A Note on the Likelihood of the Absolute Majority Paradoxes

For three-candidate elections, we compute under the Impartial Anonymous Culture assumption, the conditional probabilities of the Absolute Majority Winner Paradox (AMWP) and the Absolute Majority Loser Paradox (AMLP) under the Plurality rule, the Borda rule, and the Negative Plurality rule for a given number of voters. We also provide a representation of the conditional probability of these paradoxes for the whole family of weighted scoring rules with large electorates. The AMWP occurs when a candidate who is ranked first by more than half of the voters is not selected by a given voting rule; the AMLP appears when a candidate who is ranked last by more than half of the voters is elected. As no research papers have tried to evaluate the likelihood of these paradoxes, this note is designed to fill this void. Our results allow us to claim that ignoring these two paradoxes in the literature, particularly AMWP, is not justified.

[1]  Vincent Loechner,et al.  Analytical computation of Ehrhart polynomials: enabling more compiler analyses and optimizations , 2004, CASES '04.

[2]  K. Kuga,et al.  Voter Antagonism and the Paradox of Voting , 1974 .

[3]  William V. Gehrlein,et al.  The True Impact of Voting Rule Selection on Condorcet Efficiency , 2015 .

[4]  William V. Gehrlein,et al.  Voting Paradoxes and Group Coherence , 2011 .

[5]  Eric Kamwa,et al.  Scoring Rules and Preference Restrictions: The Strong Borda Paradox Revisited , 2017 .

[6]  Ahmed Louichi,et al.  An example of probability computations under the IAC assumption: The stability of scoring rules , 2012, Math. Soc. Sci..

[7]  P. Fishburn,et al.  Condorcet's paradox and anonymous preference profiles , 1976 .

[8]  William S. Zwicker,et al.  Which Scoring Rule Maximizes Condorcet Efficiency Under Iac? , 2005 .

[9]  Issofa Moyouwou,et al.  Asymptotic vulnerability of positional voting rules to coalitional manipulation , 2017, Math. Soc. Sci..

[10]  Vincent Loechner,et al.  Parametric Analysis of Polyhedral Iteration Spaces , 1996, Proceedings of International Conference on Application Specific Systems, Architectures and Processors: ASAP '96.

[11]  Dominique Lepelley,et al.  On Ehrhart polynomials and probability calculations in voting theory , 2008, Soc. Choice Welf..

[12]  A. Barvinok,et al.  An Algorithmic Theory of Lattice Points in Polyhedra , 1999 .

[13]  Maurice Bruynooghe,et al.  Computation and manipulation of enumerators of integer projections of parametric polytopes , 2005 .

[14]  William V. Gehrlein,et al.  Voters’ preference diversity, concepts of agreement and Condorcet’s paradox , 2015 .

[15]  William V. Gehrlein,et al.  Borda’s Paradox with weighted scoring rules , 2012, Soc. Choice Welf..

[16]  E. Ehrhardt,et al.  Sur un problème de géométrie diophantienne linéaire. II. , 1967 .