Finding Possible Winners in Spatial Voting with Incomplete Information

2025-05-18

AI Summary

This paper investigates the computational complexity of determining whether a candidate is a “possible winner” under positional scoring rules when voters’ ideal points in a spatial voting model are known only as intervals. Focusing on distance-based preference rankings in multidimensional Euclidean space, we establish the first systematic complexity boundary: the problem is in P for one-dimensional truncation rules, but NP-complete for approval, k-approval, and general weighted dichotomous scoring rules in dimensions ≥ 2. We further design a fixed-parameter tractable (FPT) algorithm parameterized by the number of candidates, fully resolving the tractability classification for weighted scoring rules in one dimension. Our core contribution lies in developing a computational complexity framework for spatial voting under incomplete geometric information, delivering tight tractability characterizations and efficient parameterized algorithms.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

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 $d=1$.

Problem

• Determining possible winners in spatial voting with incomplete voter information
• Analyzing computational complexity under positional scoring rules
• Classifying tractable and intractable cases for weighted possible winners

Innovation

• Polynomial-time solution for 1D truncated rules
• FPT algorithm for NP-complete scoring rules
• Complexity classification for weighted 1D cases