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Symmetric Spatial Games Without Majority Rule Equilibria


  • McKelvey, Richard D.
  • Ordeshook, Peter C.


The assumptions imposed in spatial models of election competition generally are restrictive in that they require either unidimensional issue spaces or symmetrically distributed electorate preferences. We attribute such assumptions to the reliance of these models on a single concept of a solution to the election game—pure strategy equilibria—and to the fact that such equilibria do not exist in general under less severe restrictions. This essay considers, then, the possibility that candidates adopt mixed minimax strategies. We show, for a general class of symmetric zero-sum two-person games, that the domain of these minimax strategies is restricted to a subset of the strategy space and that for spatial games this set not only exists, but if preferences are characterized by continuous densities, it is typically small. Thus, the hypothesis that candidates abide by mixed minimax strategies can limit considerably our expectation as to the policies candidates eventually advocate. Additionally, we examine the frequently blurred distinction between spatial conceptualizations of two-candidate elections and of committees, and we conclude that if pure strategy equilibria do not exist, this distinction is especially important since committees and elections can produce entirely different outcomes.

Suggested Citation

  • McKelvey, Richard D. & Ordeshook, Peter C., 1976. "Symmetric Spatial Games Without Majority Rule Equilibria," American Political Science Review, Cambridge University Press, vol. 70(4), pages 1172-1184, December.
  • Handle: RePEc:cup:apsrev:v:70:y:1976:i:04:p:1172-1184_17

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    Cited by:

    1. James Enelow & Melvin Hinisch, 1983. "On Plott's pairwise symmetry condition for majority rule equilibrium," Public Choice, Springer, vol. 40(3), pages 317-321, January.
    2. Brandt, Felix & Brill, Markus & Suksompong, Warut, 2016. "An ordinal minimax theorem," Games and Economic Behavior, Elsevier, vol. 95(C), pages 107-112.
    3. John Duggan & Michel Le Breton, 2014. "Choice-theoretic Solutions for Strategic Form Games," RCER Working Papers 580, University of Rochester - Center for Economic Research (RCER).
    4. Dimitrios Xefteris, 2014. "Mixed equilibriums in a three-candidate spatial model with candidate valence," Public Choice, Springer, vol. 158(1), pages 101-120, January.
    5. Paulo Pereira, 2000. "From Schumpeterian Democracy to Constitutional Democracy," Constitutional Political Economy, Springer, vol. 11(1), pages 69-86, March.
    6. Lee Dutter, 1981. "Voter preferences, simple electoral games, and equilibria in two-candidate contests," Public Choice, Springer, vol. 37(3), pages 403-423, January.
    7. Bärbel M. R. Stadler, 1998. "Abstention Causes Bifurcations in Two-Party Voting Dynamics," Working Papers 98-08-072, Santa Fe Institute.
    8. John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    9. Ivo Bischoff, 2005. "Party competition in a heterogeneous electorate: The role of dominant-issue voters," Public Choice, Springer, vol. 122(1), pages 221-243, January.
    10. Norman Schofield, 1980. "Formal political theory," Quality & Quantity: International Journal of Methodology, Springer, vol. 14(1), pages 249-275, January.
    11. Tanner, Thomas Cole, 1994. "The spatial theory of elections: an analysis of voters' predictive dimensions and recovery of the underlying issue space," ISU General Staff Papers 1994010108000018174, Iowa State University, Department of Economics.

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