Evolutionary Game Theory and Population Dynamics

Many socio-economic and biological processes can be modeled as systems of interacting individuals. The behaviour of such systems can be often described within game-theoretic models. In these lecture notes, we introduce fundamental concepts of evolutionary game theory and review basic properties of deterministic replicator dynamics and stochastic dynamics of finite populations. We discuss stability of equilibria in deterministic dynamics with migration, time-delay, and in stochastic dynamics of well-mixed populations and spatial games with local interactions. We analyze the dependence of the long-run behaviour of a population on various parameters such as the time delay, the noise level, and the size of the population.

[1]  Jörgen W. Weibull,et al.  Evolutionary Game Theory , 1996 .

[2]  P. Taylor,et al.  Fitness and evolutionary stability in game theoretic models of finite populations , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[3]  F. C. Santos,et al.  Graph topology plays a determinant role in the evolution of cooperation , 2006, Proceedings of the Royal Society B: Biological Sciences.

[4]  Glenn Ellison.  Learning, Local Interaction, and Coordination , 1993 .

[5]  M. Doebeli,et al.  Spatial evolutionary game theory: Hawks and Doves revisited , 1996, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[6]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[7]  C. Hauert,et al.  Models of cooperation based on the Prisoner's Dilemma and the Snowdrift game , 2005 .

[8]  John C. Harsanyi,et al.  Общая теория выбора равновесия в играх / A General Theory of Equilibrium Selection in Games , 1989 .

[9]  G. Szabó,et al.  Cooperation in the noisy case: Prisoner's dilemma game on two types of regular random graphs. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  D. Fudenberg,et al.  Emergence of cooperation and evolutionary stability in finite populations , 2004, Nature.

[11]  Daniel B. Neill,et al.  Evolutionary stability for large populations. , 2004, Journal of theoretical biology.

[12]  Jan Alboszta,et al.  Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay. , 2004, Journal of theoretical biology.

[13]  Jacek Miekisz,et al.  Equilibrium selection in evolutionary games with random matching of players. , 2004, Journal of theoretical biology.

[14]  E. Zeeman.  Dynamics of the evolution of animal conflicts , 1981 .

[15]  F. C. Santos,et al.  Evolutionary dynamics of social dilemmas in structured heterogeneous populations. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[16]  C Cannings,et al.  Multi-player matrix games. , 1997, Bulletin of mathematical biology.

[17]  Jacek Miekisz,et al.  Statistical mechanics of spatial evolutionary games , 2002, cond-mat/0210094.

[18]  Jacek Miekisz,et al.  Stochastic stability in three-player games , 2004, Bulletin of mathematical biology.

[19]  R. Cressman.  Evolutionary Dynamics and Extensive Form Games , 2003 .

[20]  H. Ohtsuki,et al.  A simple rule for the evolution of cooperation on graphs and social networks , 2006, Nature.

[21]  J. M. Smith.  The theory of games and the evolution of animal conflicts. , 1974, Journal of theoretical biology.

[22]  A. W. F. Edwards,et al.  The statistical processes of evolutionary theory , 1963 .

[23]  D. Fudenberg,et al.  The Theory of Learning in Games , 1998 .

[24]  J. Slawny,et al.  Phase transitions in systems with a finite number of dominant ground states , 1989 .

[25]  Tadeusz Płatkowski,et al.  Population dynamics with a stable efficient equilibrium. , 2005, Journal of theoretical biology.

[26]  Fernando Vega-Redondo,et al.  Evolution, Games, and Economic Behaviour , 1996 .

[27]  R. Axelrod,et al.  Evolutionary Dynamics , 2004 .

[28]  M. Nowak.  Five Rules for the Evolution of Cooperation , 2006, Science.

[29]  M. Nowak,et al.  Evolutionary Dynamics of Biological Games , 2004, Science.

[30]  T. Yi,et al.  Effect of time delay and evolutionarily stable strategy. , 1997, Journal of theoretical biology.

[31]  G. Szabó,et al.  Phase diagrams for an evolutionary prisoner's dilemma game on two-dimensional lattices. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  H. Ohtsuki,et al.  The replicator equation on graphs. , 2006, Journal of theoretical biology.

[33]  L. Samuelson,et al.  Evolutionary Stability in Asymmetric Games , 1992 .

[34]  T. C. Dorlas,et al.  Statistical Mechanics and Field Theory: Mathematical Aspects , 1986 .

[35]  Drew Fudenberg,et al.  Evolutionary game dynamics in finite populations , 2004, Bulletin of mathematical biology.

[36]  R. A. Fisher,et al.  On the dominance ratio , 1990 .

[37]  F. Vega-Redondo,et al.  Efficient Equilibrium Selection in Evolutionary Games with Random Matching , 1996 .

[38]  W. Hamilton,et al.  The evolution of cooperation. , 1984, Science.

[39]  M. Nowak,et al.  Evolutionary games and spatial chaos , 1992, Nature.

[40]  Arne Traulsen,et al.  Pairwise comparison and selection temperature in evolutionary game dynamics. , 2007, Journal of theoretical biology.

[41]  Martin A. Nowak,et al.  Evolutionary dynamics on graphs , 2005, Nature.

[42]  G. Szabó,et al.  Evolutionary prisoner's dilemma game on a square lattice , 1997, cond-mat/9710096.

[43]  M. Nowak,et al.  Evolutionary game dynamics in a Wright-Fisher process , 2006, Journal of mathematical biology.

[44]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[45]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[46]  M. Freidlin,et al.  ON SMALL RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS , 1970 .

