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A Necessary but Insufficient Condition for the Stochastic Binary Choice Problem

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  • Itzhak Gilboa

Abstract

The "stochastic binary choice problem" is the following: Let there be given n alternatives, to be denoted by N = {1, ..., n}. For each of the n! possible linear orderings {m}m = 1n of the alternatives, define a matrix Yn × n(m)(1 ≤ m ≤ n!) as follows: Given a real matrix Qn × n, when is Q in the convex hull of {Y(m)}m? In this paper some necessary conditions on Q--the "diagonal inequality"--are formulated and they are proved to generalize the Cohen-Falmagne conditions. A counterexample shows that the diagonal inequality is insufficient (as are hence, perforce, the Cohen-Falmagne conditions). The same example is used to show that Fishburn's conditions are also insufficient.
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  • Itzhak Gilboa, 1989. "A Necessary but Insufficient Condition for the Stochastic Binary Choice Problem," Discussion Papers 818, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    • Handle: RePEc:nwu:cmsems:818
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    References listed on IDEAS

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    1. Barbera, Salvador & Pattanaik, Prasanta K, 1986. "Falmagne and the Rationalizability of Stochastic Choices in Terms of Random Orderings," Econometrica, Econometric Society, vol. 54(3), pages 707-715, May.
    2. Daniel McFadden, 2005. "Revealed stochastic preference: a synthesis," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(2), pages 245-264, August.
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    Cited by:

    1. Yun-shil Cha & Michelle Choi & Ying Guo & Michel Regenwetter & Chris Zwilling, 2013. "Reply: Birnbaum's (2012) statistical tests of independence have unknown Type-I error rates and do not replicate within participant," Judgment and Decision Making, Society for Judgment and Decision Making, vol. 8(1), pages 55-73, January.
    2. Itzhak Gilboa & Dov Monderer, 1989. "A Game-Theoretic Approach to the Binary Stochastic Choice Problem," Discussion Papers 854, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Jerry S. Kelly & Shaofang Qi, 2016. "A conjecture on the construction of orderings by Borda’s rule," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(1), pages 113-125, June.
    4. repec:cup:judgdm:v:8:y:2013:i:1:p:55-73 is not listed on IDEAS
    5. Petri, Henrik, 2023. "Binary single-crossing random utility models," Games and Economic Behavior, Elsevier, vol. 138(C), pages 311-320.
    6. Daniele Caliari & Henrik Petri, 2024. "Irrational Random Utility Models," Papers 2403.10208, arXiv.org.
    7. Janny M. Y. Leung & Guoqing Zhang & Xiaoguang Yang & Raymond Mak & Kokin Lam, 2004. "Optimal Cyclic Multi-Hoist Scheduling: A Mixed Integer Programming Approach," Operations Research, INFORMS, vol. 52(6), pages 965-976, December.

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