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Ordinal Lindahl Equilibrium for Voting

Haoyu Song, Thanh Nguyen

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.

Ordinal Lindahl Equilibrium for Voting

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.
Paper Structure (27 sections, 1 theorem, 18 equations, 2 figures, 3 algorithms)

This paper contains 27 sections, 1 theorem, 18 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Randomized Core
  3. Deterministic Approximate Core.
  4. Alternative Approximate Core Concepts.
  5. Lindahl equilibrium approach
  6. Model
  7. Budgeted Social Choice
  8. Solution Concepts
  9. Lindahl Equilibrium with Ordinal Preferences
  10. Classical Lindahl equilibrium
  11. Lindahl Equilibrium with Ordinal Preference (LEO)
  12. Example.
  13. From LEO to a Randomized Core
  14. Proof of Lemma \ref{['theo:maintech']}
  15. From Randomized to Deterministic Core
  16. ...and 12 more sections

Key Result

Theorem 1

A Lindahl equilibrium exists, and the equilibrium allocation ${\bf x}^*$ lies in the core. That is, for any alternative ${\bf x}$,

Figures (2)

  • Figure 1: An Illustration of LEO
  • Figure 2: Division of voters into three categories according to a partial core solution $(o^*,V^*)$ and alternative outcome $o \neq o^*$.

Theorems & Definitions (1)

  • Theorem : Lindahl70