no. 2, 247–251. MR 927216, DOI 10.1016/S0022-5193(88)80100-0
  • [Mayr70]Mayr70 E. Mayr: Populations, species, and evolution. Harvard Univ. Press (1970). [Me01]Me01 G. Meszéna, È. Kisdi, U. Dieckmann, S.A.H. Geritz, J.A.J. Metz: Evolutionary optimization models and matrix games in the unified perspective of adaptive dynamics, Selection 2 (2001), 193-210.
  • S. J. van Strien and S. M. Verduyn Lunel (eds.), Stochastic and spatial structures of dynamical systems, Koninklijke Nederlandse Akademie van Wetenschappen. Verhandelingen, Afd. Natuurkunde. Eerste Reeks [Royal Netherlands Academy of Sciences. Proceedings, Physics Section. Series 1], vol. 45, North-Holland Publishing Co., Amsterdam, 1996. MR 1773292
  • [MoPa97]MoPa97 D. W. Mock and G.A. Parker: The evolution of sibling rivalry. Oxford UP (1997).
  • Dov Monderer and Lloyd S. Shapley, Potential games, Games Econom. Behav. 14 (1996), no. 1, 124–143. MR 1393599, DOI 10.1006/game.1996.0044
  • Dov Monderer and Lloyd S. Shapley, Fictitious play property for games with identical interests, J. Econom. Theory 68 (1996), no. 1, 258–265. MR 1372400, DOI 10.1006/jeth.1996.0014
  • Stephen Morris, Contagion, Rev. Econom. Stud. 67 (2000), no. 1, 57–78. MR 1745852, DOI 10.1111/1467-937X.00121
  • J. H. Nachbar, “Evolutionary” selection dynamics in games: convergence and limit properties, Internat. J. Game Theory 19 (1990), no. 1, 59–89. MR 1047348, DOI 10.1007/BF01753708
  • C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
  • [Na96]Na96 J. Nash: Essays on Game Theory, Elgar, Cheltenham (1996).
  • John F. Nash Jr., The essential John Nash, Princeton University Press, Princeton, NJ, 2002. Edited by Harold W. Kuhn and Sylvia Nasar. MR 1888522, DOI 10.1515/9781400884087
  • Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
  • Hukukane Nikaidô, Stability of equilibrium by the Brown-von-Neumann differential equation, Econometrica 27 (1959), 654–671. MR 115802, DOI 10.2307/1909356
  • Martin Nowak, An evolutionarily stable strategy may be inaccessible, J. Theoret. Biol. 142 (1990), no. 2, 237–241. MR 1037972, DOI 10.1016/S0022-5193(05)80224-3
  • Martin A. Nowak, Sebastian Bonhoeffer, and Robert M. May, More spatial games, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 4 (1994), no. 1, 33–56. MR 1276803, DOI 10.1142/S0218127494000046
  • [NoMay92]NoMay92 M.A. Nowak and R.M. May: Evolutionary games and spatial chaos, Nature 359 (1992), 826-829
  • Martin A. Nowak and Robert M. May, The spatial dilemmas of evolution, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 3 (1993), no. 1, 35–78. MR 1218718, DOI 10.1142/S0218127493000040
  • [NoS93]NoS93 M.A. Nowak and K. Sigmund: Chaos and the evolution of cooperation, Proc. Nat. Acad. Science USA 90 (1993), 5091-94. [NPS00]NPS00 M.A. Nowak, K.P. Page and K. Sigmund: Fairness versus reason in the ultimatum game, Science 289 (2000), 1773-1775. [NS97]NS97 G. Nöldeke and L. Samuelson: A dynamic model of equilibrium selection in signaling markets. J. Econ. Theory 73 (1997), 118-156.
  • Jörg Oechssler, An evolutionary interpretation of mixed-strategy equilibria, Games Econom. Behav. 21 (1997), no. 1-2, 203–237. MR 1491270, DOI 10.1006/game.1997.0550
  • Jörg Oechssler and Frank Riedel, Evolutionary dynamics on infinite strategy spaces, Econom. Theory 17 (2001), no. 1, 141–162. MR 1808098, DOI 10.1007/PL00004092
  • [OR02]OR02 J. Oechssler and F. Riedel: On the dynamic foundation of evolutionary stability in continuous models, J. Econ. Theory 107 (2002), 223-252. [Pl97]Pl97 M. Plank: Some qualitative differences between the replicator dynamics of two player and $n$ player games, Nonlinear Analysis 30 (1997), 1411-1417. [Po97]Po97 M. Posch: Cycling in a stochastic learning algorithm for normal form games, J. Evol. Economics 7 (1997), 193-207. [RS01]RS01 P.W. Rhode and M. Stegemann: Evolution through imitation: The case of duopoly, Int. J. Industrial Organization 19 (2001), 415-454.
