Voting Procedures Under a Restricted Domain

Voting paradoxes occur in particular profile domains. For the avoidance of the paradoxes it is therefore important to know if the profiles typically encountered in practice are of such nature that the paradoxes are very unlikely or downright impossible. Ever since the publication of Arrow’s theorem, the role of domain restrictions has been appreciated. However, the earlier studies have mainly focused on conditions for rational collective choices through pairwise majority comparisons. In those studies the single-peaked preferences have been found to be an important type of preference similarity that guarantees complete and transitive collective outcomes. This booklet introduces and analyzes a similar preference restriction, viz., the existence in the profile of a Condorcet winner that is elected by the procedure under study. We examine the possibilities of various voting procedures to end up with voting paradoxes under these restricted domains. Thereby we refine the results that establish the vulnerability of some procedures to various kinds of voting paradoxes.

20 voting procedures for electing a single candidate are introduced and briefly commented upon. The procedures fall into three classes in terms of the type of voter input and Condorcet consistency: non-ranked procedures, ranked procedures that are not Condorcet-consistent and ranked ones that are Condorcet-consistent. The first class consists of four procedures, the second consists of seven procedures and the third class consists of nine procedures.

This chapter focuses on the possibility that some well-known voting procedures lead to specific types of monotonicity paradoxes in preference profiles that are characterized by the presence and election of a Condorcet winner. Moulin’s (Journal of Economic Theory 45:53–64, 1988) theorem establishes the incompatibility of Condorcet-consistency and invulnerability to the No-Show paradox in voting procedures when there are more than three alternatives to be chosen from. We ask whether this conclusion would also hold in the proper subset of profiles distinguished by the property that a Condorcet winner exists and is elected in the initial profile. Our focus is on 20 voting procedures designed to elect a single candidate. These procedures include both Condorcet-consistent and non-consistent rules. The former are, however, only briefly touched upon because their invulnerability to most types of monotonicity violations in the restricted domain is obvious.

This chapter focuses on the possibility that some well-known voting procedures are vulnerable to the Inconsistency paradox even in preference profiles that are characterized by a restricted domain where a Condorcet winner exists and is elected in each disjoint subset of voters but not in their union. Our focus is on 15 voting procedures known to be vulnerable to the Inconsistency paradox in unrestricted domains. These procedures include 10 Condorcet-consistent and 5 Condorcet-non-consistent rules. The former are, however, only briefly touched upon because their invulnerability to the Inconsistency paradox in the restricted domain is obvious.

The No-Show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (Journal of Economic Theory 45:53–64, 1988) result states that if one wants to exclude the possibility of the No-Show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines 10 Condorcet-consistent and 10 Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the No-Show paradox can occur in these settings. It is found that only 2 of the 10 Condorcet-consistent procedures investigated (Minimax and Schwartz’s procedure) are invulnerable to the No-Show paradox, whereas only 3 of the 10 non-Condorcet-consistent ranked procedures investigated (Coombs’s, the Negative Plurality Elimination Rule, and the Majority Judgment procedures) are vulnerable to this paradox in the restricted domain. In other words, for a No-Show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.

The relative desirability of a voting procedure is assessed, inter alia, by verifying which axioms, or postulates, it satisfies or violates. One of these axioms is the subset choice condition (SCC). This axiom requires that if a candidate, x, is elected under a given voting procedure, f, in a profile consisting of n voters and k competing candidates (n, k > 1), then x ought to be elected by f also in such profiles over any proper subset of candidates that contain x and that preserve the pairwise preference relations of the original profile. Most known voting procedures violate, in general, this axiom. However, we were interested to find out which voting procedures satisfy or violate this axiom under a restricted domain assumption where a Condorcet winner exists and is elected in the initial profile by the investigated voting procedure. It turns out that, obviously, all conceivable Condorcet-consistent voting procedures satisfy SCC under this restricted domain assumption. However, most known non-Condorcet-consistent procedures continue to violate SCC even under the restricted domain assumption.

Responsiveness to electoral opinions is one of the hallmarks of democratic governance. We focus on a particularly strong type of unresponsiveness, viz., one where the complete inversion of all preferences in the electorate is accompanied with no change in the electoral outcome. It is known that the possibility of this extreme type of unresponsiveness, known as the Preference Inversion paradox or Reversal Bias, is associated with many voting rules. We set out to find out whether the paradox can be encountered when using various procedures under a restricted Condorcet domain, viz., one where a Condorcet winner exists and is elected by the procedure under study.

This chapter concludes the analysis of the 20 voting procedures in terms of 5 voting paradoxes in restricted domains characterized by the existence of a Condorcet winner which at the same time is elected by the procedure under investigation. The restricted domain provides a perspective to how much difference various profile types make in terms of the possibility of encountering a voting paradox. In this analysis we contrast the general (unrestricted) domain with one where the initial outcome is stable. We illustrate the problems involved in the choice of an appropriate procedure by discussing the recent proposal for electoral reform suggested by Maskin and Sen.