Forming Plurality at Minimum Cost

Abstract

In this paper, we are concerned with a kind of spatial equilibria, the plurality points, in two-dimensional Euclidean space. Given a set of voters, each of which corresponds to a point, a plurality point is defined as a point closer to at least as many voters as any other point in the space. We generalize the definition by appending weights on the voters, and define the plurality point as a point $\vartriangle$ satisfying that the sum of weights of the voters closer to $\vartriangle$ is no less than that of the voters closer to any other point in the space. To remedy the issue of the non-existence of plurality points, we investigate the problem of eliminating some voters with minimum “cost” so that there is a plurality point with respect to the remaining voters. We show that the problem is NP-hard. Moreover, if all voters’ weights are restricted to be equal, we show that the problem can be solved in O(n⁵logn) time, where n is the number of voters.