The Uncovered Set and the Core

Abstract

Recent work in social choice theory has focused on an important generalization of the core known as the uncovered set. Miller (1977, 1980), working with finite alternative spaces, and McKelvey (1986), working with infinite alternative spaces, argue that the uncovered set serves as a general solution set for majority voting games. They and others have shown that, under a variety of institutional settings, game theoretic behavior by participants leads to outcomes in the uncovered set. Perhaps the simplest example of this is two-candidate competition under the plurality rule. In this context, Miller (1980) observed that an electoral strategy is undominated (in the usual game theoretic sense) if and only if it is an element of the uncovered set, and McKelvey (1986), building on a previous paper by McKelvey and Ordeshook (1976), demonstrated that the uncovered set contains the support set of any mixed strategy equilibrium. In the context of multicandidate competition, Cox (1985) showed that, under certain Condorcet voting procedures, undominated strategies will be in the uncovered set. The relationship of the urfcovered set to sophisticated voting and agendas has also been explored. Shepsle and Weingast (1984) have applied and extended Miller's results to derive bounds on agenda reachable outcomes in multidimensional choice spaces, while McKelvey has shown in related work that the uncovered set contains any outcome reachable by sophisticated voting when agendas are endogenously generated. The importance of the uncovered set as a solution concept naturally motivates interest in its size and properties. Miller has demonstrated, in the case of tournaments, that the uncovered set coincides with the core, when a core exists. He has conjectured that the uncovered set is generally