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An example of probability computations under the IAC assumption: The stability of scoring rules

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  • Mostapha Diss

    (GATE Lyon Saint-Étienne - Groupe d'Analyse et de Théorie Economique Lyon - Saint-Etienne - ENS de Lyon - École normale supérieure de Lyon - Université de Lyon - UL2 - Université Lumière - Lyon 2 - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Ahmed Louichi

    (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • Vincent Merlin

    (CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

  • H. Smaoui

Abstract

A society facing a choice problem has also to choose the voting rule itself from a set of different possible voting rules. A voting rule is self-selective if it chooses itself when it is also used in choosing the voting rule. A set of voting rules is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this paper a society which makes a choice from a set of three alternatives {a,b,c} and a set of the three well-known scoring voting rules {Borda, Plurality, Antiplurality}. We will derive an a priori probability for the stability of this triplet of voting rules, under the Impartial Anonymous Culture assumption (IAC). In order to solve this problem, we need to specify Ehrhart polynomials, which count the number of integer points inside a (convex) polytope. We discuss briefly a recent algorithmic solution to this method before applying it. We also discuss the impact of different behavioral assumptions for the voters (consequentialist or nonconsequentialist) on the probability of stability for the triplet {Borda, Plurality, Antiplurality}.

Suggested Citation

  • Mostapha Diss & Ahmed Louichi & Vincent Merlin & H. Smaoui, 2012. "An example of probability computations under the IAC assumption: The stability of scoring rules," Post-Print halshs-00667660, HAL.
    • Handle: RePEc:hal:journl:halshs-00667660
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    Cited by:

    1. Takahiro Suzuki & Masahide Horita, 2023. "A Society Can Always Decide How to Decide: A Proof," Group Decision and Negotiation, Springer, vol. 32(5), pages 987-1023, October.
    2. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2018. "The Chamberlin-Courant Rule and the k-Scoring Rules: Agreement and Condorcet Committee Consistency," Working Papers hal-01757761, HAL.
    3. Daniela Bubboloni & Mostapha Diss & Michele Gori, 2020. "Extensions of the Simpson voting rule to the committee selection setting," Public Choice, Springer, vol. 183(1), pages 151-185, April.
    4. Mostapha Diss, 2015. "Strategic manipulability of self-selective social choice rules," Annals of Operations Research, Springer, vol. 229(1), pages 347-376, June.
    5. Eric Kamwa, 2019. "On the Likelihood of the Borda Effect: The Overall Probabilities for General Weighted Scoring Rules and Scoring Runoff Rules," Group Decision and Negotiation, Springer, vol. 28(3), pages 519-541, June.
    6. Eric Kamwa, 2022. "Scoring rules, ballot truncation, and the truncation paradox," Public Choice, Springer, vol. 192(1), pages 79-97, July.
    7. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2025. "The effect of close elections on the likelihood of voting paradoxes: Further results in three-candidate elections," Post-Print hal-04230359, HAL.
    8. Eric Kamwa, 2022. "Scoring Rules, Ballot Truncation, and the Truncation Paradox," Working Papers hal-03632662, HAL.
    9. Jansen, C. & Schollmeyer, G. & Augustin, T., 2018. "A probabilistic evaluation framework for preference aggregation reflecting group homogeneity," Mathematical Social Sciences, Elsevier, vol. 96(C), pages 49-62.
    10. William V. Gehrlein & Dominique Lepelley & Florenz Plassmann, 2018. "An Evaluation of the Benefit of Using Two-Stage Election Procedures," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 35(1), pages 53-79, June.
    11. Mostapha Diss & Eric Kamwa & Abdelmonaim Tlidi, 2018. "A Note on the Likelihood of the Absolute Majority Paradoxes," Economics Bulletin, AccessEcon, vol. 38(4), pages 1727-1734.
    12. William V. Gehrlein & Dominique Lepelley & Florenz Plassmann, 2016. "Further Support for Ranking Candidates in Elections," Group Decision and Negotiation, Springer, vol. 25(5), pages 941-966, September.
    13. Héctor Hermida‐Rivera & Toygar T. Kerman, 2025. "Binary Self‐Selective Voting Rules," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 27(3), June.
    14. Eric Kamwa, 2021. "To what extent does the model of processing sincereincomplete rankings affect the likelihood of the truncation paradox?," Working Papers hal-02879390, HAL.
    15. H'ector Hermida-Rivera, 2025. "Self-Equivalent Voting Rules," Papers 2506.15310, arXiv.org, revised Dec 2025.
    16. Fabrice Barthélémy & Mathieu Martin, 2021. "Dummy Players and the Quota in Weighted Voting Games: Some Further Results," Studies in Choice and Welfare, in: Mostapha Diss & Vincent Merlin (ed.), Evaluating Voting Systems with Probability Models, pages 299-315, Springer.

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