多赢家选举中的纳什核心
Nash Core in Multiwinner Election
- Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究多赢家选举中的纳什核心，证明分数场景下纳什核心始终存在，扩展其至离散场景并可高效验证，还通过迭代算法在真实数据上高效计算该解。
AI中文摘要：
在基于认可的委员会选举问题中，若不存在任何选民子集有动机通过选择比例规模的“阻碍”委员会来偏离，且偏离组中的每位选民都严格偏好该阻碍委员会，则称该委员会处于核心（core）中。我们考虑候选人可被分数选取的场景，在温和的正则性假设下，证明始终存在对候选人的加权方式，使得最大化候选人加权纳什社会福利（Nash Social Welfare）的分数委员会处于核心，我们将此类解称为“纳什核心（Nash core）”。此外，证明纳什核心解可在选民与候选人之间分配支付，每位选民按权重比例向其认可的候选人支付。对于候选人被完全纳入或排除的离散场景，我们通过对分数纳什核心解取整，证明每位至多含8名等权重选民的基于认可的委员会选举均存在核心委员会。尽管该场景下核心的非空性仍是开放问题，且检查核心成员资格是coNP难问题，但我们将纳什核心的概念扩展至离散场景，得到一种可高效验证的形式，为建立离散场景下的核心存在性提供了可行路径。最后，我们采用支付引导启发式方法在真实投票数据上测试所提方法，实证表明无论是分数场景还是离散场景，均可通过迭代算法高效计算纳什核心解。
英文摘要：
In the approval-based committee selection problem, a committee is said to be in the core if no subset of voters has an incentive to deviate by selecting a \emph{blocking} committee of proportional size, such that every voter in the deviating group strictly prefers the blocking committee. We consider the setting where candidates can be selected fractionally. Under a mild regularity assumption, we show that there always exists a weighting of candidates such that the fractional committee maximizing the candidate-weighted Nash Social Welfare is in the core. We refer to such a solution as being in the \emph{Nash core}. Additionally, we show that a Nash core solution admits a payment assignment between voters and candidates, where each voter pays a candidate they approve in proportion to the weight. For the discrete setting, where each candidate is either included or excluded from the committee, we prove that every approval-based committee election with at most eight equally weighted voters has a core committee by rounding the fractional Nash core solution. Although the non-emptiness of the core in this setting remains an open question and checking core membership is coNP-hard, we extend the notion of the Nash core to the discrete case, yielding a formulation that is efficiently verifiable and offers a promising path toward establishing core existence in discrete settings. Finally, we test our approach on real voting data using a payment-guided heuristic. We empirically show that the Nash core solution can be efficiently computed through an iterative algorithm in both the fractional and discrete settings.
