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Adjacencies on random ordering polytopes and flow polytopes

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  • Jean-Paul Doignon
  • Kota Saito

Abstract

The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak (1960). Fishburn (1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). Apart for a recognition of the facets by Suck (2002), the geometric structure of the MCP was apparently not much investigated. Recently, Chang, Narita and Saito (2022) refer to the adjacency of vertices while Turansick (2022) uses a condition which we show to be equivalent to the non-adjacency of two vertices. We characterize the adjacency of vertices and the adjacency of facets. To derive a more enlightening proof of Falmagne Theorem and of Suck result, Fiorini (2004) assimilates the MCP with the flow polytope of some acyclic network. Our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).

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  • Jean-Paul Doignon & Kota Saito, 2022. "Adjacencies on random ordering polytopes and flow polytopes," Papers 2207.06925, arXiv.org.
    • Handle: RePEc:arx:papers:2207.06925
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    References listed on IDEAS

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    1. Haoge Chang & Yusuke Narita & Kota Saito, 2022. "Approximating Choice Data by Discrete Choice Models," Papers 2205.01882, arXiv.org, revised Dec 2023.
    2. Regenwetter, Michel & Marley, A. A. J. & Grofman, Bernard, 2002. "A general concept of majority rule," Mathematical Social Sciences, Elsevier, vol. 43(3), pages 405-428, July.
    3. Turansick, Christopher, 2022. "Identification in the random utility model," Journal of Economic Theory, Elsevier, vol. 203(C).
    4. Fishburn, Peter C., 1992. "Induced binary probabilities and the linear ordering polytope: a status report," Mathematical Social Sciences, Elsevier, vol. 23(1), pages 67-80, February.
    5. Suck, Reinhard, 2002. "Independent random utility representations," Mathematical Social Sciences, Elsevier, vol. 43(3), pages 371-389, July.
    6. Morgan McClellon, 2015. "Unique Random Utility Representations," Working Paper 262661, Harvard University OpenScholar.
    7. 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.
    8. Clintin P. Davis-Stober & Jean-Paul Doignon & Samuel Fiorini & François Glineur & Michel Regenwetter, 2018. "Extended formulations for order polytopes through network flows," LIDAM Reprints CORE 2987, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Irène Charon & Olivier Hudry, 2010. "An updated survey on the linear ordering problem for weighted or unweighted tournaments," Annals of Operations Research, Springer, vol. 175(1), pages 107-158, March.
    10. Fishburn, Peter C. & Falmagne, Jean-Claude, 1989. "Binary choice probabilities and rankings," Economics Letters, Elsevier, vol. 31(2), pages 113-117, December.
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    Cited by:

    1. Christopher Turansick, 2023. "An Alternative Approach for Nonparametric Analysis of Random Utility Models," Papers 2303.14249, arXiv.org, revised May 2024.
    2. Haruki Kono & Kota Saito & Alec Sandroni, 2023. "Axiomatization of Random Utility Model with Unobservable Alternatives," Papers 2302.03913, arXiv.org, revised Aug 2023.

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