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Generalizing Liquid Democracy to multi-agent delegation: A Voting Weight Measure and Equilibrium Analysis

Francisco M. Bersetche

TL;DR

This work addresses the lack of equilibria in classical liquid democracy by introducing a fractional, multi-agent delegation framework that penalizes long delegation chains. It develops a voting weight measure $V$ via a particle-system interpretation and a small-parameter penalty $\varepsilon>0\$, ensuring both the delegation properties and the existence of Nash equilibria in the delegation game, with a well-defined limiting case $V(P)=\lim_{\varepsilon\to0}V^{\varepsilon}(P)$. The authors prove key properties, including self-selection, conservation of total voting weight, and a generalization property aligning with the classic model on binary delegations, while guaranteeing equilibrium existence for $\varepsilon>0$ through fixed-point arguments. This framework supports stable, multi-agent delegation dynamics with computable voting weights, offering a rigorous path toward practical, scalable liquid democracy implementations that retain core democratic guarantees.

Abstract

In this study, we propose a generalization of the classic model of liquid democracy that allows fractional delegation of voting weight, while simultaneously allowing for the existence of equilibrium states. Our approach empowers agents to partition and delegate their votes to multiple representatives, all while retaining a fraction of the voting power for themselves. We introduce a penalty mechanism for the length of delegation chains. We discuss the desirable properties of a reasonable generalization of the classic model, and prove that smaller penalty factors bring the model closer to satisfying these properties. In the subsequent section, we explore the presence of equilibrium states in a general delegation game utilizing the proposed voting measure. In contrast to the classical model, we demonstrate that this game exhibits pure strategy Nash equilibria, contingent upon the imposition of a penalty on the length of delegation chains.

Generalizing Liquid Democracy to multi-agent delegation: A Voting Weight Measure and Equilibrium Analysis

TL;DR

This work addresses the lack of equilibria in classical liquid democracy by introducing a fractional, multi-agent delegation framework that penalizes long delegation chains. It develops a voting weight measure via a particle-system interpretation and a small-parameter penalty , ensuring both the delegation properties and the existence of Nash equilibria in the delegation game, with a well-defined limiting case . The authors prove key properties, including self-selection, conservation of total voting weight, and a generalization property aligning with the classic model on binary delegations, while guaranteeing equilibrium existence for through fixed-point arguments. This framework supports stable, multi-agent delegation dynamics with computable voting weights, offering a rigorous path toward practical, scalable liquid democracy implementations that retain core democratic guarantees.

Abstract

In this study, we propose a generalization of the classic model of liquid democracy that allows fractional delegation of voting weight, while simultaneously allowing for the existence of equilibrium states. Our approach empowers agents to partition and delegate their votes to multiple representatives, all while retaining a fraction of the voting power for themselves. We introduce a penalty mechanism for the length of delegation chains. We discuss the desirable properties of a reasonable generalization of the classic model, and prove that smaller penalty factors bring the model closer to satisfying these properties. In the subsequent section, we explore the presence of equilibrium states in a general delegation game utilizing the proposed voting measure. In contrast to the classical model, we demonstrate that this game exhibits pure strategy Nash equilibria, contingent upon the imposition of a penalty on the length of delegation chains.
Paper Structure (23 sections, 14 theorems, 98 equations, 4 figures, 1 algorithm)

This paper contains 23 sections, 14 theorems, 98 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and main ideas
  3. Preliminaries
  4. Motivation
  5. The voting weight measure
  6. Solving delegation cycles
  7. Formal framework
  8. Well-definedness
  9. Generalization
  10. Self-selection
  11. Conservation of total voting weight
  12. Numerical computation
  13. Alternative definitions of $V$
  14. The generalization property
  15. The delegation property
  16. ...and 8 more sections

Key Result

Lemma 3.1

Let $\{a_k\}_{k \in \mathbb{N}_0}$ and $\{b_k\}_{k \in \mathbb{N}_0} \in \ell^{\infty}$ be two uniformly bounded sequences, where $a_k$ is periodic with period $q \in \mathbb{N}_0$, and $\sum_k | b_k | < \infty$. Then:

Figures (4)

  • Figure 2.1: An example illustrating a simple delegation configuration in the classical LD model.
  • Figure 3.1: According to Theorem \ref{['teo:delegation']}, the delegation configurations depicted above are equivalent. It can be observed that the act of delegating all the voting weight to agent $2$ is equivalent to replicating her delegation weighting.
  • Figure 3.2: According to Theorem \ref{['teo:delegation']}, the delegation setting on the left is equivalent to the one on the right. That is, when agent $3$ delegates the voting weight to agent $2$, the effect is the same as copying her delegation weighting and redistributing the voting weight that agent $2$ delegates back to $3$ among her proxies in proportion to their respective weights.
  • Figure A.1: We consider two settings with $N = \{1,2,3\}$, where the delegation property establishes that the voting weight distribution should be the same for both settings. However, it can be verified that this distribution differs when evaluated in each of the aforementioned settings using the measure $V^c$.

Theorems & Definitions (34)

  • Definition 2.1: Delegation cycle
  • Remark 1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Definition 3.1: Self-selection property
  • Lemma 3.2
  • proof
  • ...and 24 more