The core is a central concept in multi-winner social choice, ensuring that no coalition of voters can support an alternative outcome whose size or cost exceeds the group's share of the electorate. This idea originates from the Lindahl equilibrium in classical public goods theory. Yet Lindahl equilibria may fail to exist when voters have ordinal preferences over a finite set of outcomes and monetary transfers are not allowed. We introduce Lindahl Equilibrium with Ordinal Preferences (LEO), extending the equilibrium framework to discrete collective choice. Using LEO, we construct randomized outcomes that satisfy (approximate) core constraints for a probabilistic set of voters, while ensuring that each voter is represented with high probability. We also provide a deterministic approximate core guarantee with a factor of 6.24, improving on the previous bound of 32. In structured environments, these outcomes can be computed efficiently. Overall, our results extend classical equilibrium concepts, providing a normative foundation for proportional representation and practical algorithms for applications in voting and fair machine learning.

翻译：核心是多赢家社会选择中的一个核心概念，它确保不存在任何选民联盟能够支持一个规模或成本超过该群体在全体选民中所占份额的替代结果。这一思想源于经典公共物品理论中的林达尔均衡。然而，当选民对有限结果集具有序数偏好且不允许货币转移时，林达尔均衡可能不存在。我们引入了序数偏好下的林达尔均衡，将均衡框架扩展到离散集体选择。利用LEO，我们构建了满足（近似）核心约束的随机化结果，该约束针对一个概率性的选民集合，同时确保每位选民以高概率被代表。我们还提供了一个确定性近似核心保证，其因子为6.24，改进了先前32的界限。在结构化环境中，这些结果可以被高效计算。总体而言，我们的研究扩展了经典均衡概念，为比例代表制提供了规范性基础，并为投票和公平机器学习中的应用提供了实用算法。