The Condorcet Efficiency of Voting Rules with Mutually Coherent Voter Preferences: A Borda Compromise

The Condorcet Efficiency of a voting rule is defined as the conditional probability that the voting rule elects the Pairwise Majority Rule Winner (PMRW), given that a PMRW exists. Five simple voting rules are considered in this paper: Plurality Rule, Negative Plurality Rule, Borda Rule, Plurality Elimination Rule and Negative Plurality Elimination Rule. In order to study the impact that the presence of degrees of group mutual coherence in voting situations will have on the probability of selecting the PMRW for each of these rules, we develop representations for their Condorcet Efficiency as a function of the proximity of voters' preferences on candidates to being perfectly singlepeaked, perfectly single-troughed or perfectly polarized. The results we obtain lead us to appeal for a Borda Compromise.

[1]  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.

[2]  G Saari Donald.  THE MATHEMATICAL SYMMETRY OF CHOOSING , 1996 .

[3]  Peter C. Fishburn,et al.  Probabilities of election outcomes for large electorates , 1978 .

[4]  William V. Gehrlein,et al.  On the probability of observing Borda’s paradox , 2010, Soc. Choice Welf..

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

[6]  William V. Gehrlein,et al.  The Unexpected Behavior of Plurality Rule , 2008 .

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

[8]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[9]  William V. Gehrlein,et al.  The Condorcet efficiency of Borda Rule with anonymous voters , 2001, Math. Soc. Sci..

[10]  Williams V. Gehrlein.  Condorcet efficiency of two stage constant scoring rules , 1993 .

[11]  A. A. J. Marley,et al.  Behavioral Social Choice - Probabilistic Models, Statistical Inference, and Applications , 2006 .

[12]  G. Thompson,et al.  The Theory of Committees and Elections. , 1959 .

[13]  Dominique Lepelley.  Condorcet efficiency of positional voting rules with single-peaked preferences , 1994 .

[14]  D. Saari.  Explaining All Three-Alternative Voting Outcomes , 1999 .

[15]  Donald G. Saari,et al.  The Borda dictionary , 1990 .

[16]  Donald G. Saari,et al.  A chaotic Exploration of Aggregation Paradoxes , 1995, SIAM Rev..

[17]  W. Gehrlein.  Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences* , 2002 .

[18]  William V. Gehrlein,et al.  Condorcet efficiencies under the maximal culture condition , 1999 .

[19]  William V. Gehrlein.  Strong measures of group coherence and the probability that a pairwise majority rule winner exists , 2011 .

[20]  William V. Gehrlein.  The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences , 2006, Soc. Choice Welf..

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

[22]  Peter C. Fishburn,et al.  Majority efficiencies for simple voting procedures: Summary and interpretation , 1982 .

[23]  William Vickrey,et al.  Utility, Strategy, and Social Decision Rules , 1960 .

[24]  Dominique Lepelley.  Michel Regenwetter, Bernard Grofman, A.A.J. Marley, and Ilia M. Tsetlin: Behavioral social choice. Probabilistic models, statistical inference and applications , 2008, Soc. Choice Welf..

[25]  William V. Gehrlein.  Probabilities of election outcomes with two parameters: The relative impact of unifying and polarizing candidates , 2005 .

[26]  Peter C. Fishburn,et al.  Coincidence probabilities for simple majority and positional voting rules , 1978 .