2 research outputs found

    On beta-Plurality Points in Spatial Voting Games

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    Let VV be a set of nn points in Rd\mathbb{R}^d, called voters. A point p∈Rdp\in \mathbb{R}^d is a plurality point for VV when the following holds: for every q∈Rdq\in\mathbb{R}^d the number of voters closer to pp than to qq is at least the number of voters closer to qq than to pp. Thus, in a vote where each v∈Vv\in V votes for the nearest proposal (and voters for which the proposals are at equal distance abstain), proposal pp will not lose against any alternative proposal qq. For most voter sets a plurality point does not exist. We therefore introduce the concept of Ξ²\beta-plurality points, which are defined similarly to regular plurality points except that the distance of each voter to pp (but not to qq) is scaled by a factor Ξ²\beta, for some constant 0<β≀10<\beta\leq 1. We investigate the existence and computation of Ξ²\beta-plurality points, and obtain the following. * Define \beta^*_d := \sup \{ \beta : \text{any finite multiset Vin in \mathbb{R}^dadmitsa admits a \beta-plurality point} \}. We prove that Ξ²2βˆ—=3/2\beta^*_2 = \sqrt{3}/2, and that 1/d≀βdβˆ—β‰€3/21/\sqrt{d} \leq \beta^*_d \leq \sqrt{3}/2 for all dβ‰₯3d\geq 3. * Define \beta(p, V) := \sup \{ \beta : \text{pisa is a \betaβˆ’pluralitypointfor-plurality point for V}\}. Given a voter set V∈R2V \in \mathbb{R}^2, we provide an algorithm that runs in O(nlog⁑n)O(n \log n) time and computes a point pp such that Ξ²(p,V)β‰₯Ξ²2βˆ—\beta(p, V) \geq \beta^*_2. Moreover, for dβ‰₯2d\geq 2 we can compute a point pp with Ξ²(p,V)β‰₯1/d\beta(p,V) \geq 1/\sqrt{d} in O(n)O(n) time. * Define \beta(V) := \sup \{ \beta : \text{Vadmitsa admits a \beta-plurality point}\}. We present an algorithm that, given a voter set VV in Rd\mathbb{R}^d, computes an (1βˆ’Ξ΅)β‹…Ξ²(V)(1-\varepsilon)\cdot \beta(V) plurality point in time O(n2Ξ΅3dβˆ’2β‹…log⁑nΞ΅dβˆ’1β‹…log⁑21Ξ΅)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
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