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Domains, Ranges and Strategy-Proofness: The Case of Single-dipped Preferences

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  • Salvador Barberà
  • Dolors Berga
  • Bernardo Moreno

Abstract

We characterize the set of all individual and group strategy-proof rules on the domain of all single-dipped preferences on a line. For rules defined on this domain, and on several of its subdomains, we explore the implications of these strategy-proofness requirements on the maximum size of the rules' range. We show that when all single-dipped preferences are admissible, the range must contain two alternatives at most. But this bound changes as we consider different subclasses of single-dipped preferences: we provide examples of subdomains admitting strategy-proof rules with larger ranges. We establish exact bounds on the maximal size of strategy-proof functions on each of these domains, and prove that the relationship between the sizes of the subdomains and those of the ranges of strategy-proof functions on them need not be monotonic. Our results exhibit a sharp contrast between the structure of strategy-proof rules defined on subdomains of single-dipped preferences and those defined on subsets of single-peaked ones.

Suggested Citation

  • Salvador Barberà & Dolors Berga & Bernardo Moreno, 2009. "Domains, Ranges and Strategy-Proofness: The Case of Single-dipped Preferences," Working Papers 418, Barcelona Graduate School of Economics.
  • Handle: RePEc:bge:wpaper:418
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    References listed on IDEAS

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    1. Barberà, Salvador & Berga, Dolors & Moreno, Bernardo, 2010. "Individual versus group strategy-proofness: When do they coincide?," Journal of Economic Theory, Elsevier, vol. 145(5), pages 1648-1674, September.
    2. Salvador Barberà & Dolors Berga & Bernardo Moreno, 2012. "Group strategy-proof social choice functions with binary ranges and arbitrary domains: characterization results," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 791-808, November.
    3. Barbera, Salvador & Sonnenschein, Hugo & Zhou, Lin, 1991. "Voting by Committees," Econometrica, Econometric Society, vol. 59(3), pages 595-609, May.
    4. Satterthwaite, Mark Allen, 1975. "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions," Journal of Economic Theory, Elsevier, vol. 10(2), pages 187-217, April.
    5. Barbera, Salvador & Sonnenschein, Hugo & Zhou, Lin, 1991. "Voting by Committees," Econometrica, Econometric Society, vol. 59(3), pages 595-609, May.
    6. Sen, Amartya & Pattanaik, Prasanta K., 1969. "Necessary and sufficient conditions for rational choice under majority decision," Journal of Economic Theory, Elsevier, vol. 1(2), pages 178-202, August.
    7. Saari, Donald G. & Valognes, Fabrice, 1999. "The geometry of Black's single peakedness and related conditions," Journal of Mathematical Economics, Elsevier, vol. 32(4), pages 429-456, December.
    8. Manjunath, Vikram, 2012. "Group strategy-proofness and voting between two alternatives," Mathematical Social Sciences, Elsevier, vol. 63(3), pages 239-242.
    9. Gibbard, Allan, 1973. "Manipulation of Voting Schemes: A General Result," Econometrica, Econometric Society, vol. 41(4), pages 587-601, July.
    10. H. Moulin, 1980. "On strategy-proofness and single peakedness," Public Choice, Springer, vol. 35(4), pages 437-455, January.
    11. Larsson, Bo & Svensson, Lars-Gunnar, 2006. "Strategy-proof voting on the full preference domain," Mathematical Social Sciences, Elsevier, vol. 52(3), pages 272-287, December.
    12. Klaus, Bettina & Peters, Hans & Storcken, Ton, 1997. "Strategy-proof division of a private good when preferences are single-dipped," Economics Letters, Elsevier, vol. 55(3), pages 339-346, September.
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    Cited by:

    1. Thomson, William, 2011. "Chapter Twenty-One - Fair Allocation Rules," Handbook of Social Choice and Welfare,in: K. J. Arrow & A. K. Sen & K. Suzumura (ed.), Handbook of Social Choice and Welfare, edition 1, volume 2, chapter 21, pages 393-506 Elsevier.
    2. Öztürk, Murat & Peters, Hans & Storcken, Ton, 2013. "Strategy-proof location of a public bad on a disc," Economics Letters, Elsevier, vol. 119(1), pages 14-16.
    3. Bossert, Walter & Peters, Hans, 2014. "Single-basined choice," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 162-168.
    4. Stefan Ambec & Yann Kervinio, 2016. "Cooperative decision-making for the provision of a locally undesirable facility," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(1), pages 119-155, January.
    5. Peters, Hans & Roy, Souvik & Sadhukhan, Soumyarup & Storcken, Ton, 2017. "An extreme point characterization of strategy-proof and unanimous probabilistic rules over binary restricted domains," Journal of Mathematical Economics, Elsevier, vol. 69(C), pages 84-90.
    6. repec:eee:matsoc:v:90:y:2017:i:c:p:150-159 is not listed on IDEAS
    7. Rebelo, S., 1997. "On the Determinant of Economic Growth," RCER Working Papers 443, University of Rochester - Center for Economic Research (RCER).
    8. Yamamura, Hirofumi, 2016. "Coalitional stability in the location problem with single-dipped preferences: An application of the minimax theorem," Journal of Mathematical Economics, Elsevier, vol. 65(C), pages 48-57.
    9. Murat Öztürk & Hans Peters & Ton Storcken, 2014. "On the location of public bads: strategy-proofness under two-dimensional single-dipped preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(1), pages 83-108, May.
    10. Grisel Ayllón & Diego M. Caramuta, 2016. "Single-dipped preferences with satiation: strong group strategy-proofness and unanimity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(2), pages 245-264, August.
    11. Mishra, Debasis, 2016. "Ordinal Bayesian incentive compatibility in restricted domains," Journal of Economic Theory, Elsevier, vol. 163(C), pages 925-954.
    12. Lahiri, Abhinaba & Peters, Hans & Storcken, Ton, 2017. "Strategy-proof location of public bads in a two-country model," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 150-159.
    13. Jorge Alcalde Unzu & Marc Vorsatz, 2015. "Strategy-proof location of public facilities," Documentos de Trabajo - Lan Gaiak Departamento de Economía - Universidad Pública de Navarra 1502, Departamento de Economía - Universidad Pública de Navarra.
    14. Ahmed Doghmi, 2013. "Nash Implementation in an Allocation Problem with Single-Dipped Preferences," Games, MDPI, Open Access Journal, vol. 4(1), pages 1-12, January.

    More about this item

    Keywords

    Strategy-proof; group strategy-proof; binary range rules; single-dipped;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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