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Basins of attraction and equilibrium selection under different learning rules

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An Erratum to this article was published on 28 May 2009

Abstract

A deterministic learning model applied to a game with multiple equilibria produces distinct basins of attraction for those equilibria. In symmetric two-by-two games, basins of attraction are invariant to a wide range of learning rules including best response dynamics, replicator dynamics, and fictitious play. In this paper, we construct a class of three-by-three symmetric games for which the overlap in the basins of attraction under best response learning and replicator dynamics is arbitrarily small. We then derive necessary and sufficient conditions on payoffs for these two learning rules to create basins of attraction with vanishing overlap. The necessary condition requires that with probability one the initial best response is not an equilibrium to the game. The existence of parasitic or misleading actions allows subtle differences in the learning rules to accumulate.

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Notes

  1. These categories also go by the terms epistemic learning and behavioral learning respectively (Walliser 1998).

  2. The connection between fictitious play and best response dynamics requires the view that in fictitious play, a new agent enters the population each round with an action that is fixed forever. The state variable must then take on an interpretation as the opponent’s population mixed strategy.

  3. The aforementioned Nash learning rule, or what is now called the Brown–von Neumann–Nash (BNN) dynamics also can be interpreted as a form of reinforcement learning (Brown and von Neumann 1950; Skyrms 1990).

  4. Similar logic applies to repelling equilibria: if the mean payoff function is strictly convex, then a possible interior Nash Equilibrium must be repelling for each dynamic. Hofbauer and Sigmund’s theorem (2003) follows from earlier work with each dynamic (Hofbauer and Sigmund 1998; Hofbauer 2000; Hopkins 1999).

  5. Another set of sufficient conditions might allow π 13 > 0, but would then require additional conditions to ensure that the best response dynamics avoids selecting (1, 0, 0).

  6. These dynamics are also termed sign-preserving (Nachbar 1990).

  7. Theorem 5 would hold without these requirements, but with the possibility that the one-sided payoff positive dynamics have measure zero basins of attraction for all strict equilibria. We want to focus on the case that the one-sided payoff positive dynamic selects a different equilibria than the threshold dynamic.

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Authors and Affiliations

  1. Department of Applied and Interdisciplinary Mathematics, University of Michigan, 2082 East Hall, 530 Church Street, Ann Arbor, MI, 48109, USA

    Russell Golman

  2. Center for the Study of Complex Systems, University of Michigan, 321A West Hall, 1085 S. University Ave., Ann Arbor, MI, 48109, USA

    Scott E. Page

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  1. Russell Golman
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  2. Scott E. Page
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Correspondence to Russell Golman.

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This work was funded by an NSF-IGERT grant and a U.S. Air Force MURI grant.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00191-009-0151-y

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Golman, R., Page, S.E. Basins of attraction and equilibrium selection under different learning rules. J Evol Econ 20, 49–72 (2010). https://doi.org/10.1007/s00191-009-0136-x

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