Identifying Strong Voter Support: Condorcet and Smith Revisited

The conditions of strong Condorcet winner consistency and strong Condorcet loser consistency are, in essence, universally accepted as attractive criteria to evaluate the performance of social choice functions. However, there are many situations in which these conditions are silent because such winners and losers may not exist. Hence, weakening these desiderata in order to extend the domain of profiles where they apply is an appealing task. Yet, the often-proposed and accepted weak counterparts of these properties suffer from the shortcoming that a weak Condorcet winner can be a weak Condorcet loser at the same time. We propose new notions of Condorcet-type winners and losers that are between these two extremes: they share the intuitive appeal of strong Condorcet winner consistency and strong Condorcet loser consistency and avoid the contradictory recommendations that would derive from the double identification of candidates as being weak Condorcet winners and losers at the same time. We provide a thorough examination of the extent to which some important reference properties are satisfied by social choice functions that are consistent with our new proposals. In addition, we revisit the concept of Smith sets (Smith, 1973) and examine a possible modification. As is the case for our intermediate Condorcet winners and losers, these notions are intended to generalize Condorcet’s ideas. Using our reference properties again, we discuss the social choice functions that are consistent with the selection of candidates from these sets. By contrasting the consequences of using the suggestions inspired by Smith with those that are implied by the intermediate Condorcet consistency conditions that we propose, we hope to shed new light on the possibilities of extending Condorcet’s principles to a larger set of circumstances.