A two-player voting game in Euclidean space

Abstract

Given a finite set $S$ of points in $\mathbb{R}^d$, which we regard as the locations of voters on a $d$-dimensional political `spectrum', two candidates (Alice and Bob) select one point in $\mathbb{R}^d$ each, in an attempt to get as many votes as possible. Alice goes first and Bob goes second, and then each voter simply votes for the candidate closer to them in terms of Euclidean distance. If a voter's distance from the two candidates is the same, they vote for nobody. We give a geometric characterization of the sets $S$ for which each candidate wins, assuming that Alice wins if they get an equal number of votes. We also show that, if not all the voters lie on a single line, then, whenever Alice has a winning strategy, there is a unique winning point for her. We also provide an algorithm which decides whether Alice has a winning point, and determines the location of that point, both in finite (in fact polynomial) time.