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Simple equilibria in finite games with convexity properties

Volume 42 / 2015

Tadeusz Radzik, Piotr Więcek Applicationes Mathematicae 42 (2015), 83-109 MSC: 91A05, 91A06, 91A10, 91B52. DOI: 10.4064/am42-1-6

Abstract

This review paper gives a characterization of non-coalitional zero-sum and non-zero-sum games with finite strategy spaces and payoff functions having some concavity or convexity properties. The characterization is given in terms of the existence of two-point Nash equilibria, that is, equilibria consisting of mixed strategies with spectra consisting of at most two pure strategies. The structure of such simple equilibria is discussed in various cases. In particular, many of the results discussed can be seen as discrete counterparts of classical theorems about the existence of pure (or “almost pure”) Nash equilibria in continuous concave (convex) games with compact convex spaces of pure strategies. The paper provides many examples illustrating the results presented and ends with four related open problems.

Authors

  • Tadeusz RadzikInstitute of Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Piotr WięcekInstitute of Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland

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