Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banks J, Duggan J, Le Breton M (2002) Bounds for mixed strategy equilibria and the spatial model of elections. J Econ Theory 103:88–105
Banks JS (1985) Sophisticated voting outcomes and agenda control. Soc Choice Welf 1:295–306
Bianco W, Jeliazkov I, Sened I (2004) The uncovered set and the limits of legislative action. Polit Anal 12(3):256–276
Bordes G (1983) On the possibility of reasonable consistent majoritarian choice: some positive results. J Econ Theory 31:122–132
Bordes G, Le Breton M, Salles M (1992) Gillies and Miller’s subrelations of a relation over an infinite set of alternatives: General results and applications to voting games. Math Oper Res 17(3):509–518
Cox GW (1987) The uncovered set and the core. Am J Polit Sci 31(2):408–422
Duggan J (2005) Uncovered sets. University of Rochester, Mimeo
Duggan J, Jackson M (2004) Mixed strategy equilibrium and deep covering in multidimensional electoral competition. California Institute of Technology, Mimeo
Dutta B, Jackson M, Le Breton M (2004) The Banks set and the uncovered set in budget allocation problems. In: Austen-Smith D, Duggan J (eds) Forthcoming in a volume in honor of Jeffrey Banks
Epstein D (1998) Uncovering some subtleties of the uncovered set: social choice theory and distributive politics. Soc Choice Welf 15:81–93
Gillies DB (1959) Solutions to general non-zero-sum games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games, IV, Annals of mathematic studies no. 40. Princeton University Press, Princeton, NJ
Laslier J-F, Picard N (2002) Distributive politics and electoral competition. J Econ Theory 103:106–130
McKelvey R (1986) Covering, dominance, and institution-free properties of social choice. Am J Polit Sci 30(2):283–314
Miller N (1980) A new solution set for tournaments and majority voting: Further graph-theoretic approaches to the theory of voting. Am J Polit Sci 24:68–96
Penn EM (2004) The Banks set in infinite spaces. California Institute of Technology, Mimeo
Penn EM (2006) The Banks set in infinite spaces. Soc Choice Welf (in press)
Shepsle K, Weingast B (1984) Uncovered sets and sophisticated voting outcomes with implications for agenda institutions. Am J Polit Sci 28(1):49–74
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Penn, E.M. Alternate Definitions of the Uncovered Set and Their Implications. Soc Choice Welfare 27, 83–87 (2006). https://doi.org/10.1007/s00355-006-0114-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00355-006-0114-2