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Propagational Proxy Voting

Yasushi Sakai, Parfait Atchade-Adelomou, Ryan Jiang, Luis Alonso, Kent Larson, Ken Suzuki

TL;DR

Propagational Proxy Voting (PPV) extends Liquid Democracy by enabling fractional delegation across proposals, participants, and intermediaries, modeled via absorbing Markov chains to obtain a stable consensus. The framework combines a CES-based utility for delegation with an influence term and evolves over discrete periods, capturing both policy alignment and network effects through a limit distribution $\mathcal{W}$ and Net Proxy Vote $p_i$. Empirically, a Cambridge Participatory Budgeting study with $n=69$ participants demonstrated that intermediaries such as categories can dominate delegation flows and that PPV yields nuanced rankings and fractional vote totals that preserve the one-person-one-vote principle. The work highlights potential for real-world participatory platforms to handle complex, hierarchical knowledge structures, while noting computational scalability as a key area for further development. Overall, PPV provides a flexible, principled approach to richer democratic expression with quantifiable influence propagation.

Abstract

This paper proposes a voting process in which voters allocate fractional votes to their expected utility in different domains: over proposals, other participants, and sets containing proposals and participants. This approach allows for a more nuanced expression of preferences by calculating the result and relevance within each node. We modeled this by creating a voting matrix that reflects their preference. We use absorbing Markov chains to gain the consensus, and also calculate the influence within the participating nodes. We illustrate this method in action through an experiment with 69 students using a budget allocation topic.

Propagational Proxy Voting

TL;DR

Propagational Proxy Voting (PPV) extends Liquid Democracy by enabling fractional delegation across proposals, participants, and intermediaries, modeled via absorbing Markov chains to obtain a stable consensus. The framework combines a CES-based utility for delegation with an influence term and evolves over discrete periods, capturing both policy alignment and network effects through a limit distribution and Net Proxy Vote . Empirically, a Cambridge Participatory Budgeting study with participants demonstrated that intermediaries such as categories can dominate delegation flows and that PPV yields nuanced rankings and fractional vote totals that preserve the one-person-one-vote principle. The work highlights potential for real-world participatory platforms to handle complex, hierarchical knowledge structures, while noting computational scalability as a key area for further development. Overall, PPV provides a flexible, principled approach to richer democratic expression with quantifiable influence propagation.

Abstract

This paper proposes a voting process in which voters allocate fractional votes to their expected utility in different domains: over proposals, other participants, and sets containing proposals and participants. This approach allows for a more nuanced expression of preferences by calculating the result and relevance within each node. We modeled this by creating a voting matrix that reflects their preference. We use absorbing Markov chains to gain the consensus, and also calculate the influence within the participating nodes. We illustrate this method in action through an experiment with 69 students using a budget allocation topic.

Paper Structure

This paper contains 22 sections, 3 theorems, 42 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Prior Work
  4. The model
  5. Step one: Liquid Democracy
  6. Step two: Propagational Proxy Voting
  7. Step 3: Generalization with Intermediaries
  8. Influence: Net Proxy Vote
  9. Utility Function
  10. Utility Function Definition
  11. Simplified Utility Function
  12. Constant Elasticity of Substitution Function for Vote Delegation
  13. Incorporating Influence into the Utility Function
  14. Dynamic Model in Discrete Periods
  15. Total Utility
  16. ...and 7 more sections

Key Result

Theorem 4.3

Let $\mathcal{V}$ be a voting matrix. Then, the sequence of its powers $\mathcal{V}^x$ converges as $x \to \infty$. Specifically, there exists a matrix $\mathcal{W}$ such that:

Figures (2)

  • Figure 1: An example of PPV in action. The left shows the information as a network graph structure. Edges indicate partial trust between Nodes which are individuals like Alice, Bob, Charlie, and the parents group, family trip locations: Seoul, Hanoi, and Barcelona, and the category 'Asia'. The accompanying matrix on the right explains the identical information with real values ranging from 0 to 1. Notably, Alice (red) has given a higher preference (0.6) to visit Seoul, while Bob shows a preference for Hanoi, but delegating 30% and 20% of voting power to Daniel and Alice respectively. It indicates varying preferences among individuals, note that the sum of the columns is $1.0.$
  • Figure 2: The PPV webapp interface. Participants individually rated using a slider interface.

Theorems & Definitions (7)

  • Definition 4.1: Liquid Democracy Model
  • Definition 4.2: Propagational Proxy Voting (PPV)
  • Theorem 4.3
  • Corollary 4.4: Optimized Computation of the Limit Matrix
  • Lemma 4.5: Parallelism of Linear Functionals under a Rank-One Update
  • proof
  • proof : Net Proxy Vote