The Price of Proportional Representation in Temporal Voting

TL;DR

This paper investigates the trade-off between proportional representation and social welfare in temporal voting, revealing that enforcing proportionality can lead to significant, sometimes diminishing, welfare loss over time.

Contribution

It formalizes the welfare-proportionality tension in temporal voting, analyzes the asymptotic behavior of welfare loss, and proves computational hardness results for welfare maximization under proportionality axioms.

Findings

Welfare loss due to proportionality can grow sublinearly with voters or rounds.

For justified representation, welfare loss diminishes over time and vanishes asymptotically.

Welfare maximization under proportionality axioms is NP-complete and APX-hard.

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.