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Uncovered sets

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  • John Duggan

Abstract

This paper covers the theory of the uncovered set used in the literatures on tournaments and spatial voting. I discern three main extant definitions, and I introduce two concepts that bound existing sets from above and below: the deep uncovered set and the shallow uncovered set. In a general topological setting, I provide relationships to other solutions and give results on existence and external stability for all of the covering concepts, and I establish continuity properties of the two new uncovered sets. Of note, I characterize each of the uncovered sets in terms of a decomposition into choices from externally stable sets, and I define the minimal generalized covering solution, a nonempty refinement of the deep uncovered set that employs both of the new relations. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • John Duggan, 2013. "Uncovered sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(3), pages 489-535, September.
    • Handle: RePEc:spr:sochwe:v:41:y:2013:i:3:p:489-535
    DOI: 10.1007/s00355-012-0696-9
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    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. Duggan, John, 1999. "A General Extension Theorem for Binary Relations," Journal of Economic Theory, Elsevier, vol. 86(1), pages 1-16, May.
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    7. Laffond G. & Laslier J. F. & Le Breton M., 1993. "The Bipartisan Set of a Tournament Game," Games and Economic Behavior, Elsevier, vol. 5(1), pages 182-201, January.
    8. Bordes, Georges, 1983. "On the possibility of reasonable consistent majoritarian choice: Some positive results," Journal of Economic Theory, Elsevier, vol. 31(1), pages 122-132, October.
    9. Josep E. Peris & BegoÓa Subiza, 1999. "Condorcet choice correspondences for weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(2), pages 217-231.
    10. Georges Bordes & Michel Le Breton & Maurice Salles, 1992. "Gillies and Miller's Subrelations of a Relation over an Infinite Set of Alternatives: General Results and Applications to Voting Games," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 509-518, August.
    11. Bhaskar Dutta & Jean-Francois Laslier, 1999. "Comparison functions and choice correspondences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 16(4), pages 513-532.
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    14. Brandt, Felix & Fischer, Felix, 2008. "Computing the minimal covering set," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 254-268, September.
    15. Elizabeth Penn, 2006. "The Banks Set in Infinite Spaces," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(3), pages 531-543, December.
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    Cited by:

    1. Hannu Vartiainen, 2015. "Dynamic stable set as a tournament solution," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 309-327, September.
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    3. Weibin Han & Adrian Deemen & D. Ary A. Samsura, 2016. "A note on extended stable sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(2), pages 265-275, August.
    4. Nicholas R. Miller, 2015. "The spatial model of social choice and voting," Chapters, in: Jac C. Heckelman & Nicholas R. Miller (ed.), Handbook of Social Choice and Voting, chapter 10, pages 163-181, Edward Elgar Publishing.

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