Let
V
be a set of
n
points in mathcal R
d
, called
voters
. A point
p
∈ mathcal R
d
is a
plurality point
for
V
when the following holds: For every
q
∈ mathcal 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
∈
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
:= {β : any finite multiset
V
in mathcal R
d
admits a β-plurality point. We prove that β
*
d
= √3/2, and that 1/√
d
⩽ β
*
d
⩽ √ 3/2 for all
d
⩾ 3.
• Define β (
p, V
) := sup {β :
p
is a β -plurality point for
V
}. Given a voter set
V
in mathcal R
2
, we provide an algorithm that runs in
O
(
n
log
n
) time and computes a point
p
such that β (
p
,
V
) ⩾ β
*
b
. Moreover, for
d
⩾ 2, we can compute a point
p
with β (
p
,
V
) ⩾ 1/√
d
in
O
(
n
) time.
• Define β (
V
) := sup { β :
V
admits a β -plurality point}. We present an algorithm that, given a voter set
V
in mathcal R
d
, computes an ((1-ɛ)ċ β (
V
))-plurality point in time
O
n
2
ɛ
3d-2
ċ log
n
ɛ
d-1
ċ log
2
1ɛ).
Reconciling Candidate Extremism and Spatial Voting