Spatial Voting with Incomplete Voter Information

TL;DR

This paper investigates the computational complexity of determining possible and necessary winners in spatial voting models with incomplete voter location information, revealing tractable cases and hardness results across dimensions and voting rules.

Contribution

It characterizes the complexity of winner determination in spatial voting with incomplete data, identifying tractable and intractable scenarios for various rules and dimensions.

Findings

Possible-winner problem is tractable in 1D for positional scoring rules.

Necessary-winner problem is tractable in up to 2D for approval voting.

Possible-winner problem is hard in any dimension for approval voting.

Abstract

We consider spatial voting where candidates are located in the Euclidean $d$-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.