Computations of volumes and Ehrhart series in four candidates elections

  • Winfried Bruns
  • Bogdan IchimEmail author
  • Christof Söger
Original Research


We describe several analytical results obtained in four candidates social choice elections under the assumption of the Impartial Anonymous Culture. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.


Rational polytope Volume Ehrhart series Social choice Condorcet paradox Borda paradox 

Mathematics Subject Classification

52B20 91B12 



The authors like to thank Achill Schürmann for several test examples that were used during the development of Normaliz. They are also grateful to the anonymous referees for helpful comments. Bogdan Ichim was partially supported by a grant of Romanian Ministry of Research and Innovation, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2016-0157, within PNCDI III.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.FB Mathematik/InformatikUniversität OsnabrückOsnabrückGermany
  2. 2.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania

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