  • Klaus Ritzberger and Jörgen W. Weibull, Evolutionary selection in normal-form games, Econometrica 63 (1995), no. 6, 1371–1399. MR 1361238, DOI 10.2307/2171774
  • C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
  • Joachim Rosenmüller, Über Periodizitätseigenschften spieltheoretischer Lernprozesse, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 259–308 (German). MR 289156, DOI 10.1007/BF00536300
  • Aldo Rustichini, Optimal properties of stimulus-response learning models, Games Econom. Behav. 29 (1999), no. 1-2, 244–273. Learning in games: a symposium in honor of David Blackwell. MR 1729319, DOI 10.1006/game.1999.0712
  • Larry Samuelson, Evolutionary games and equilibrium selection, MIT Press Series on Economic Learning and Social Evolution, vol. 1, MIT Press, Cambridge, MA, 1997. MR 1447191
  • Larry Samuelson and Jianbo Zhang, Evolutionary stability in asymmetric games, J. Econom. Theory 57 (1992), no. 2, 363–391. MR 1180002, DOI 10.1016/0022-0531(92)90041-F
  • William H. Sandholm, Simple and clever decision rules for a model of evolution, Econom. Lett. 61 (1998), no. 2, 165–170. MR 1657826, DOI 10.1016/S0165-1765(98)00163-3
  • William H. Sandholm, Potential games with continuous player sets, J. Econom. Theory 97 (2001), no. 1, 81–108. MR 1816280, DOI 10.1006/jeth.2000.2696
  • [Sa02]Sa02 W.H. Sandholm: Excess payoff dynamics, potential dynamics and stable games. Preprint.
  • Yuzuru Sato, Eizo Akiyama, and J. Doyne Farmer, Chaos in learning a simple two-person game, Proc. Natl. Acad. Sci. USA 99 (2002), no. 7, 4748–4751. MR 1895748, DOI 10.1073/pnas.032086299
  • Mark E. Schaffer, Evolutionarily stable strategies for a finite population and a variable contest size, J. Theoret. Biol. 132 (1988), no. 4, 469–478. MR 949816, DOI 10.1016/S0022-5193(88)80085-7
  • [S89]S89 M.E. Schaffer: Are profit-maximisers the best survivors?, J. Econ. Behav. Org. 12 (1989), 29-45.
  • Karl H. Schlag, Why imitate, and if so, how? A boundedly rational approach to multi-armed bandits, J. Econom. Theory 78 (1998), no. 1, 130–156. MR 1612239, DOI 10.1006/jeth.1997.2347
  • Sebastian J. Schreiber, Urn models, replicator processes, and random genetic drift, SIAM J. Appl. Math. 61 (2001), no. 6, 2148–2167. MR 1856886, DOI 10.1137/S0036139999352857
  • Peter Schuster and Karl Sigmund, Replicator dynamics, J. Theoret. Biol. 100 (1983), no. 3, 533–538. MR 693413, DOI 10.1016/0022-5193(83)90445-9
  • Aner Sela, Fictitious play in $2\times 3$ games, Games Econom. Behav. 31 (2000), no. 1, 152–162. MR 1748363, DOI 10.1006/game.1999.0731
  • Reinhard Selten, A note on evolutionarily stable strategies in asymmetric animal conflicts, J. Theoret. Biol. 84 (1980), no. 1, 93–101. MR 577174, DOI 10.1016/S0022-5193(80)81038-1
  • Rajiv Sethi, Strategy-specific barriers to learning and nonmonotonic selection dynamics, Games Econom. Behav. 23 (1998), no. 2, 284–304. MR 1627022, DOI 10.1006/game.1997.0613
  • L. S. Shapley, Some topics in two-person games, Advances in Game Theory, Princeton Univ. Press, Princeton, N.J., 1964, pp. 1–28. MR 0198990
  • Karl Sigmund, Game dynamics, mixed strategies, and gradient systems, Theoret. Population Biol. 32 (1987), no. 1, 114–126. MR 901938, DOI 10.1016/0040-5809(87)90043-8
  • [Si93]Si93 K. Sigmund: Games of Life. Penguin, Harmondsworth. 1993. [SHN01]SHN01 K. Sigmund, C. Hauert and M.A. Nowak: Reward and punishment in minigames, Proc. Nat. Acad. Sci. 98 (2001), 10757-10762. [SL96]SL96 B. Sinervo and C.M. Lively: The rock-paper-scissors game and the evolution of alternative male strategies, Nature 380 (1996), 240-243.
