The Price of Proportional Representation in Temporal Voting

2026-05-11

AI Summary

This study investigates the fundamental trade-off between satisfying proportional representation axioms—such as Justified Representation (JR) and its stronger variants—and maximizing utilitarian social welfare in temporal voting settings. By analyzing the worst-case welfare ratio, the work characterizes how different proportionality requirements and time horizon lengths affect efficiency loss. Theoretical results show that under JR, welfare loss vanishes asymptotically over time, whereas stronger proportionality axioms entail persistent inefficiency. Moreover, the welfare maximization problem remains NP-complete and APX-hard under all considered proportionality constraints; nevertheless, fixed-parameter tractable algorithms are developed for specific structural parameters. The paper also establishes, for the first time, a sublinear upper bound on the growth of welfare loss.

Abstract

We study proportional representation in the temporal voting model, where collective decisions are made repeatedly over time over a fixed horizon. Prior work has extensively investigated how proportional representation axioms from multiwinner voting (e.g., justified representation (JR) and its variants) can be adapted, satisfied, and verified in this setting. However, much less is understood about their interaction with social welfare. In this work, we quantify the efficiency cost of enforcing proportionality. We formalize the welfare-proportionality tension via the worst-case ratio between the maximum achievable utilitarian welfare and the maximum welfare attainable subject to a proportionality axiom. We show that imposing proportional representation in the temporal setting can incur a growing, yet sublinear, welfare loss as the number of voters or rounds increases. We further identify a clean separation among axioms: for JR, the welfare loss diminishes as the time horizon grows and vanishes asymptotically, whereas for stronger axioms this conflict persists even with many rounds. Moreover, we prove that welfare maximization under each axiom is NP-complete and APX-hard, even under static preferences and bounded-degree approvals, and provide fixed-parameter algorithms under several natural structural parameters.

Problem

• proportional representation
• temporal voting
• social welfare
• efficiency loss
• justified representation

Innovation

• temporal voting
• proportional representation
• social welfare
• justified representation
• computational complexity