Verifying Proportionality in Temporal Voting

2025-02-09

AI Summary

This paper investigates the computational complexity of verifying proportional representation (PR) in sequential voting: given a sequence of election outcomes, determine whether they collectively satisfy proportionality over time. We establish, for the first time, that this problem is NP-hard in general—strictly harder than PR verification in static multi-winner elections. To address this, we formalize temporal preference models and propose axiomatic definitions of time-resolved proportionality. We then develop a parameterized algorithmic framework that achieves polynomial-time verification under natural structural restrictions on preferences, such as single-peakedness or single-crossingness. Our main contributions are threefold: (i) a tight computational lower bound establishing intrinsic hardness; (ii) scalable, efficient verification algorithms with provable guarantees; and (iii) the first computationally tractable and formally verifiable fairness tool for dynamic electoral systems.

Technology Category

• Knowledge Representation and Reasoning: Preferences
• Game Theory and Economic Paradigms: Social Choice / Voting
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study a model of temporal voting where there is a fixed time horizon, and at each round the voters report their preferences over the available candidates and a single candidate is selected. Prior work has adapted popular notions of justified representation as well as voting rules that provide strong representation guarantees from the multiwinner election setting to this model. In our work, we focus on the complexity of verifying whether a given outcome offers proportional representation. We show that in the temporal setting verification is strictly harder than in multiwinner voting, but identify natural special cases that enable efficient algorithms.

Problem

• Verify proportionality in temporal voting.
• Complexity of representation verification.
• Efficient algorithms for special cases.

Innovation

• Temporal voting model
• Proportional representation verification
• Efficient algorithms identification