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Adaptive learning and p-best response sets


A product set of strategies is a p-best response set if for each agent it contains all best responses to any distribution placing at least probability p on his opponents’ profiles belonging to the product set. A p-best response set is minimal if it does not properly contain another p-best response set. We study a perturbed joint fictitious play process with bounded memory and sample and a perturbed independent fictitious play process as in Young (Econometrica 61:57–84, 1993). We show that in n-person games only strategies contained in the unique minimal p-best response set can be selected in the long run by both types of processes provided that the rate of perturbations and p are sufficiently low. For each process, an explicit bound of p is given and we analyze how this critical value evolves when n increases. Our results are robust to the degree of incompleteness of sampling relative to memory.

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  • Basu K, Weibull JW (1991) Strategy subsets closed under rational behavior. Econ Lett 36: 141–146

    Article  Google Scholar 

  • Ellison G (2000) Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution. Rev Econ Stud 67: 17–45

    Article  Google Scholar 

  • Harsanyi JC, Selten R (1988) A general theory of equilibrium selection in games. MIT Press Books, Cambridge, MA

    Google Scholar 

  • Hurkens S (1995) Learning by forgetful players. Games Econ Behav 11: 304–329

    Article  Google Scholar 

  • Kajii A, Morris S (1997) The robustness of equilibria to incomplete information. Econometrica 65: 1283–1309

    Article  Google Scholar 

  • Klimm M, Sandholm T, Weibull JW (2010) Finding all minimal sCURB sets in finite games. Working Paper, Stockholm School of Economics

  • Kojima F, Takahashi S (2008) p-Dominance and perfect foresight dynamics. J Econ Behav Organ 67: 689–701

    Article  Google Scholar 

  • Maruta T (1997) On the relationship between risk-dominance and stochastic stability. Games Econ Behav 19: 221–234

    Article  Google Scholar 

  • Matsui A, Matsuyama K (1995) An approach to equilibrium selection. J Econ Theory 65: 415–434

    Article  Google Scholar 

  • Monderer D, Sela A (1997) Fictitious play and no-cycling conditions. Working Paper, The Technion

  • Morris S, Rob R, Shin HS (1995) p-Dominance and belief potential. Econometrica 63: 145–157

    Article  Google Scholar 

  • Morris S, Ui T (2005) Generalized potentials and robust sets of equilibria. J Econ Theory 124: 45–78

    Google Scholar 

  • Oyama D, Takahashi S, Hofbauer J (2008) Monotone methods for equilibrium selection under perfect foresight dynamics. Theor Econ 3: 155–192

    Google Scholar 

  • Tercieux O (2006a) p-Best response set. J Econ Theory 131: 45–70

    Article  Google Scholar 

  • Tercieux O (2006b) p-Best response sets and the robustness of equilibria to incomplete information. Games Econ Behav 56: 371–384

    Article  Google Scholar 

  • Young HP (1993) The evolution of conventions. Econometrica 61: 57–84

    Article  Google Scholar 

  • Young HP (1998) Individual strategy and social structure. An evolutionary theory of institutions. Princeton University Press, Princeton

    Google Scholar 

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Correspondence to O. Tercieux.

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Durieu, J., Solal, P. & Tercieux, O. Adaptive learning and p-best response sets. Int J Game Theory 40, 735–747 (2011).

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  • Evolutionary game theory
  • Fictitious play process
  • p-Dominance
  • Stochastic stability

JEL Classification

  • C72
  • C73