Generalized medians and a political center

Building on properties of the median in one dimension under simple majority rule, I propose two generalizations for multi-dimensional environments and general voting rules. A P-ball (for pivotal) is a smallest radius ball such that, for every pair of alternatives, if all its members prefer one alternative over another then all members of some winning coalition share that preference (and there does not exist a winning coalition all the members of which have the opposite strict preference). A W-ball (for winning) is a smallest radius ball such that, if all members of some winning coalition prefer one alternative over another, then there exists a member of that ball that shares that preference. P-balls and W-balls coincide with the unicameral yolk (McKelvey in Am J Polit Sci 30(2):283–314, 1986; Ferejohn et al., in Soc Choice Welf 1:45–67, 1984) under simple majority rule. Using these constructs, I generalize and sharpen McKelvey’s (Am J Polit Sci 30(2):283–314, 1986) circular bounds on the set of alternatives socially preferred to any alternative, and bound the core and the uncovered set for general voting rules. I study comparative statics on the effect of changes in the voting rule.