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Understanding some recent existence results for discontinuous games

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Abstract

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (Econometrica 67:1029–1056, 1999) and Carmona (J Econ Theory 144:1333–1340, 2009) and that it is equivalent to a recent result of Barelli and Soza (On the Existence of Nash Equilibria in Discontinuous and Qualitative Games, University of Rochester, Rochester, 2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property.

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Authors and Affiliations

  1. Faculty of Economics and King’s College, University of Cambridge, Sidgwick Avenue, Cambridge, CB3 9DD, UK

    Guilherme Carmona

  2. Faculdade de Economia, Universidade Nova de Lisboa, Lisbon, Portugal

    Guilherme Carmona

Authors
  1. Guilherme Carmona
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Correspondence to Guilherme Carmona.

Additional information

I wish to thank Paulo Barelli, John Campbell, Luciano de Castro, Bob Evans, Andy McLennan, Pavlo Prokopovych, Philip Reny and participants at the Creta Workshop 2009 (University of Warwick) and the 18th European Workshop on General Equilibrium (Universitat Pompeu Fabra) for very helpful comments.

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Carmona, G. Understanding some recent existence results for discontinuous games. Econ Theory 48, 31–45 (2011). https://doi.org/10.1007/s00199-010-0532-3

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  • DOI: https://doi.org/10.1007/s00199-010-0532-3

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