Abstract
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.
Similar content being viewed by others
References
Abbott, J., Bigatti, A. M., & Lagorio, G. (2018). CoCoA-5: A system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it.
Almendra, V., & Ichim, B. (2018). jNormaliz: A graphical interface for Normaliz. www.math.uos.de/normaliz.
Avis, D. (2018). lrs: A revised implementation of the reverse search vertex enumeration algorithm. http://cgm.cs.mcgill.ca/~avis/C/lrs.html.
Baldoni, V., Berline, N., De Loera, J. A., Dutra, B., Köppe, M., Moreinis, S., Pinto, G., Vergne, M., & Wu, J. (2013). A user’s guide for LattE integrale v1.7.2, 2013. Software package LattE is available at http://www.math.ucdavis.edu/~latte/.
Baldoni, V., Berline, N., De Loera, J. A., Köppe, M., & Vergne, M. (2011). How to integrate a polynomial over a simplex. Mathematics of Computation, 80, 297–325.
Baldoni, V., Berline, N., De Loera, J. A., Köppe, M., & Vergne, M. (2012). Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Foundations of Computational Mathematics, 12, 435–469.
Brandt, F., Geist, C., & Strobel, M. (2016). Analyzing the practical relevance of voting paradoxes via Ehrhart theory, computer simulations, and empirical data. In Proceedings of the 2016 international conference on autonomous agents and multiagent systems (pp. 385–393).
Brandt, F., Hofbauer, J., & Strobel, M. (2019). Exploring the no-show paradox for condorcet extensions using Ehrhart theory and computer simulations. In Proceedings of the 18th International Conference on Autonomous Agents and Multiagent Systems (AAMAS). IFAAMAS.
Bruns, W., & Gubeladze, J. (2009). Polytopes, rings and K-theory. Berlin: Springer.
Bruns, W., Hemmecke, R., Ichim, B., Köppe, M., & Söger, C. (2011). Challenging computations of Hilbert bases of cones associated with algebraic statistics. Experimental Mathematics, 20, 25–33.
Bruns, W., & Ichim, B. (2010). Normaliz: Algorithms for affine monoids and rational cones. Journal of Algebra, 324, 1098–1113.
Bruns, W., & Ichim, B. (2018). Polytope volume by descent in the face lattice and applications in social choice. Preprint arXiv:1807.02835.
Bruns, W., Ichim, B., Römer, T., Sieg, R., & Söger, C. (2018). Normaliz: Algorithms for rational cones and affine monoids. http://normaliz.uos.de.
Bruns, W., Ichim, B., & Söger, C. (2016). The power of pyramid decomposition in Normaliz. Journal of Symbolic Computation, 74, 513–536.
Bruns, W., & Koch, R. (2001). Computing the integral closure of an affine semigroup. Universitatis Iagellonicae Acta Mathematica, 39, 59–70.
Bruns, W., Sieg, R., & Söger, C. (2017). Normaliz 2013–2016. In G. Böckle, W. Decker, & G. Malle (Eds.), Algorithmic and experimental methods in algebra, geometry, and number theory (pp. 123–146). Cham: Springer.
Bruns, W., & Söger, C. (2015). Generalized Ehrhart series and integration in Normaliz. Journal of Symbolic Computation, 68, 75–86.
Chevalier de Borda, J.-C. (1781). Mémoire sur les élections au scrutin. Histoire de’Académie Royale Des Science, 102, 657–665.
De Loera, J. A., Dutra, B., Köppe, M., Moreinis, S., Pinto, G., & Wu, J. (2013). Software for exact integration of polynomials over polyhedra. Computational Geometry, 46, 232–252.
El Ouafdi, A., Lepelley, D., & Smaoui, H. (2018). Probabilities of electoral outcomes in four-candidate elections. Preprint https://doi.org/10.13140/RG.2.2.27775.87201.
Gehrlein, W. V. (2001). Condorcet winners on four candidates with anonymous voters. Economics Letters, 71, 335–340.
Gehrlein, W. V., & Fishburn, P. (1976). Condorcet’s paradox and anonymous preference profiles. Public Choice, 26, 1–18.
Gehrlein, W. V., & Lepelley, D. (2010). On the probability of observing Borda’s paradox. Social Choice and Welfare, 35, 1–23.
Gehrlein, W. V., & Lepelley, D. (2011). Voting paradoxes and group coherence. Berlin: Springer.
Gehrlein, W. V., & Lepelley, D. (2017). Elections, voting rules and paradoxical outcomes. Berlin: Springer.
Lepelley, D., Louichi, A., & Smaoui, H. (2008). On Ehrhart polynomials and probability calculations in voting theory. Social Choice and Welfare, 30, 363–383.
Marquis de Condorcet, N. (1785). Éssai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Paris: Imprimerie Royale.
R Core Team. (2013). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. http://www.R-project.org/
Schürmann, A. (2013). Exploiting polyhedral symmetries in social choice. Social Choice and Welfare, 40, 1097–1110.
Wilson, M. C., & Pritchard, G. (2007). Probability calculations under the IAC hypothesis. Mathematical Social Sciences, 54, 244–256.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bruns, W., Ichim, B. & Söger, C. Computations of volumes and Ehrhart series in four candidates elections. Ann Oper Res 280, 241–265 (2019). https://doi.org/10.1007/s10479-019-03152-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-019-03152-y