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Choice on the simplex domain

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  • Bossert, Walter
  • Peters, Hans

Abstract

One unit of a good has to be divided among a group of agents who each are entitled to a minimal share, and these shares sum up to less than one. The associated set of choice problems consists of the unit simplex and all its full-dimensional subsimplices with the same orientation. We characterize all choice rules that are independent of irrelevant alternatives, continuous, and monotonic; the last condition means that if an agent receives its minimal share and that share increases, then no other agent benefits. In line with Kıbrıs (2012) we show that these rules are rationalizable and representable by a real-valued function. On the issue of rationalizability, we also consider weakenings of our conditions. In particular, we show that in general, excluding cycles of any fixed length does not imply the strong axiom of revealed preference, that is, the exclusion of cycles of any length. For continuous three-agent choice rules, however, excluding cycles of length three implies the strong axiom of revealed preference.

Suggested Citation

  • Bossert, Walter & Peters, Hans, 2019. "Choice on the simplex domain," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 63-72.
    • Handle: RePEc:eee:mateco:v:84:y:2019:i:c:p:63-72
    DOI: 10.1016/j.jmateco.2019.07.002
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    References listed on IDEAS

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    1. Walter Bossert & Hans Peters, 2009. "Single-peaked choice," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 41(2), pages 213-230, November.
    2. Stovall, John E., 2014. "Collective rationality and monotone path division rules," Journal of Economic Theory, Elsevier, vol. 154(C), pages 1-24.
    3. Bossert, Walter, 1994. "Rational choice and two-person bargaining solutions," Journal of Mathematical Economics, Elsevier, vol. 23(6), pages 549-563, November.
    4. Bossert, Walter & Suzumura, Kotaro, 2010. "Consistency, Choice, and Rationality," Economics Books, Harvard University Press, number 9780674052994, Spring.
    5. Blackorby, Charles & Bossert, Walter & Donaldson, David, 1995. "Multi-valued demand and rational choice in the two-commodity case," Economics Letters, Elsevier, vol. 47(1), pages 5-10, January.
    6. Hugh Rose, 1958. "Consistency of Preference: The Two-Commodity Case," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(2), pages 124-125.
    7. Peters, Hans & Wakker, Peter, 1991. "Independence of Irrelevant Alternatives and Revealed Group Preferences," Econometrica, Econometric Society, vol. 59(6), pages 1787-1801, November.
    8. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    9. Thomson, William, 2015. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update," Mathematical Social Sciences, Elsevier, vol. 74(C), pages 41-59.
    10. Peters Hans & Wakker Peter, 1994. "WARP Does Not Imply SARP for More Than Two Commodities," Journal of Economic Theory, Elsevier, vol. 62(1), pages 152-160, February.
    11. Özgür Kıbrıs, 2012. "A revealed preference analysis of solutions to simple allocation problems," Theory and Decision, Springer, vol. 72(4), pages 509-523, April.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Choice rules; Simplex domain; Rationalizability; Representability;
    All these keywords.

    JEL classification:

    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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