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Counting Combinatorial Choice Rules

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  • Federico Echenique

    (California Institute of Technology)

Abstract

I count the number of combinatorial choice rules that satisfy certain properties: Kelso-Crawford substitutability, and independence of irrelevant alternatives. The results are important for two-sided matching theory, where agents are modeled by combinatorial choice rules with these properties. The rules are a small, and asymptotically vanishing, fraction of all choice rules. But they are still exponentially more than the preference relations over individual agents- --which has positive implications for the Gale-Shapley algorithm of matching theory.

Suggested Citation

  • Federico Echenique, 2004. "Counting Combinatorial Choice Rules," Game Theory and Information 0404004, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpga:0404004
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    References listed on IDEAS

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    1. Kim, Taesung & Richter, Marcel K., 1986. "Nontransitive-nontotal consumer theory," Journal of Economic Theory, Elsevier, vol. 38(2), pages 324-363, April.
    2. Bevia, Carmen & Quinzii, Martine & Silva, Jose A., 1999. "Buying several indivisible goods," Mathematical Social Sciences, Elsevier, vol. 37(1), pages 1-23, January.
    3. Charles Blair, 1988. "The Lattice Structure of the Set of Stable Matchings with Multiple Partners," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 619-628, November.
    4. Gul, Faruk & Stacchetti, Ennio, 2000. "The English Auction with Differentiated Commodities," Journal of Economic Theory, Elsevier, vol. 92(1), pages 66-95, May.
    5. Gul, Faruk & Stacchetti, Ennio, 1999. "Walrasian Equilibrium with Gross Substitutes," Journal of Economic Theory, Elsevier, vol. 87(1), pages 95-124, July.
    6. Roth, Alvin E. & Sotomayor, Marilda, 1992. "Two-sided matching," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 16, pages 485-541, Elsevier.
    7. Moulin,Hervi, 1991. "Axioms of Cooperative Decision Making," Cambridge Books, Cambridge University Press, number 9780521424585.
    8. Ralph W. Bailey, 1998. "The number of weak orderings of a finite set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(4), pages 559-562.
    9. Roth, Alvin E, 1984. "Stability and Polarization of Interests in Job Matching," Econometrica, Econometric Society, vol. 52(1), pages 47-57, January.
    10. Kelso, Alexander S, Jr & Crawford, Vincent P, 1982. "Job Matching, Coalition Formation, and Gross Substitutes," Econometrica, Econometric Society, vol. 50(6), pages 1483-1504, November.
    11. John W. Hatfield & Paul Milgrom, 2005. "Auctions, Matching and the Law of Aggregate Demand," Levine's Bibliography 122247000000000780, UCLA Department of Economics.
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    Cited by:

    1. Samson Alva & Battal Dou{g}an, 2021. "Choice and Market Design," Papers 2110.15446, arXiv.org, revised Nov 2021.
    2. Echenique, Federico & Yenmez, M. Bumin, 2007. "A solution to matching with preferences over colleagues," Games and Economic Behavior, Elsevier, vol. 59(1), pages 46-71, April.
    3. Hans Peters & Panos Protopapas, 2021. "Set and revealed preference axioms for multi-valued choice," Theory and Decision, Springer, vol. 90(1), pages 11-29, February.
    4. Hideo Konishi & M. Utku Ünver, 2003. "Credible Group Stability in Multi-Partner Matching Problems," Working Papers 2003.115, Fondazione Eni Enrico Mattei.
    5. Juan F. Fung & Chia-Ling Hsu, 2021. "A cumulative offer process for supply chain networks," Review of Economic Design, Springer;Society for Economic Design, vol. 25(1), pages 93-109, June.
    6. Koji Yokote & Isa E. Hafalir & Fuhito Kojima & M. Bumin Yenmez, 2023. "Representation Theorems for Path-Independent Choice Rules," Papers 2303.00892, arXiv.org.
    7. Tamás Fleiner & Zsuzsanna Jankó & Ildikó Schlotter & Alexander Teytelboym, 2023. "Complexity of stability in trading networks," International Journal of Game Theory, Springer;Game Theory Society, vol. 52(3), pages 629-648, September.
    8. Battal Dou{g}an & Kenzo Imamura & M. Bumin Yenmez, 2022. "Market Design with Deferred Acceptance: A Recipe for Policymaking," Papers 2209.06777, arXiv.org.
    9. M. Bumin Yenmez, 2014. "College Admissions," GSIA Working Papers 2014-E24, Carnegie Mellon University, Tepper School of Business.
    10. Yenmez, M. Bumin, 2018. "A college admissions clearinghouse," Journal of Economic Theory, Elsevier, vol. 176(C), pages 859-885.
    11. Orhan Aygün & Tayfun Sönmez, 2012. "The Importance of Irrelevance of Rejected Contracts in Matching under Weakened Substitutes Conditions," Boston College Working Papers in Economics 805, Boston College Department of Economics.
    12. Mehmet Ekmekci & M. Bumin Yenmez, "undated". "Integrating Schools for Centralized Admissions," GSIA Working Papers 2014-E20, Carnegie Mellon University, Tepper School of Business.
    13. Tam'as Fleiner & Zsuzsanna Jank'o & Ildik'o Schlotter & Alexander Teytelboym, 2018. "Complexity of Stability in Trading Networks," Papers 1805.08758, arXiv.org, revised Feb 2019.
    14. Segal, Ilya, 2007. "The communication requirements of social choice rules and supporting budget sets," Journal of Economic Theory, Elsevier, vol. 136(1), pages 341-378, September.
    15. Doğan, Battal & Yenmez, M. Bumin, 2019. "Unified versus divided enrollment in school choice: Improving student welfare in Chicago," Games and Economic Behavior, Elsevier, vol. 118(C), pages 366-373.
    16. Tam'as Fleiner & Zsuzsanna Jank'o & Akihisa Tamura & Alexander Teytelboym, 2015. "Trading Networks with Bilateral Contracts," Papers 1510.01210, arXiv.org, revised May 2018.
    17. Erdil, Aytek & Kumano, Taro, 2019. "Efficiency and stability under substitutable priorities with ties," Journal of Economic Theory, Elsevier, vol. 184(C).
    18. Ilya Segal, 2004. "The Communication Requirements of of Social Choice Rules and Supporting Budget Sets," Economics Working Papers 0039, Institute for Advanced Study, School of Social Science.
    19. Alva, Samson, 2018. "WARP and combinatorial choice," Journal of Economic Theory, Elsevier, vol. 173(C), pages 320-333.
    20. Stefano Vannucci, 2011. "Widwast Choice," Department of Economics University of Siena 629, Department of Economics, University of Siena.

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    More about this item

    Keywords

    Substitutability; Choice rules; Matching markets; Gale-Shapley Algorithm;
    All these keywords.

    JEL classification:

    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory

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