Monotonicity paradoxes in three-candidate elections using scoring elimination rules

Scoring elimination rules (SER), that give points to candidates according to their rank in voters’ preference orders and eliminate the candidate(s) with the lowest number of points, constitute an important class of voting rules. This class of rules, that includes some famous voting methods such as Plurality Runoff or Coombs Rule, suffers from a severe pathology known as monotonicity paradox or monotonicity failure, that is, getting more points from voters can make a candidate a loser and getting fewer points can make a candidate a winner. In this paper, we study three-candidate elections and we identify, under various conditions, which SER minimizes the probability that a monotonicity paradox occurs. We also analyze some strategic aspects of these monotonicity failures. The probability model on which our results are based is the impartial anonymous culture condition, often used in this kind of study.

[1]  William V. Gehrlein.  Condorcet efficiency and constant scoring rules , 1982, Math. Soc. Sci..

[2]  Elena Dubrova,et al.  Polynomial-time algorithm for computing bound sets , 2006 .

[3]  Dan S. Felsenthal,et al.  Interacting double monotonicity failure with direction of impact under five voting methods , 2014, Math. Soc. Sci..

[4]  W. Gehrlein.  Condorcet's paradox , 1983 .

[5]  William V. Gehrlein,et al.  A comparison of theoretical and empirical evaluations of the Borda Compromise , 2014, Soc. Choice Welf..

[6]  Gideon Doron,et al.  Single Transferrable Vote: An Example of a Perverse Social Choice Function , 1977 .

[7]  Dominique Lepelley,et al.  Borda elimination rule and monotonicity paradoxes in three-candidate elections , 2016 .

[8]  E. Ehrhart,et al.  Polynômes arithmétiques et méthode des polyèdres en combinatoire , 1974 .

[9]  Dan S. Felsenthal,et al.  Varieties of failure of monotonicity and participation under five voting methods , 2013 .

[10]  Nicholas R. Miller.  Closeness matters: monotonicity failure in IRV elections with three candidates , 2017 .

[11]  A. Barvinok.  A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed , 1994 .

[12]  Alexander I. Barvinok,et al.  A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[13]  Geoffrey Pritchard,et al.  Probability calculations under the IAC hypothesis , 2007, Math. Soc. Sci..

[14]  Jacques Cohen,et al.  Two Algorithms for Determining Volumes of Convex Polyhedra , 1979, JACM.

[15]  Sven Berg,et al.  The likelihood of monotonicity paradoxes in run-off elections , 1996 .

[16]  Florenz Plassmann,et al.  How frequently do different voting rules encounter voting paradoxes in three-candidate elections? , 2014, Soc. Choice Welf..

[17]  Peter C. Fishburn,et al.  Paradoxes of Preferential Voting , 1983 .

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

[19]  Peter C. Fishburn,et al.  Monotonicity paradoxes in the theory of elections , 1982, Discret. Appl. Math..

[20]  J. H. Smith.  AGGREGATION OF PREFERENCES WITH VARIABLE ELECTORATE , 1973 .

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