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