The Robustness of Hybrid Equilibria in Costly Signaling Games

Abstract

Recent work on costly signaling games has identified new Nash equilibria in addition to the standard costly signaling equilibrium as a possible explanation for signaling behavior. These so-called hybrid equilibria are Liapunov stable, but not asymptotically stable for the replicator dynamics. Since some eigenvalues of the hybrid equilibria have zero real part, this result is not structurally stable. The purpose of this paper is to show that under one reasonable perturbation of the replicator dynamics—the selection–mutation dynamics—rest points close to the hybrid equilibrium exist and are asymptotically stable. Moreover, for another plausible version of the replicator dynamics—Maynard Smith’s adjusted replicator dynamics—the same is true. This reinforces the significance of hybrid equilibria for signaling.

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Notes

  1. 1.

    The order of the publications is somewhat misleading since Wagner was actually the first to recognize the significance of hybrid equilibria for evolutionary games.

  2. 2.

    In the context of signaling games, see [12] and [13] for a general selection–mutation dynamics.

  3. 3.

    The Jacobian matrix of the replicator dynamics is the matrix of partial derivatives of (1) with respect to \(x_1, \ldots , x_4,y_1, \ldots , y_4\). A rest point of the replicator dynamics (2) is hyperbolic if no eigenvalue of the Jacobian matrix of (1) evaluated at the rest point has zero real part.

  4. 4.

    By the eigenvalues of the selection–mutation dynamics, we mean the eigenvalues of the Jacobian matrix of (2). The role of the implicit function theorem is explained in, e.g., [6].

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Acknowledgments

We would like to thank Josef Hofbauer and Carl Bergstrom for helpful comments. This material is based upon work supported by the National Science Foundation under Grant No. EF 1038456. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Correspondence to Simon M. Huttegger.

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Huttegger, S.M., Zollman, K.J.S. The Robustness of Hybrid Equilibria in Costly Signaling Games. Dyn Games Appl 6, 347–358 (2016). https://doi.org/10.1007/s13235-015-0159-x

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Keywords

  • Costly signaling games
  • Hybrid equilibrium
  • Replicator dynamics
  • Structural stability
  • Selection–mutation dynamics