zbMATH — the first resource for mathematics

Probabilities of electoral outcomes: from three-candidate to four-candidate elections. (English) Zbl 1433.91065
Summary: The main purpose of this paper is to compute the theoretical likelihood of some electoral outcomes under the impartial anonymous culture in four-candidate elections by using the last versions of software like LattE or Normaliz. By comparison with the three-candidate case, our results allow to analyze the impact of the number of candidates on the occurrence of these voting outcomes.
91B12 Voting theory
91B14 Social choice
Convex; LattE; Normaliz
Full Text: DOI
[1] Barvinok, A., A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Mathematics of Operations Research, 19, 769-779 (1994) · Zbl 0821.90085
[2] Brandt, F., Geist, C., & Strobel, M. (2016). Analyzing the practical relevance of voting paradoxes via Ehrhart theory, computer simulations and empirical data. In J. Thangarajah et al. (Eds.) Proceedings of the 15th international conference on autonomous agents and multiagent systems (AAMAS 2016).
[3] Brandt, F., Hofbauer, J., & Strobel, M. (2019). Exploring the no-show paradox for Condorcet extensions using Ehrhart theory and computer simulations. http://dss.in.tum.de/files/brandt-research/noshow_study.pdf. Accessed 15 Oct 2019.
[4] Bruns, W., & Ichim, B. (2018). Polytope volumes by descent in the face lattice and applications in social choice. arXiv preprint arXiv:1807.02835.
[5] Bruns, W.; Ichim, B.; Söger, C., Computations of volumes and Ehrhart series in four candidate elections, Annals of Operations Research, 280, 241-265 (2019) · Zbl 1430.91039
[6] Bruns, W.; Söger, C., The computation of generalized Ehrhart series in Normaliz, Journal of Symbolic Computation, 68, 75-86 (2015) · Zbl 1320.52018
[7] Bubboloni, D., Diss, M., & Gori, M. (2018). Extensions of the Simpson voting rule to committee selection setting, forthcoming in Public Choice.
[8] Clauss, P.; Loechner, V., Parametric analysis of polyhedral iteration spaces, Journal of VLSI Signal Processing, 19, 2, 179-194 (1998)
[9] De Loera, Ja; Dutra, B.; Koeppe, M.; Moreinis, S.; Pinto, G.; Wu, J., Software for exact integration of polynomials over polytopes, Computational Geometry: Theory and Applications, 46, 232-252 (2013) · Zbl 1259.65054
[10] De Loera, Ja; Hemmecke, R.; Tauzer, J.; Yoshida, R., Effective lattice point counting in rational convex polytopes, Journal of Symbolic Computation, 38, 1273-1302 (2004) · Zbl 1137.52303
[11] Diss, M.; Doghmi, A., Multi-winner scoring election methods: Condorcet consistency and paradoxes, Public Choice, 169, 97-116 (2016)
[12] Diss, M.; Kamwa, E.; Tlidi, A., A note on the likelihood of the absolute majority paradoxes, Economic Bulletin, 38, 1727-1734 (2018)
[13] Diss, M., Kamwa, E., & Tlidi, A. (2019). On some k-scoring rules for committees elections: agreement and Condorcet principle. https://hal.univ-antilles.fr/hal-02147735/document. Accessed 15 Oct 2019.
[14] Diss, M., & Mahajne, M. (2019). Social acceptability of Condorcet committees. https://halshs.archives-ouvertes.fr/halshs-02003292/document. Accessed 15 Oct 2019.
[15] Ehrhart, E., Sur les polyèdres rationnels homothétiques à n dimensions, Comptes Rendus de l’académie des Sciences paris, 254, 616-618 (1962) · Zbl 0100.27601
[16] Gehrlein, Wv, Condorcet winners on four candidates with anonymous voters, Economics Letters, 71, 335-340 (2001) · Zbl 0983.91019
[17] Gehrlein, Wv, Obtaining representations for probabilities of voting outcomes with effectively unlimited precision integer arithmetic, Social Choice and Welfare, 19, 503-512 (2002) · Zbl 1072.91530
[18] Gehrlein, Wv, Condorcet’s paradox (2006), Berlin: Springer, Berlin · Zbl 1122.91027
[19] Gehrlein, Wv; Fishburn, Pc, The probability of the paradox of voting: a computable solution, Journal of Economic Theory, 13, 14-25 (1976) · Zbl 0351.90002
[20] Gehrlein, Wv; Lepelley, D., Voting paradoxes and group coherence (2011), Berlin: Springer, Berlin · Zbl 1252.91001
[21] Gehrlein, Wv; Lepelley, D., Elections, voting rules and paradoxical outcomes (2017), Berlin: Springer, Berlin · Zbl 1429.91003
[22] Gehrlein, Wv; Lepelley, D.; Plassmann, F., An evaluation of the benefit of using two-stage election procedures, Homo Oeconomicus, 35, 53-79 (2018)
[23] Huang, Hc; Chua, V., Analytical representation of probabilities under the IAC condition, Social Choice and Welfare, 17, 143-155 (2000) · Zbl 1069.91529
[24] Lepelley, D. (1989). Contribution à l’analyse des procédures de décision collective, unpublished dissertation, université de Caen.
[25] Lepelley, D.; Louichi, A.; Smaoui, H., On Ehrhart polynomials and probability calculations in voting theory, Social Choice and Welfare, 30, 363-383 (2008) · Zbl 1149.91028
[26] Lepelley, D.; Mbih, B., The proportion of coalitionally unstable situations under the plurality rule, Economics Letters, 24, 311-315 (1987) · Zbl 1328.91073
[27] Moulin, H., Axioms of cooperative decision making (1988), Cambridge: Cambridge University Press, Cambridge · Zbl 0699.90001
[28] Schürmann, A., Exploiting polyhedral symmetries in social choice, Social Choice and Welfare, 40, 1097-1110 (2013) · Zbl 1288.91076
[29] Verdoolaege, S., & Bruynooghe, M. (2008). Algorithms for weighted counting over parametric polytopes: a survey and a practical comparison. In Proceedings of the 2008 international conference on information theory and statistical learning (ITSL).
[30] Wilson, Mc; Pritchard, G., Probability calculations under the IAC hypothesis, Mathematical Social Sciences, 54, 244-256 (2007) · Zbl 1141.91379
[31] Barvinok by Verdoolaege S, ver. 0.34. (2011). http://freshmeat.net/projects/barvinok.
[32] LattE integrale by De Loera, J.A., Hemmecke, R., Tauzer, J., Yoshida, R., & Köppe M., ver. 1.7.3. (2016). http://www.math.ucdavis.edu/ latte/.
[33] Normaliz by Bruns W, Ichim B, and Söger C, ver. 3.6.2. (2018). http://www.mathmatik.uni-osnabrueck.de/normaliz.
[34] lrs by Avis D, ver. 6.2. (2016). cgm.cs.mcgill.ca/ avis/C/lrs.html.
[35] Convex, a Maple package for convex geometry, Franz. (2017). http://www.math.uwo.ca/faculty/franz/convex/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.