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

Issofa Moyouwou (), Nicolas Gabriel Andjiga () and Boniface Mbih
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Issofa Moyouwou: Université de Yaoundé1
Nicolas Gabriel Andjiga: Université de Yaoundé1

A chapter in Evaluating Voting Systems with Probability Models, 2021, pp 367-398 from Springer

Abstract: 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.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:spr:stcchp:978-3-030-48598-6_16

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DOI: 10.1007/978-3-030-48598-6_16

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