On beta-Plurality Points in Spatial Voting Games

Let $V$ be a set of $n$ points in $\mathbb{R}^d$, called voters. A point $p\in \mathbb{R}^d$ is a plurality point for $V$ when the following holds: for every $q\in\mathbb{R}^d$ the number of voters closer to $p$ than to $q$ is at least the number of voters closer to $q$ than to $p$. Thus, in a vote where each $v\in V$ votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal $p$ will not lose against any alternative proposal $q$. For most voter sets a plurality point does not exist. We therefore introduce the concept of $\beta$-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to $p$ (but not to $q$) is scaled by a factor $\beta$, for some constant $0<\beta\leq 1$. We investigate the existence and computation of $\beta$-plurality points, and obtain the following. * Define \beta^*_d := \sup \{ \beta : \text{any finite multiset V$in$\mathbb{R}^d$admits a$\beta-plurality point} \}. We prove that $\beta^*_2 = \sqrt{3}/2$, and that $1/\sqrt{d} \leq \beta^*_d \leq \sqrt{3}/2$ for all $d\geq 3$. * Define \beta(p, V) := \sup \{ \beta : \text{p$is a$\beta$-plurality point for$V}\}. Given a voter set $V \in \mathbb{R}^2$, we provide an algorithm that runs in $O(n \log n)$ time and computes a point $p$ such that $\beta(p, V) \geq \beta^*_2$. Moreover, for $d\geq 2$ we can compute a point $p$ with $\beta(p,V) \geq 1/\sqrt{d}$ in $O(n)$ time. * Define \beta(V) := \sup \{ \beta : \text{V$admits a$\beta-plurality point}\}. We present an algorithm that, given a voter set $V$ in $\mathbb{R}^d$, computes an $(1-\varepsilon)\cdot \beta(V)$ plurality point in time $O(\frac{n^2}{\varepsilon^{3d-2}} \cdot \log \frac{n}{\varepsilon^{d-1}} \cdot \log^2 \frac {1}{\varepsilon})$.Comment: 21 pages, 10 figures, SoCG'2

On beta-Plurality Points in Spatial Voting Games

Let V be a set of n points in Rd, called voters. A point p∈Rd is a plurality point for V when the following holds: for every q∈Rd the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v∈V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0&lt;β≤1. We investigate the existence and computation of β-plurality points, and obtain the following results. * Define β∗d:=sup{β:any finite multiset V in Rd admits a β-plurality point}. We prove that β∗2=3–√/2, and that 1/d−−√≤β∗d≤3–√/2 for all d≥3. * Define β(V):=sup{β:V admits a β-plurality point}. We present an algorithm that, given a voter set V in Rd, computes an (1−ε)⋅β(V) plurality point in time O(n2ε3d−2⋅lognεd−1⋅log21ε)

On beta-Plurality Points in Spatial Voting Games

Let V be a set of n points in Rd, called voters. A point p∈Rd is a plurality point for V when the following holds: for every q∈Rd the number of voters closer to p than to q is at least the number of voters closer to q than to p. Thus, in a vote where each v∈V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal p will not lose against any alternative proposal q. For most voter sets a plurality point does not exist. We therefore introduce the concept of β-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to p (but not to q) is scaled by a factor β, for some constant 0<β≤1. We investigate the existence and computation of β-plurality points, and obtain the following results. * Define β∗d:=sup{β:any finite multiset V in Rd admits a β-plurality point}. We prove that β∗2=3–√/2, and that 1/d−−√≤β∗d≤3–√/2 for all d≥3. * Define β(V):=sup{β:V admits a β-plurality point}. We present an algorithm that, given a voter set V in Rd, computes an (1−ε)⋅β(V) plurality point in time O(n2ε3d−2⋅lognεd−1⋅log21ε)