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Discontinuity and non-existence of equilibrium in the probabilistic spatial voting model

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  • Richard Ball

    () (Department of Economics, Haverford College, Haverford, PA 19041, USA)

Abstract

This paper shows that in the simplest one-dimensional, two-candidate probabilistic spatial voting model (PSVM), a pure strategy Nash equilibrium may fail to exist. The existence problem studied here is the result of a discontinuity in the function mapping the candidates' platforms into their probabilities of winning. Proposition 1 of the paper shows that, whenever this probability of winning function satisfies a certain monotonicity property, it must be discontinuous on the diagonal. As an immediate consequence of the discontinuity in the probability of winning function, the candidates' objective functions are discontinuous as well. It is therefore impossible to invoke standard theorems guaranteeing the existence of a pure strategy equilibrium, and an example is developed in which in fact there is no pure strategy equilibrium. Finally, however, it is demonstrated that, for a large class of probability of winning functions, the PSVM satisfies all the conditions of a theorem of Dasgupta and Maskin (1986a) which guarantees that it will always have an equilibrium in mixed strategies.

Suggested Citation

  • Richard Ball, 1999. "Discontinuity and non-existence of equilibrium in the probabilistic spatial voting model," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(4), pages 533-555.
  • Handle: RePEc:spr:sochwe:v:16:y:1999:i:4:p:533-555
    Note: Received: 31 December 1996/Accepted: 12 May 1998
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    Cited by:

    1. Prokopovych, Pavlo & Yannelis, Nicholas C., 2014. "On the existence of mixed strategy Nash equilibria," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 87-97.
    2. Peter J. Coughlin, 2015. "Probabilistic voting in models of electoral competition," Chapters,in: Handbook of Social Choice and Voting, chapter 13, pages 218-234 Edward Elgar Publishing.
    3. Drouvelis, Michalis & Saporiti, Alejandro & Vriend, Nicolaas J., 2014. "Political motivations and electoral competition: Equilibrium analysis and experimental evidence," Games and Economic Behavior, Elsevier, vol. 83(C), pages 86-115.
    4. Alejandro Saporiti, 2008. "Existence and Uniqueness of Nash Equilibrium in Electoral Competition Games: The Hybrid Case," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 10(5), pages 827-857, October.
    5. repec:eee:mateco:v:74:y:2018:i:c:p:99-107 is not listed on IDEAS
    6. Bagh, Adib, 2014. "Candidates' Uncertainty and Error Distribution Models in Electoral Competitions," MPRA Paper 77631, University Library of Munich, Germany.
    7. Alejandro Saporiti, 2010. "Power, ideology, and electoral competition," The School of Economics Discussion Paper Series 1003, Economics, The University of Manchester.
    8. Marcus Berliant & Hideo Konishi, 2005. "Salience: Agenda choices by competing candidates," Public Choice, Springer, vol. 125(1), pages 129-149, July.
    9. Alejandro Saporiti, 2014. "Power sharing and electoral equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 55(3), pages 705-729, April.
    10. Duggan, John, 2007. "Equilibrium existence for zero-sum games and spatial models of elections," Games and Economic Behavior, Elsevier, vol. 60(1), pages 52-74, July.
    11. Alejandro Saporiti, 2005. "On the existence of Nash equilibrium in electoral competition," Game Theory and Information 0504005, EconWPA.
    12. Dolmas, Jim, 2014. "Almost orthogonal outcomes under probabilistic voting: A cautionary example," MPRA Paper 53628, University Library of Munich, Germany.

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