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Families of abstract decision problems whose admissible sets intersect in a singleton

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  • Michele Gori

    (Università degli Studi di Firenze)

Abstract

An abstract decision problem is an ordered pair where the first component is a nonempty and finite set of alternatives and the second component is an irreflexive relation on that set, called dominance relation. The admissible set of an abstract decision problem is the set of the maximal elements of the reflexive and transitive closure of the dominance relation. Given a finite sequence of abstract decision problems on the same set of alternatives, we give conditions on the dominance relations that guarantee that the intersection of all the admissible sets of the considered problems is a singleton as well as conditions that guarantee that the intersection is nonempty. We show then that such results allow to deduce some interesting facts about the resoluteness of the Schulze network solution and the Schulze social choice correspondence as well as some information about the existence of a (unique) common recurrent state for finite families of discrete-time homogeneous Markov chains.

Suggested Citation

  • Michele Gori, 2023. "Families of abstract decision problems whose admissible sets intersect in a singleton," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 61(1), pages 131-154, July.
    • Handle: RePEc:spr:sochwe:v:61:y:2023:i:1:d:10.1007_s00355-022-01443-1
    DOI: 10.1007/s00355-022-01443-1
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    References listed on IDEAS

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    1. Robert Delver & Herman Monsuur, 2001. "Stable sets and standards of behaviour," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 555-570.
    2. Kalai, Ehud & Pazner, Elisha A & Schmeidler, David, 1976. "Collective Choice Correspondences as Admissible Outcomes of Social Bargaining Processes," Econometrica, Econometric Society, vol. 44(2), pages 233-240, March.
    3. Julio González-Díaz & Ruud Hendrickx & Edwin Lohmann, 2014. "Paired comparisons analysis: an axiomatic approach to ranking methods," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 139-169, January.
    4. Peris, Josep E. & Subiza, Begoña, 2013. "A reformulation of von Neumann–Morgenstern stability: m-stability," Mathematical Social Sciences, Elsevier, vol. 66(1), pages 51-55.
    5. Daniela Bubboloni & Michele Gori, 2018. "The flow network method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(4), pages 621-656, December.
    6. Markus Schulze, 2011. "A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(2), pages 267-303, February.
    7. Kalai, Ehud & Schmeidler, David, 1977. "An admissible set occurring in various bargaining situations," Journal of Economic Theory, Elsevier, vol. 14(2), pages 402-411, April.
    8. Han, Weibin & Van Deemen, Adrian, 2016. "On the solution of w-stable sets," Mathematical Social Sciences, Elsevier, vol. 84(C), pages 87-92.
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