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A two-player voting game in Euclidean space

Stelios Stylianou

TL;DR

This work analyzes a two-player voting game in Euclidean space, where voters vote for the nearest candidate and ties abstain. It provides a complete geometric characterization of when Alice has a winning strategy: in general, a single point $x$ with the property that every affine hyperplane through $x$ cuts $S$ into at most half on each side determines a winning $a=x$, with higher-dimensional versions framed in terms of good/perfect hyperplanes and balanced lines; odd/even cardinalities yield distinct structural criteria. The authors give polynomial-time algorithms to decide the existence of a winning point and to locate it (uniqueness holds when not all voters are collinear), and they analyze the size of the Alice-winning set, showing it is typically nowhere dense except in a few low-dimensional, highly symmetric cases. They also establish that Alice can guarantee at least $|S|/(d+1)$ votes, linking the problem to Radon-type partition results and providing a tight, dimension-dependent bound with potential implications for geometric voting theory and related optimization problems.

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.

A two-player voting game in Euclidean space

TL;DR

This work analyzes a two-player voting game in Euclidean space, where voters vote for the nearest candidate and ties abstain. It provides a complete geometric characterization of when Alice has a winning strategy: in general, a single point with the property that every affine hyperplane through cuts into at most half on each side determines a winning , with higher-dimensional versions framed in terms of good/perfect hyperplanes and balanced lines; odd/even cardinalities yield distinct structural criteria. The authors give polynomial-time algorithms to decide the existence of a winning point and to locate it (uniqueness holds when not all voters are collinear), and they analyze the size of the Alice-winning set, showing it is typically nowhere dense except in a few low-dimensional, highly symmetric cases. They also establish that Alice can guarantee at least votes, linking the problem to Radon-type partition results and providing a tight, dimension-dependent bound with potential implications for geometric voting theory and related optimization problems.

Abstract

Given a finite set of points in , which we regard as the locations of voters on a -dimensional political `spectrum', two candidates (Alice and Bob) select one point in 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 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.

Paper Structure

This paper contains 11 sections, 17 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The result in two dimensions
  3. Odd number of voters
  4. Even number of voters
  5. The result in higher dimensions
  6. Odd number of voters
  7. Even number of voters
  8. How 'big' is the set of Alice wins?
  9. Algorithms for determining the winning point for Alice
  10. Odd number of voters
  11. Even number of voters

Key Result

Theorem 1.1

Alice wins if and only if there exists some point $x \in \mathbb{R}^d$ such that any affine hyperplane through $x$ contains at most half the points of $S$ on either (strict) side. Moreover, provided that not all voters lie on a single line, Alice cannot have two distinct winning points.

Figures (3)

  • Figure 1: A winning set for Alice with $\abs{S}$ odd and $a \in S$, or with $\abs{S}$ even and $a \notin S$.
  • Figure 2: A winning set for Alice with $\abs{S}$ even and $a \in S$.
  • Figure 3: Finding a perfect line that does not contain $x$.

Theorems & Definitions (34)

  • Theorem 1.1
  • proof
  • Remark 1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Lemma 2.7
  • ...and 24 more