Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tracking Truth with Liquid Democracy

Adam Berinsky, Daniel Halpern, Joseph Y. Halpern, Ali Jadbabaie, Elchanan Mossel, Ariel D. Procaccia, Manon Revel

TL;DR

The results suggest that the concerns raised about fluid democracy can be overcome, thereby bolstering the case for this emerging paradigm and proving that delegations can be treated as stochastic processes.

Abstract

The dynamics of random transitive delegations on a graph are of particular interest when viewed through the lens of an emerging voting paradigm, liquid democracy. This paradigm allows voters to choose between directly voting and transitively delegating their votes to other voters, so that those selected cast a vote weighted by the number of delegations they received. In the epistemic setting, where voters decide on a binary issue for which there is a ground truth, previous work showed that a few voters may amass such a large amount of influence that liquid democracy is less likely to identify the ground truth than direct voting. We quantify the amount of permissible concentration of power and examine more realistic delegation models, showing they behave well by ensuring that (with high probability) there is a permissible limit on the maximum number of delegations received. Our theoretical results demonstrate that the delegation process is similar to well-known processes on random graphs that are sufficiently bounded for our purposes. Along the way, we prove new bounds on the size of the largest component in an infinite Pólya urn process, which may be of independent interest. In addition, we empirically validate the theoretical results, running six experiments (for a total of $N=168$ participants, $62$ delegation graphs and over $11k$ votes collected). We find that empirical delegation behaviors meet the conditions for our positive theoretical guarantees. Overall, our work alleviates concerns raised about liquid democracy and bolsters the case for the applicability of this emerging paradigm.

Tracking Truth with Liquid Democracy

TL;DR

The results suggest that the concerns raised about fluid democracy can be overcome, thereby bolstering the case for this emerging paradigm and proving that delegations can be treated as stochastic processes.

Abstract

The dynamics of random transitive delegations on a graph are of particular interest when viewed through the lens of an emerging voting paradigm, liquid democracy. This paradigm allows voters to choose between directly voting and transitively delegating their votes to other voters, so that those selected cast a vote weighted by the number of delegations they received. In the epistemic setting, where voters decide on a binary issue for which there is a ground truth, previous work showed that a few voters may amass such a large amount of influence that liquid democracy is less likely to identify the ground truth than direct voting. We quantify the amount of permissible concentration of power and examine more realistic delegation models, showing they behave well by ensuring that (with high probability) there is a permissible limit on the maximum number of delegations received. Our theoretical results demonstrate that the delegation process is similar to well-known processes on random graphs that are sufficiently bounded for our purposes. Along the way, we prove new bounds on the size of the largest component in an infinite Pólya urn process, which may be of independent interest. In addition, we empirically validate the theoretical results, running six experiments (for a total of participants, delegation graphs and over votes collected). We find that empirical delegation behaviors meet the conditions for our positive theoretical guarantees. Overall, our work alleviates concerns raised about liquid democracy and bolsters the case for the applicability of this emerging paradigm.

Paper Structure

This paper contains 75 sections, 7 theorems, 99 equations, 24 figures, 19 tables.

Table of Contents

  1. Introduction
  2. Our Contributions and Techniques
  3. Stochastic Delegations
  4. Delegation Models
  5. Upward Delegation:
  6. Confidence-Based Delegation.
  7. General Continuous Delegation.
  8. Component Sizes in Infinite Pólya Urn Processes
  9. Consistency With Experiments
  10. Related work
  11. Model
  12. Delegation graphs
  13. Delegation instances
  14. Gain of a delegation instance
  15. Randomization over delegation instances
  16. ...and 60 more sections

Key Result

Lemma 1

If $M$ is a model, $\mathfrak{D}$ a class of distributions, $n$ a number of persons, and for all distributions $\mathcal{D} \in \mathfrak{D}$, there is an $\alpha \in (0, 1)$ and $C: \mathbb{N} \to \mathbb{N}$ with $C(n) \in o(n)$ such that then $M$ satisfies probabilistic do no harm. If in addition, there exists a distribution $\mathcal{D} \in \mathfrak{D}$ and an $\alpha \in (0, 1)$ such that

Figures (24)

  • Figure 1: Delegation graphs for task $T_7$ ("You will be given upcoming European men soccer games and asked to predict the games' outcome.") from Experiment $6.$ Each node is a voter and the node's number represents the rounded expertise $\eta_{i, t}$ of a given voter $i$ for task $t,$ computed using Item Response Theory, see \ref{['app:IRT']}.
  • Figure 2: Pooled estimates of $\varphi^{\ell}_{e, t}$, both for each bucket individually, and grouped together. The blue crosses show the values computed for $\varphi_{e,t}^{\ell}(\eta_k)$. The pink dots show the average across all values for that $\eta_k$, and the pink lines correspond to a linear regression over the mean values. We observe increasing trends across the board, with slope (coefficient of determination) being $0.53 (0.90), 0.28 (0.46), 0.29 (0.47)$ and $0.60 (0.92)$, respectively, for individual buckets, and $0.38 (0.85)$ for the pooled test. The shaded area represents the $95\%$ confidence interval.
  • Figure 3: Example of a survey flow with three tasks.
  • Figure 4: Screenshots from the survey of both task prompts and specific questions.
  • Figure 5: Delegation graphs for task $T_3$ from Experiment $2$ (left) and task $T_2$ from Experiment $3$ (right).
  • ...and 19 more figures

Theorems & Definitions (9)

  • Definition 1: Probabilistic do no harm
  • Definition 2: Probabilistic positive gain
  • Lemma 1
  • Theorem 1: Upward Delegation Model
  • Lemma 2
  • Theorem 2: Confidence-Based Delegation Model
  • Theorem 3: Continuous General Delegation Model
  • Lemma 3: Hoeffding's Inequality
  • Proposition 1