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From Gehrlein-Fishburn’s Method on Frequency Representation to a Direct Proof of Ehrhart’s extended Conjecture

In: Evaluating Voting Systems with Probability Models

Author

Listed:
  • Issofa Moyouwou

    (Université de Yaoundé1)

  • Nicolas Gabriel Andjiga

    (Université de Yaoundé1)

  • Boniface Mbih

    (Université de Caen Normandie)

Abstract

Deriving a closed-form formula for the exact number of integer solutions to a system of linear inequalities involving integer coefficients of bounded integer free variables and integer parameters as a function of parameters is a general problem encountered in various fields, especially in social choice theory when analyzing how frequent an event is. From Gehrlein and Fishburn’s approach of computation [Gehrlein W.V., Fishburn P.C., 1976. Condorcet’s Paradox and Anonymous Preference Profiles. Public Choice 26, 1–18], we give a straightforward proof that such a closed-form formula is a piecewise defined polynomial function in parameters as stated in Ehrhart’s extended conjecture. We even extend this result to the sum of a multivariate polynomial function over the set of integer points in a rational polytope.

Suggested Citation

  • Issofa Moyouwou & Nicolas Gabriel Andjiga & Boniface Mbih, 2021. "From Gehrlein-Fishburn’s Method on Frequency Representation to a Direct Proof of Ehrhart’s extended Conjecture," Studies in Choice and Welfare, in: Mostapha Diss & Vincent Merlin (ed.), Evaluating Voting Systems with Probability Models, pages 367-398, Springer.
    • Handle: RePEc:spr:stcchp:978-3-030-48598-6_16
    DOI: 10.1007/978-3-030-48598-6_16
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