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

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  • Richard Ball

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

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    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.

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    Bibliographic Info

    Article provided by Springer in its journal Social Choice and Welfare.

    Volume (Year): 16 (1999)
    Issue (Month): 4 ()
    Pages: 533-555

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    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. Michalis Drouvelis & Alejandro Saporiti & Nicolaas J. Vriend, 2011. "Political Motivations and Electoral Competition: Equilibrium Analysis and Experimental Evidence," The School of Economics Discussion Paper Series 1119, Economics, The University of Manchester.
    2. Pavlo Prokopovych & Nicholas C.Yannelis, 2013. "On the Existence of Mixed Strategy Nash equilibria," Discussion Papers 50, Kyiv School of Economics.
    3. Dolmas, Jim, 2014. "Almost orthogonal outcomes under probabilistic voting: A cautionary example," MPRA Paper 53628, University Library of Munich, Germany.
    4. Alejandro Saporiti, 2013. "Power Sharing and Electoral Equilibrium," The School of Economics Discussion Paper Series 1301, Economics, The University of Manchester.
    5. 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.
    6. Alejandro Saporiti, 2005. "On the existence of Nash equilibrium in electoral competition," Game Theory and Information 0504005, EconWPA.
    7. Marcus Berliant & Hideo Konishi, 2004. "Salience: Agenda Choices by Competing Candidates," Game Theory and Information 0407003, EconWPA.

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