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Mixed refinements of Shapley’s saddles and weak tournaments

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  • DUGGAN, John

    (Department of Political Science and Department of Economics University of Rochester Rochester, NY 14627 U.S.A.)

  • LE BRETON, Michel

    (Center for Operations Research and Econometrics (CORE), Université catholique de Louvain (UCL), 1348 Louvain la Neuve, Belgium)

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

  • DUGGAN, John & LE BRETON, Michel, 1999. "Mixed refinements of Shapley’s saddles and weak tournaments," LIDAM Discussion Papers CORE 1999021, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1999021
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1999.html
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    Cited by:

    1. 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.
    2. 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.
    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. Brandt, Felix & Brill, Markus & Suksompong, Warut, 2016. "An ordinal minimax theorem," Games and Economic Behavior, Elsevier, vol. 95(C), pages 107-112.
    5. John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    6. 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.
    7. 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.
    8. Lavi, Ron & Nisan, Noam, 2015. "Online ascending auctions for gradually expiring items," Journal of Economic Theory, Elsevier, vol. 156(C), pages 45-76.

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