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Closure and Preferences

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  • Christopher Chambers
  • Alan Miller
  • M. Bumin Yenmez

Abstract

We investigate the results of Kreps (1979), dropping his completeness axiom. As an added generalization, we work on arbitrary lattices, rather than a lattice of sets. We show that one of the properties of Kreps is intimately tied with representation via a closure operator. That is, a preference satis es Kreps' axiom (and a few other mild conditions) if and only if there is a closure operator on the lattice, such that preferences over elements of the lattice coincide with dominance of their closures. We tie the work to recent literature by Richter and Rubinstein (2015). Finally, we carry the concept to the theory of path-independent choice functions.

Suggested Citation

  • Christopher Chambers & Alan Miller & M. Bumin Yenmez, 2015. "Closure and Preferences," GSIA Working Papers 2015-E36, Carnegie Mellon University, Tepper School of Business.
    • Handle: RePEc:cmu:gsiawp:-942662566
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    References listed on IDEAS

    as
    1. Christopher P. Chambers & Alan D. Miller, 2014. "Inefficiency Measurement," American Economic Journal: Microeconomics, American Economic Association, vol. 6(2), pages 79-92, May.
    2. Christopher P. Chambers & M. Bumin Yenmez, 2017. "Choice and Matching," American Economic Journal: Microeconomics, American Economic Association, vol. 9(3), pages 126-147, August.
    3. Grant, Simon & Kajii, Atsushi & Polak, Ben, 1998. "Intrinsic Preference for Information," Journal of Economic Theory, Elsevier, vol. 83(2), pages 233-259, December.
    4. Richter, Michael & Rubinstein, Ariel, 2019. ""Convex preferences": a new definition," Theoretical Economics, Econometric Society, vol. 14(4), November.
    5. Michael Richter & Ariel Rubinstein, 2015. "Back to Fundamentals: Equilibrium in Abstract Economies," American Economic Review, American Economic Association, vol. 105(8), pages 2570-2594, August.
    6. Georg Nöldeke & Larry Samuelson, 2018. "The Implementation Duality," Econometrica, Econometric Society, vol. 86(4), pages 1283-1324, July.
    7. Koshevoy, Gleb A., 1999. "Choice functions and abstract convex geometries," Mathematical Social Sciences, Elsevier, vol. 38(1), pages 35-44, July.
    8. Klaus Nehring, 1999. "Preference for Flexibility in a Savage Framework," Econometrica, Econometric Society, vol. 67(1), pages 101-120, January.
    9. Epstein, Larry G. & Marinacci, Massimo, 2007. "Mutual absolute continuity of multiple priors," Journal of Economic Theory, Elsevier, vol. 137(1), pages 716-720, November.
    10. Dekel, Eddie & Lipman, Barton L & Rustichini, Aldo, 2001. "Representing Preferences with a Unique Subjective State Space," Econometrica, Econometric Society, vol. 69(4), pages 891-934, July.
    11. Chambers, Christopher P. & Echenique, Federico, 2008. "Ordinal notions of submodularity," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1243-1245, December.
    12. Ok, Efe A., 2002. "Utility Representation of an Incomplete Preference Relation," Journal of Economic Theory, Elsevier, vol. 104(2), pages 429-449, June.
    13. Chambers, Christopher P. & Miller, Alan D., 2014. "Scholarly influence," Journal of Economic Theory, Elsevier, vol. 151(C), pages 571-583.
    14. Chambers, Christopher P. & Echenique, Federico, 2009. "Supermodularity and preferences," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1004-1014, May.
    15. Chambers, Christopher P. & Miller, Alan D., 2018. "Benchmarking," Theoretical Economics, Econometric Society, vol. 13(2), May.
    16. Danilov, V. & Koshevoy, G., 2005. "Mathematics of Plott choice functions," Mathematical Social Sciences, Elsevier, vol. 49(3), pages 245-272, May.
    17. Plott, Charles R, 1973. "Path Independence, Rationality, and Social Choice," Econometrica, Econometric Society, vol. 41(6), pages 1075-1091, November.
    18. Johnson, Mark R. & Dean, Richard A., 2001. "Locally complete path independent choice functions and their lattices," Mathematical Social Sciences, Elsevier, vol. 42(1), pages 53-87, July.
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    Cited by:

    1. Chambers, Christopher P. & Miller, Alan D., 2018. "Benchmarking," Theoretical Economics, Econometric Society, vol. 13(2), May.
    2. Vladimir Danilov, 2022. "Complementary choice functions," Papers 2209.06514, arXiv.org.
    3. Brian Duricy, 2023. "Preferences on Ranked-Choice Ballots," Papers 2301.02697, arXiv.org.
    4. Hamed Hamze Bajgiran & Federico Echenique, 2022. "Closure operators: Complexity and applications to classification and decision-making," Papers 2202.05339, arXiv.org, revised May 2022.

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