Abstract
Animal behavior and evolution can often be described by game-theoretic models. Although in many situations the number of players is very large, their strategic interactions are usually decomposed into a sum of two-player games. Only recently were evolutionarily stable strategies defined for multi-player games and their properties analyzed [Broom, M., Cannings, C., Vickers, G.T., 1997. Multi-player matrix games. Bull. Math. Biol. 59, 931–952]. Here we study the long-run behavior of stochastic dynamics of populations of randomly matched individuals playing symmetric three-player games. We analyze the stochastic stability of equilibria in games with multiple evolutionarily stable strategies. We also show that, in some games, a population may not evolve in the long run to an evolutionarily stable equilibrium.
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References
Broom, M., Cannings, C., Vickers, G.T., 1997. Multi-player matrix games. Bull. Math. Biol. 59, 931–952.
Bukowski, M., Miekisz, J., 2004. Evolutionary and asymptotic stability in multi-player games with two strategies. Int. J. Game Theory 33, 41–54.
Foster, D., Young, P.H., 1990. Stochastic evolutionary game dynamics. Theor. Popul. Biol. 38, 219–232.
Freidlin, M., Wentzell, A., 1970. On small random perturbations of dynamical systems. Russian Math. Surveys 25, 1–55.
Freidlin, M., Wentzell, A., 1984. Random Perturbations of Dynamical Systems. Springer Verlag, New York.
Hofbauer, J., Schuster, P., Sigmund, K., 1979. A note on evolutionarily stable strategies and game dynamics. J. Theor. Biol. 81, 609–612.
Hofbauer, J., Sigmund, K., 1998. Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge.
Hofbauer, J., Sigmund, K., 2003. Evolutionary game dynamics. Bull. Amer. Math. Soc. 40, 479–519.
Kandori, M., Mailath, G.J., Rob, R., 1993. Learning, mutation, and long-run equilibria in games. Econometrica 61, 29–56.
Kim, Y., 1996. Equilibrium selection in n-person coordination games. Games Econom. Behav. 15, 203–277.
Maynard Smith, J., Price, G.R., 1973. The logic of animal conflicts. Nature 246, 15–18.
Maynard Smith, J., 1982. Evolution and the Theory of Games. Cambridge University Press, Cambridge.
Miekisz, J., 2004. Stochastic stability in spatial three-player games. Physica A 343, 175–184.
Miekisz, J., 2005. Equilibrium selection in evolutionary games with random matching of players. J. Theor. Biol. 232, 47–53.
Robson, A., Vega-Redondo, F., 1996. Efficient equilibrium selection in evolutionary games with random matching. J. Econom. Theory 70, 65–92.
Samuelson, L., 1997. Evolutionary Games and Equilibrium Selection. MIT Press, Cambridge.
Taylor, P.D., Jonker, L.B., 1978. Evolutionarily stable strategy and game dynamics. Math. Biosci. 40, 145–156.
Vega-Redondo, F., 1996. Evolution, Games, and Economic Behaviour. Oxford University Press, Oxford.
Weibull, J., 1995. Evolutionary Game Theory. MIT Press, Cambridge.
Zeeman, E., 1981. Dynamics of the evolution of animal conflicts. J. Theor. Biol. 89, 249–270.
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Kamiński, D., Miekisz, J. & Zaborowski, M. Stochastic stability in three-player games. Bull. Math. Biol. 67, 1195–1205 (2005). https://doi.org/10.1016/j.bulm.2005.01.004
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DOI: https://doi.org/10.1016/j.bulm.2005.01.004