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Mixed refinements of Shapley's saddles and weak tournaments

Author

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  • Michel Le Breton

    (CORE, Voie du Roman Pays, 34, 1348, Louvain La Neuve, Belgium)

  • John Duggan

    (Department of Political Science and Department of Economics, University of Rochester, Rochester, NY 14627, USA)

Abstract

We investigate refinements of two solutions, the saddle and the weak saddle, defined by Shapley (1964) for two-player zero-sum games. Applied to weak tournaments, the first refinement, the mixed saddle, is unique and gives us a new solution, generally lying between the GETCHA and GOTCHA sets of Schwartz (1972, 1986). In the absence of ties, all three solutions reduce to the usual top cycle set. The second refinement, the weak mixed saddle, is not generally unique, but, in the absence of ties, it is unique and coincides with the minimal covering set.

Suggested Citation

  • Michel Le Breton & John Duggan, 2001. "Mixed refinements of Shapley's saddles and weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(1), pages 65-78.
    • Handle: RePEc:spr:sochwe:v:18:y:2001:i:1:p:65-78
    Note: Received: 14 August 1998/Accepted: 12 November 1999
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    Cited by:

    1. Vincent Anesi, 2012. "A new old solution for weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 919-930, October.
    2. Brandt, Felix & Brill, Markus & Suksompong, Warut, 2016. "An ordinal minimax theorem," Games and Economic Behavior, Elsevier, vol. 95(C), pages 107-112.
    3. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2002. "Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections," Journal of Economic Theory, Elsevier, vol. 103(1), pages 88-105, March.
    4. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2006. "Social choice and electoral competition in the general spatial model," Journal of Economic Theory, Elsevier, vol. 126(1), pages 194-234, January.
    5. Lavi, Ron & Nisan, Noam, 2015. "Online ascending auctions for gradually expiring items," Journal of Economic Theory, Elsevier, vol. 156(C), pages 45-76.
    6. John Duggan, 2007. "A systematic approach to the construction of non-empty choice sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(3), pages 491-506, April.
    7. John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    8. Felix Brandt & Markus Brill & Paul Harrenstein, 2018. "Extending tournament solutions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(2), pages 193-222, August.

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