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Gains from switching and evolutionary stability in multi-player matrix games

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  • Laurent Lehmann
  • Georg Nöldeke

    ()

  • Jorge Peña

    (University of Basel)

Abstract

In this paper we unify, simplify, and extend previous work on the evolutionary dynamics of symmetric N-player matrix games with two pure strategies. In such games, gains from switching strategies depend, in general, on how many other individuals in the group play a given strategy. As a consequence, the gain function determining the gradient of selection can be a polynomial of degree N-1. In order to deal with the intricacy of the resulting evolutionary dynamics, we make use of the theory of polynomials in Bernstein form. This theory implies a tight link between the sign pattern of the gains from switching on the one hand and the number and stability properties of the rest points of the replicator dynamics on the other hand. While this relationship is a general one, it is most informative if gains from switching have at most two sign changes, as it is the case for most multi-player matrix games considered in the literature. We demonstrate that previous results for public goods games are easily recovered and extended using this observation. Further examples illustrate how focusing on the sign pattern of the gains from switching obviates the need for a more involved analysis.

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Bibliographic Info

Paper provided by Faculty of Business and Economics - University of Basel in its series Working papers with number 2013/13.

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Date of creation: 2013
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Handle: RePEc:bsl:wpaper:2013/13

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Keywords: evolutionary game theory; multi-player matrix games; replicator dynamics; public goods games; gains from switching; polynomials in Bernstein form 2;

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  1. Florian Herold, 2012. "Carrot or Stick? The Evolution of Reciprocal Preferences in a Haystack Model," American Economic Review, American Economic Association, vol. 102(2), pages 914-40, April.
  2. J. A. Cuesta & R. Jimenez & H. Lugo & A. Sanchez, 2007. "Rewarding cooperation in social dilemmas," Economics Working Papers we075227, Universidad Carlos III, Departamento de Economía.
  3. Ross Cressman, 2003. "Evolutionary Dynamics and Extensive Form Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262033054, December.
  4. U. Dieckmann & R. Law, 1996. "The Dynamical Theory of Coevolution: A Derivation from Stochastic Ecological Processes," Working Papers wp96001, International Institute for Applied Systems Analysis.
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