5. The (In)Vulnerability of 20 Voting Procedures to the No-Show Paradox in a Restricted Domain

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.