An existence result for discontinuous games

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, 1: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity. (C) 2008 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)1333-1340
JournalJournal of Economic Theory
Volume144
Issue number3
DOIs
Publication statusPublished - 1 Jan 2009

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Discontinuous games
Upper semicontinuity
Existence of equilibrium
Economics
Pure strategy equilibrium
Mixed strategy equilibrium
Pure strategies
Nash equilibrium
Normal form games
Guarantee

Keywords

  • Discontinuous games
  • Nash equilibrium
  • Upper semicontinuity
  • Lower semicontinuity

Cite this

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abstract = "We introduce a notion of upper semicontinuity, weak upper semicontinuity, and show that it, together with a weak form of payoff security, is enough to guarantee the existence of Nash equilibria in compact, quasiconcave normal form games. We show that our result generalizes the pure strategy existence theorem of Dasgupta and Maskin [P. Dasgupta, E. Maskin, The existence of equilibrium in discontinuous economic games, 1: Theory, Rev. Econ. Stud. 53 (1986) 1-26] and that it is neither implied nor does it imply the existence theorems of Baye, Tian, and Zhou [M. Baye, G. Tian, J. Zhou, Characterizations of the existence of equilibria in games with discontinuous and non-quasiconcave payoffs, Rev. Econ. Stud. 60 (1993) 935-948] and Reny [P. Reny, On the existence of pure and mixed strategy equilibria in discontinuous games, Econometrica 67 (1999) 1029-1056]. Furthermore, we show that an equilibrium may fail to exist when, while maintaining weak payoff security, weak upper semicontinuity is weakened to reciprocal upper semicontinuity. (C) 2008 Elsevier Inc. All rights reserved.",
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An existence result for discontinuous games. / Carmona, Guilherme.

In: Journal of Economic Theory, Vol. 144, No. 3, 01.01.2009, p. 1333-1340.

Research output: Contribution to journalArticle

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