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Condorcet domains, median graphs and the single-crossing property

Author

Listed:
  • Clemens Puppe

    () (Karlsruhe Institute of Technology (KIT))

  • Arkadii Slinko

    () (The University of Auckland)

Abstract

Abstract Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters of a society belong to this set, their majority relation has no cycles. We observe that, without loss of generality, every such domain can be assumed to be closed in the sense that it contains the majority relation of every profile with an odd number of voters whose preferences belong to this domain. We show that every closed Condorcet domain can be endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (in general, not uniquely). Condorcet domains that have linear graphs (chains) associated with them are exactly the preference domains with the classical single-crossing property. As a corollary, we obtain that a domain with the so-called ‘representative voter property’ is either a single-crossing domain or a very special domain containing exactly four different preference orders whose associated median graph is a 4-cycle. Maximality of a Condorcet domain imposes additional restrictions on the associated median graph. We prove that among all trees only (some) chains can be associated graphs of maximal Condorcet domains, and we characterize those single-crossing domains which are maximal Condorcet domains. Finally, using the characterization of Nehring and Puppe (J Econ Theory 135:269–305, 2007) of monotone Arrovian aggregation, we show that any closed Condorcet domain admits a rich class of strategy-proof social choice functions.

Suggested Citation

  • Clemens Puppe & Arkadii Slinko, 2019. "Condorcet domains, median graphs and the single-crossing property," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(1), pages 285-318, February.
  • Handle: RePEc:spr:joecth:v:67:y:2019:i:1:d:10.1007_s00199-017-1084-6
    DOI: 10.1007/s00199-017-1084-6
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    References listed on IDEAS

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    1. Gabrielle Demange, 2012. "Majority relation and median representative ordering," SERIEs: Journal of the Spanish Economic Association, Springer;Spanish Economic Association, vol. 3(1), pages 95-109, March.
    2. Saporiti, Alejandro, 2009. "Strategy-proofness and single-crossing," Theoretical Economics, Econometric Society, vol. 4(2), June.
    3. Gans, Joshua S. & Smart, Michael, 1996. "Majority voting with single-crossing preferences," Journal of Public Economics, Elsevier, vol. 59(2), pages 219-237, February.
    4. Peter C. Fishburn, 2002. "Acyclic sets of linear orders: A progress report," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(2), pages 431-447.
    5. Nehring, Klaus & Puppe, Clemens, 2007. "The structure of strategy-proof social choice -- Part I: General characterization and possibility results on median spaces," Journal of Economic Theory, Elsevier, vol. 135(1), pages 269-305, July.
    6. Ádám Galambos & Victor Reiner, 2008. "Acyclic sets of linear orders via the Bruhat orders," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(2), pages 245-264, February.
    7. H. Moulin, 1980. "On strategy-proofness and single peakedness," Public Choice, Springer, vol. 35(4), pages 437-455, January.
    8. Gabrielle Demange, 2011. "Majority relation and median representative ordering," PSE Working Papers halshs-00581310, HAL.
    9. Grandmont, Jean-Michel, 1978. "Intermediate Preferences and the Majority Rule," Econometrica, Econometric Society, vol. 46(2), pages 317-330, March.
    10. Roberts, Kevin W. S., 1977. "Voting over income tax schedules," Journal of Public Economics, Elsevier, vol. 8(3), pages 329-340, December.
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    More about this item

    Keywords

    Social choice; Condorcet domains; Acyclic sets of linear orders; Median graphs; Single-crossing property; Distributive lattice; Arrovian aggregation; Strategy-proofness; Intermediate preferences;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

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