  • Brian Skyrms, The dynamics of rational deliberation, Harvard University Press, Cambridge, MA, 1990. MR 1102463
  • [Sk01]Sk01 B. Skyrms: The stag hunt, Proceedings and Addresses of the American Philosophical Association. To appear.
  • Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
  • Jeroen M. Swinkels, Evolutionary stability with equilibrium entrants, J. Econom. Theory 57 (1992), no. 2, 306–332. MR 1179999, DOI 10.1016/0022-0531(92)90038-J
  • Jeroen M. Swinkels, Adjustment dynamics and rational play in games, Games Econom. Behav. 5 (1993), no. 3, 455–484. MR 1227919, DOI 10.1006/game.1993.1025
  • T. Takada and J. Kigami, The dynamical attainability of ESS in evolutionary games, J. Math. Biol. 29 (1991), no. 6, 513–529. MR 1118754, DOI 10.1007/BF00164049
  • Peter D. Taylor, Evolutionary stability in one-parameter models under weak selection, Theoret. Population Biol. 36 (1989), no. 2, 125–143. MR 1020493, DOI 10.1016/0040-5809(89)90025-7
  • Peter D. Taylor and Leo B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Biosci. 40 (1978), no. 1-2, 145–156. MR 489983, DOI 10.1016/0025-5564(78)90077-9
  • Bernhard Thomas, On evolutionarily stable sets, J. Math. Biol. 22 (1985), no. 1, 105–115. MR 802738, DOI 10.1007/BF00276549
  • [Tr74]Tr74 R.L. Trivers: Parent-offspring conflict, Amer. Zool. 14 (1974), 249-264.
  • Eric van Damme, Stability and perfection of Nash equilibria, 2nd ed., Springer-Verlag, New York, 1991. MR 1293126, DOI 10.1007/978-3-642-58242-4
  • [V96]V96 F. Vega-Redondo: Evolution, games, and economic theory. Oxford UP (1996). [V97]V97 F. Vega-Redondo: The evolution of Walrasian behavior. Econometrica 65 (1997), 375-384.
  • G. T. Vickers, Spatial patterns and ESS’s, J. Theoret. Biol. 140 (1989), no. 1, 129–135. MR 1014395, DOI 10.1016/S0022-5193(89)80033-5
  • Jörgen W. Weibull, Evolutionary game theory, MIT Press, Cambridge, MA, 1995. With a foreword by Ken Binmore. MR 1347921
  • [Wei91]Wei91 F. Weissing: Evolutionary stability and dynamic stability in a class of evolutionary normal form games. In R. Selten (ed.) Game Equilibrium Models I, Berlin, Springer (1991), 29-97. [Wei96]Wei96 F. Weissing: Genetic versus phenotypic models of selection: can genetics be neglected in a long-term perspective?, J. Math. Biol. 34 (1996), 533-555.
  • H. Peyton Young, The evolution of conventions, Econometrica 61 (1993), no. 1, 57–84. MR 1201703, DOI 10.2307/2951778
  • [Y98]Y98 H.P. Young: Individual strategy and social structure, Princeton UP (1998).
  • E. C. Zeeman, Population dynamics from game theory, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 471–497. MR 591205
  • M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dynam. Stability Systems 8 (1993), no. 3, 189–217. MR 1246002, DOI 10.1080/02681119308806158
  • [ZZ02]ZZ02 E.C. Zeeman, M.L. Zeeman: An $n$-dimensional competitive Lotka-Volterra system is generically determined by its edges, Nonlinearity 15 (2002), 2019-2032. [ZZ03]ZZ03 E.C. Zeeman, M.L. Zeeman: From local to global behavior in competitive Lotka-Volterra systems, Trans. Amer. Math. Soc. 355 (2003) 713–734.
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Additional Information
  • Josef Hofbauer
  • Affiliation: (Hofbauer) Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Vienna, Austria
  • Email: Josef.Hofbauer@univie.ac.at
  • Karl Sigmund
  • Affiliation: (Sigmund) Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Vienna, Austria; (Sigmund) IIASA, A-2361 Laxenburg, Austria
  • Email: Karl.Sigmund@univie.ac.at
  • Received by editor(s): March 7, 2003
  • Received by editor(s) in revised form: April 12, 2003
  • Published electronically: July 10, 2003
  • © Copyright 2003 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 40 (2003), 479-519
  • MSC (2000): Primary 91A22; Secondary 91-02, 92-02, 34D20
  • DOI: https://doi.org/10.1090/S0273-0979-03-00988-1
  • MathSciNet review: 1997349