时间投票中的完全正当代表：高效计算与验证

中文摘要 AI 辅助

在时间投票中，一组固定的选民在若干轮次中分别做出集体决策，而比例性要求具有共同利益的群体在这些决策中得到代表。完全正当代表（FJR）是该设定下已知可满足的最强比例性公理之一，因为它允许群体中的不同成员由不同的当选候选人来满足。然而，FJR结果是否能在多项式时间内计算仍然是一个开放问题。我们通过引入时间FJR+（temporal FJR+）解决了这个问题，它是FJR和扩展正当代表+（EJR+）的加强版，可以在多项式时间内计算和验证。该公理通过群体中批准同一候选人的最大成员数来衡量群体在每一轮中的一致性，并要求某个成员达到该群体按其比例份额所能获得的最佳平均满意度的整数部分，当该份额为整数时，允许最多一个单位的短缺。我们证明验证问题可归结为统计各满意度阈值以下选民的批准数。对于计算，我们观察到比例批准投票（PAV）的局部搜索可能在违反FJR的结果处停止。有点反直觉的是，解决方法是：从PAV得分中减去最小选民满意度的一个小倍数；这样每个FJR+违反都允许一个单轮更改，其改进保证了多项式运行时间。我们进一步证明，时间FJR+严格强于FJR和EJR+的组合，并且在政党名单偏好上与低配额一致。最后，时间FJR+和多赢家FJR+在一般情况下不可比较，但当批准是静态的且每个候选-轮次对被视为独立候选时，它们是一致的。

英文摘要

In temporal voting, a fixed set of voters makes a collective decision in each of several rounds, and proportionality requires that groups with shared interests be represented across these decisions. Full justified representation (FJR) is among the strongest proportionality axioms known to be satisfiable in this setting, since it allows different members of a group to be satisfied by different selected candidates. However, whether an FJR outcome can be computed in polynomial time has remained open. We resolve this question by introducing temporal FJR+, a strengthening of both FJR and extended justified representation+ (EJR+) that can be computed and verified in polynomial time. The axiom measures a group's agreement in each round by the largest number of its members who approve a common candidate, and requires some member to attain the integer part of the group's proportional share of its best attainable average satisfaction, with a shortfall of at most one unit when this share is an integer. We show that verification reduces to counting approvals among the voters below each satisfaction threshold. For computation, we observe that local search for Proportional Approval Voting (PAV) can stop at an outcome violating FJR. Somewhat counterintuitively, the remedy is to subtract a small multiple of the minimum voter satisfaction from the PAV score: every FJR+ violation then admits a single-round change whose improvement guarantees polynomial running time. We further show that temporal FJR+ is strictly stronger than FJR and EJR+ combined, and coincides with lower quota on party-list preferences. Finally, temporal FJR+ and multiwinner FJR+ are incomparable in general, but coincide when approvals are static and each candidate-round pair is treated as a separate candidate.

发表机构

• University of Oxford（牛津大学）

机构由 AI 辅助整理，请以论文原文为准。