Finding Possible Winners in Spatial Voting with Incomplete Information

TL;DR

This paper studies the computational complexity of identifying possible winners in spatial voting models with incomplete location data, providing polynomial-time algorithms and complexity classifications for various voting rules and dimensions.

Contribution

It offers new polynomial-time algorithms for one-dimensional cases and complexity classifications, including NP-complete and fixed-parameter tractable results, for various scoring rules.

Findings

Possible winner problem in 1D is polynomial for all constant-k truncated rules.

NP-completeness for approval voting in any dimension and k-approval in 2D or higher.

FPT algorithms for certain scoring rules parameterized by candidate count.

Abstract

We consider a spatial voting model where both candidates and voters are positioned in the $d$-dimensional Euclidean space, and each voter ranks candidates based on their proximity to the voter's ideal point. We focus on the scenario where the given information about the locations of the voters' ideal points is incomplete; for each dimension, only an interval of possible values is known. In this context, we investigate the computational complexity of determining the possible winners under positional scoring rules. Our results show that the possible winner problem in one dimension is solvable in polynomial time for all $k$-truncated voting rules with constant $k$. Moreover, for some scoring rules for which the possible winner problem is NP-complete, such as approval voting for any dimension or $k$-approval for $d \geq 2$ dimensions, we give an FPT algorithm parameterized by the number of candidates. Finally, we classify tractable and intractable settings of the weighted possible winner problem in one dimension, and resolve the computational complexity of the weighted case for all two-valued positional scoring rules when $d=1$.