The Unexpected Behavior of Plurality Rule

Abstract

International audienceWhen voters’ preferences on candidates are mutually coherent, in the sense that they are at all close to being perfectly single-peaked, perfectly single-troughed, or perfectly polarized, there is a large probability that a Condorcet Winner exists in elections with a small number of candidates. Given this fact, the study develops representations for Condorcet Efficiency of plurality rule as a function of the proximity of voters’ preferences on candidates to being perfectly single-peaked, perfectly single-troughed or perfectly polarized. We find that the widely used plurality rule has Condorcet Efficiency values that behave in very different ways under each of these three models of mutual coherence

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

“…The logic behind this procedure is quite straightforward to develop, but it can become extremely cumbersome to implement. Gehrlein and Lepelley [2009;2010] use the EUPIA2 procedure to develop a number of related representations under the IACx(k) assumption. The representations that are obtained for general n become so complex that analysis is restricted to the limiting case as n ->■ °o.…”

“…When it was not feasible to use eupia2 in Gehrlein and Lepelley [2009;2010], a procedure from Lepelley et al [2008] was used. Lepelley et al [2008] point out that the EUPIA2 procedure is a special case of a more general problem, which is equivalent to counting the integral points in polyhedra, that was first studied by Ehrhart [1962].…”

“…Some new and efficient algorithms have been recently proposed by computer scientists to solve this problem. Gehrlein and Lepelley [2009;2010], an algorithm that was developed by Barvinok [1994] and extended and implemented by Verdoolaege et al [2004] (see Lepelley et al ., 2008, for a presentation of this procedure) was used to obtain limiting representations for probabilities as n -> ©o.…”

“…Concluding Remark and Future Research Gehrlein and Lepelley [2009] introduced the general hypothesis, denoted by H2, that the Condorcet efficiency of voting rules should tend to increase as voters' preferences show an increased degree of mutual coherence in the way in which the individual voters form their preferences on the candidates. According to this rather intuitively appealing assumption, the curves shown in Figures 1-6 should be decreasing as the Parameters b, t and с increase.…”

Aggregating out of indeterminacy

“…As a result, Lepelly et al show that when we have a never worst restriction on preferences and the population gets large enough, then there is perfect convergence on the Condorcet winner . Gehrlein and Lepelly further show that, at least when it comes to the plurality rule, when preferences satisfy the single-troughed domain restriction (a close relative of the single-peaked assumption), then, again, there are high rates of convergence (2009: 291). Thus, when we start looking at the kinds of preference distributions Gaus must assume over the set of all proposed moral rules for the eligible set to be non-empty we find that, if preferences over the optimal eligible set have a similar structure, then the convergence solution to Nozick’s objection is quite compelling – all major social choice mechanisms end up selecting the same moral rule.…”