Ordinal Lindahl Equilibrium for Voting

TL;DR

This paper introduces a new equilibrium concept, LEO, for discrete collective choice with ordinal preferences, enabling fair probabilistic outcomes and improving core guarantees in voting and fair machine learning.

Contribution

The paper extends Lindahl equilibrium to ordinal preferences, providing a new framework for proportional representation with efficient computation and improved core guarantees.

Findings

Constructs randomized outcomes satisfying approximate core constraints.

Provides a deterministic approximate core guarantee with a factor of 6.24.

Outcomes can be computed efficiently in structured environments.

Abstract

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.