The No-Show Paradox Under a Restricted Domain

Abstract

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 (J Econ 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 ten Condorcet-consistent and six 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 two of the ten Condorcet-consistent procedures investigated (Maximin and Schwartz’s procedure) are not vulnerable to the no-show paradox, whereas only two of the six non-Condorcet-consistent ranked procedures investigated (Coombs’ and the Negative Plurality Elimination Rule 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.