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Infinite Supermodularity and Preferences

In: Game Theory - Applications in Logistics and Economy

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  • Alain Chateauneuf
  • Vassili Vergopoulos
  • Jianbo Zhang

Abstract

This chapter studies the ordinal content of supermodularity on lattices. This chapter is a generalization of the famous study of binary relations over finite Boolean algebras obtained by Wong, Yao and Lingras. We study the implications of various types of supermodularity for preferences over finite lattices. We prove that preferences on a finite lattice merely respecting the lattice order cannot disentangle these usual economic assumptions of supermodularity and infinite supermodularity. More precisely, the existence of a supermodular representation is equivalent to the existence of an infinitely supermodular representation. In addition, the strict increasingness of a complete preorder on a finite lattice is equivalent to the existence of a strictly increasing and infinitely supermodular representation. For wide classes of binary relations, the ordinal contents of quasisupermodularity, supermodularity and infinite supermodularity are exactly the same. In the end, we extend our results from finite lattices to infinite lattices.

Suggested Citation

  • Alain Chateauneuf & Vassili Vergopoulos & Jianbo Zhang, 2018. "Infinite Supermodularity and Preferences," Chapters, in: Danijela Tuljak-Suban (ed.), Game Theory - Applications in Logistics and Economy, IntechOpen.
    • Handle: RePEc:ito:pchaps:154359
    DOI: 10.5772/intechopen.79150
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    References listed on IDEAS

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    1. Klaus Nehring & Clemens Puppe, 2002. "A Theory of Diversity," Econometrica, Econometric Society, vol. 70(3), pages 1155-1198, May.
    2. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    3. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-180, January.
    4. Chambers, Christopher P. & Echenique, Federico, 2008. "Ordinal notions of submodularity," Journal of Mathematical Economics, Elsevier, vol. 44(11), pages 1243-1245, December.
    5. Kreps, David M, 1979. "A Representation Theorem for "Preference for Flexibility"," Econometrica, Econometric Society, vol. 47(3), pages 565-577, May.
    6. Chambers, Christopher P. & Echenique, Federico, 2009. "Supermodularity and preferences," Journal of Economic Theory, Elsevier, vol. 144(3), pages 1004-1014, May.
    7. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
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    Cited by:

    1. Brian Duricy, 2023. "Preferences on Ranked-Choice Ballots," Papers 2301.02697, arXiv.org.

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    More about this item

    Keywords

    supermodularity; ?-supermodularity; lattice; JEL Classifications: D11; D12; C65;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory

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