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.

翻译：我们研究时间投票模型中的比例代表制，其中集体决策在一段固定时间跨期内反复进行。先前工作已广泛探讨了多赢家投票中的比例代表制公理（例如正当代表制（JR）及其变体）如何在此环境下被适配、满足及验证。然而，这些公理与社会福利之间的相互作用尚不明确。本文量化了执行比例原则的效率成本。我们通过无比例约束下可实现的最大功利主义社会福利与受比例公理约束下可实现的最大社会福利之间的最坏情况比率，形式化了福利与比例之间的张力。研究表明，在时间投票环境中施加比例代表制会导致福利损失随选民或轮次数量增加而增长，但呈次线性趋势。我们进一步揭示了各公理间的显著区分：对于JR公理，福利损失随时间跨度增长而逐渐减少并渐近消失；而对于更强的公理，即使经过多轮投票，这一冲突依然存在。此外，我们证明即使在静态偏好和有界度认可条件下，各公理下的福利最大化问题均为NP完全和APX-hard问题，并提供了基于若干自然结构参数的固定参数算法。