[47]  Bruno O. Shubert,et al.  A Flow-Graph Formula for the Stationary Distribution of a Markov Chain , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[48]  Martin A Nowak,et al.  Evolutionary games on cycles , 2006, Proceedings of the Royal Society B: Biological Sciences.

[49]  R. Buerger.  The Mathematical Theory of Selection, Recombination, and Mutation , 2000 .

[50]  M. Nowak,et al.  MORE SPATIAL GAMES , 1994 .

[51]  Glenn Ellison.  Basins of Attraction, Long-Run Stochastic Stability, and the Speed of Step-by-Step Evolution , 2000 .

[52]  Jacek Miekisz.  Long-Run Behavior of Games with Many Players , 2004 .

[53]  J. Miȩkisz.  Equilibrium transitions in finite populations of players , 2004, q-bio/0412038.

[54]  R. Punnett,et al.  The Genetical Theory of Natural Selection , 1930, Nature.

[55]  Attila Szolnoki,et al.  Phase transitions for rock-scissors-paper game on different networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[57]  S Redner,et al.  Evolutionary dynamics on degree-heterogeneous graphs. , 2006, Physical review letters.

[58]  László Gulyás,et al.  On Evolutionary 3-Person Prisoner's Dilemma Games on 2-D Lattice , 2004, ACRI.

[59]  C. Hauert,et al.  Coevolutionary dynamics: from finite to infinite populations. , 2004, Physical review letters.

[60]  F. C. Santos,et al.  Scale-free networks provide a unifying framework for the emergence of cooperation. , 2005, Physical review letters.

[61]  Iva Dostálková,et al.  Evolutionarily stable strategies for stochastic processes. , 2004, Theoretical population biology.

[62]  Arne Traulsen,et al.  Stochastic payoff evaluation increases the temperature of selection. , 2007, Journal of theoretical biology.

[63]  David P. Myatt,et al.  A multinomial probit model of stochastic evolution , 2003 .

[64]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

[65]  Brauchli,et al.  Evolution of cooperation in spatially structured populations , 1999, Journal of theoretical biology.

[66]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[67]  L. Shapley,et al.  Potential Games , 1994 .

[68]  A V Herz,et al.  Collective phenomena in spatially extended evolutionary games. , 1994, Journal of theoretical biology.

[69]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[70]  Christoph Hauert,et al.  Spatial effects in social dilemmas. , 2006, Journal of theoretical biology.

[71]  Michael Doebeli,et al.  Spatial structure often inhibits the evolution of cooperation in the snowdrift game , 2004, Nature.

[72]  Peter C. Ordeshook,et al.  Game Theory And Political Theory , 1987 .

[73]  R. Rob,et al.  Learning, Mutation, and Long Run Equilibria in Games , 1993 .

[74]  E. Akin.  Domination or equilibrium , 1980 .

[75]  H. Peyton Young,et al.  Individual Strategy and Social Structure , 2020 .

[76]  K Sigmund,et al.  A note on evolutionary stable strategies and game dynamics. , 1979, Journal of theoretical biology.

[77]  K. Lindgren,et al.  Evolutionary dynamics of spatial games , 1994 .

[78]  Jacek Miekisz,et al.  Stochastic stability in spatial three-player games , 2004, cond-mat/0409649.

[79]  Arne Traulsen,et al.  Coevolutionary dynamics in large, but finite populations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[80]  S. Wright,et al.  Evolution in Mendelian Populations. , 1931, Genetics.

[81]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[82]  M. Cripps.  The theory of learning in games. , 1999 .

[83]  J. Slawny,et al.  First order phase transitions and perturbation theory , 1986 .

[84]  Jacek Miękisz,et al.  Stochastic Stability in Spatial Games , 2004, cond-mat/0409647.

[85]  K. Sigmund.  Games of Life: Explorations in Ecology, Evolution and Behaviour , 1993 .

[86]  L. Imhof,et al.  Stochasticity and evolutionary stability. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[87]  György Szabó,et al.  Evolutionary prisoner's dilemma game on hierarchical lattices. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[88]  Jacek Miekisz,et al.  Evolutionary and asymptotic stability in symmetric multi-player games , 2004, Int. J. Game Theory.

[89]  L. Blume.  The Statistical Mechanics of Strategic Interaction , 1993 .

[90]  Attila Szolnoki,et al.  Rock-scissors-paper game on regular small-world networks , 2004 .

[91]  Tadeusz Platkowski,et al.  ESSs in n-player mixed games , 2005, Appl. Math. Comput..

[92]  H. Young,et al.  The Evolution of Conventions , 1993 .

[93]  L. Samuelson.  Evolutionary Games and Equilibrium Selection , 1997 .

[94]  Āṇṭāḷ,et al.  Spatial evolutionary prisoner's dilemma game with three strategies and external constraints , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[95]  T. Platkowski,et al.  Evolution of populations playing mixed multiplayer games , 2004 .

[96]  Lawrence E. Blume,et al.  How noise matters , 2003, Games Econ. Behav..

[97]  Toshimasa Maruta,et al.  On the Relationship Between Risk-Dominance and Stochastic Stability , 1997 .

[98]  G. Szabó,et al.  Evolutionary games on graphs , 2006, cond-mat/0607344.

[99]  Noble Frankland,et al.  Common Sense and Nuclear Warfare , 1959 .

[100]  Youngse Kim,et al.  Equilibrium Selection inn-Person Coordination Games , 1996 .

[101]  M. Nowak,et al.  THE SPATIAL DILEMMAS OF EVOLUTION , 1993 .

[102]  H. Peyton Young,et al.  Stochastic Evolutionary Game Dynamics , 1990 .