Alternate Definitions of the Uncovered Set and Their Implications

Different definitions of the uncovered set are commonly, and often interchangeably, used in the literature. If we assume individual preferences are strict over all alternatives, these definitions are equivalent. However, if one or more voters are indifferent between alternatives these definitions may not yield the same uncovered set. This note examines how these definitions differ in a distributive setting, where each voter can be indifferent between any number of alternatives. I show that, defined one way, the uncovered set is equal to the set of Pareto allocations that give over half the voters a strictly positive payoff, while alternate definitions yield an uncovered set that is equal to the entire Pareto set. These results highlight a small error in Epstein (Soc Choice Welf 15, 81–93, 1998) in which the author characterizes the uncovered set for a different definition of covering than claimed.