On the likelihood of Condorcet's profiles

Abstract. Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, wj, to the alternative she ranks in jth position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases.

[1]  Vincent Merlin,et al.  On the relationship of the Condorcet winner and positional voting rules , 1997 .

[2]  Ruth Kellerhals,et al.  On the volume of hyperbolic polyhedra , 1989 .

[3]  S. Berg.  Paradox of voting under an urn model: The effect of homogeneity , 1985 .

[4]  Scoring rule sensitivity to weight selection , 1983 .

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

[6]  H. S. M. Coxeter,et al.  THE FUNCTIONS OF SCHLÄFLI AND LOBATSCHEFSKY , 1935 .

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

[8]  H. Young.  Social Choice Scoring Functions , 1975 .

[9]  H. Moulin.  Condorcet's principle implies the no show paradox , 1988 .

[10]  Donald G. Saari,et al.  Copeland Method II: Manipulation, Monotonicity, and Paradoxes☆ , 1997 .

[11]  Sven Berg,et al.  On probability models in voting theory , 1994 .

[12]  William V. Gehrlein.  On the Probability that all Weighted Scoring Rules Elect the Condorcet Winner , 1999 .

[13]  D. Lepelley.  Constant scoring rules, condorcet criteria and single-peaked preferences , 1996 .

[14]  V Gehrlein William,et al.  CONDORCET'S PARADOX AND THE CONDORCET EFFICIENCY OF VOTING RULES. , 1997 .

[15]  D. Saari,et al.  The Copeland method , 1996 .

[16]  H. Young.  Condorcet's Theory of Voting , 1988, American Political Science Review.

[17]  William V. Gehrlein.  The sensitivity of weight selection on the Condorcet efficiency of weighted scoring rules , 1998 .

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

[19]  Fabrice Valognes,et al.  On the probability that all decision rules select the same winner , 2000 .

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

[21]  Donald G. Saari,et al.  The likelihood of dubious election outcomes , 1999 .

[22]  Donald G. Saari,et al.  The Copeland method: I.: Relationships and the dictionary , 1996 .

[23]  Jill Van Newenhizen.  The Borda method is most likely to respect the Condorcet principle , 1992 .

[24]  L. Schläfli.  Theorie der vielfachen Kontinuität , 1901 .