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A Natural Adaptive Process for Collective Decision-Making

Florian Brandl, Felix Brandt

TL;DR

It is proved that the empirical distribution of collective decisions produced by this process approximates a maximal lottery, a celebrated probabilistic voting rule proposed by Peter C. Fishburn.

Abstract

Consider an urn filled with balls, each labeled with one of several possible collective decisions. Now, let a random voter draw two balls from the urn and pick her more preferred as the collective decision. Relabel the losing ball with the collective decision, put both balls back into the urn, and repeat. Once in a while, relabel a randomly drawn ball with a random collective decision. We prove that the empirical distribution of collective decisions produced by this process approximates a maximal lottery, a celebrated probabilistic voting rule proposed by Peter C. Fishburn (Rev. Econ. Stud., 51(4), 1984). In fact, the probability that the collective decision in round $n$ is made according to a maximal lottery increases exponentially in $n$. The proposed procedure is more flexible than traditional voting rules and bears strong similarities to natural processes studied in biology, physics, and chemistry as well as algorithms proposed in machine learning.

A Natural Adaptive Process for Collective Decision-Making

TL;DR

It is proved that the empirical distribution of collective decisions produced by this process approximates a maximal lottery, a celebrated probabilistic voting rule proposed by Peter C. Fishburn.

Abstract

Consider an urn filled with balls, each labeled with one of several possible collective decisions. Now, let a random voter draw two balls from the urn and pick her more preferred as the collective decision. Relabel the losing ball with the collective decision, put both balls back into the urn, and repeat. Once in a while, relabel a randomly drawn ball with a random collective decision. We prove that the empirical distribution of collective decisions produced by this process approximates a maximal lottery, a celebrated probabilistic voting rule proposed by Peter C. Fishburn (Rev. Econ. Stud., 51(4), 1984). In fact, the probability that the collective decision in round is made according to a maximal lottery increases exponentially in . The proposed procedure is more flexible than traditional voting rules and bears strong similarities to natural processes studied in biology, physics, and chemistry as well as algorithms proposed in machine learning.

Paper Structure

This paper contains 16 sections, 17 theorems, 82 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Maximal Lotteries, Dynamic Voting, and Approximate Axiomatics
  3. Applications Beyond Collective Decision-Making
  4. The Model
  5. Maximal Lotteries
  6. Markov Chains
  7. The Urn Process
  8. The Result
  9. The Case of a Condorcet Winner
  10. Discussion
  11. Equilibrium learning
  12. Evolutionary Game Theory
  13. A Continuous Vector Field Induced by the Markov Chain
  14. Properties of the Deterministic Process
  15. Properties of the Stochastic Process
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\delta,\tau > 0$. Then, there is $r_0>0$ such that for all $0< r\le r_0$, there are $p^*\in\mathit{ML}\xspace({R})$ and $N_0\in\mathbb N$ such that for all $N \ge N_0$ and $p_0\in \Delta^{(N)}$, almost surely Moreover, there is $C>0$ such that for all $n\in \mathbb N_0$,

Figures (4)

  • Figure 1: Illustration of one round of the urn process (i and ii) and the main result (iii).
  • Figure 2: Simulations of the urn process. The left diagram shows the urn process for the profile given in \ref{['ex:condorcetwinner']} using an urn with $N = 50$ balls for 1,000 rounds and mutation rate $r = 0.02$, starting from an almost uniform distribution. Each intersection of the grid lines corresponds to a configuration of the urn. The right diagram shows the urn process for the profile given in \ref{['ex:condorcetcycle']} using an urn with $N = 5,000$ balls for 500,000 rounds and mutation rate $r = 0.04$, starting from the degenerate distribution in which all balls are labeled with Alternative 2. The green lines depict the actual distribution of balls while the red lines depict the temporal average of urn distributions until the given round.
  • Figure 3: Simulation of the urn process for the profile in \ref{['ex:3cycle+condorcetloser']} on an urn with $N = 50,000$ balls for $10^7$ rounds and mutation rate $r = 0.01$. The solid lines show the fraction of balls in the urn. The dashed lines show the temporal average of the fraction of balls in the urn until the given round. The unique maximal lottery is $p=(1/3,1/6,1/2,0)$. The dotted line shows the relative entropy $D(p\mid q) = \sum_{i\in[d]} p_i \log(\frac{p_i}{q_i})$ of $p$ with respect to the distribution in the urn $q$.
  • Figure 4: The continuous deterministic process $y(t)$ solving \ref{['eq:diffeq1']} for the profile in \ref{['ex:3cycle+condorcetloser']} with $r = 0.01$ on the left and $r = 0$ on the right. For strictly positive $r$, $y(t)$ converges to a zero of $f^{(r)}$ (see \ref{['thm:continuous']}). For $r = 0$, it approaches an orbit of constant entropy relative to a zero of $f^{(r)}$.

Theorems & Definitions (37)

  • Example 1: Condorcet winner
  • Theorem 1
  • Corollary 1
  • proof
  • Remark 1: Decoupling collective decisions
  • Remark 2: Non-uniform mutation rates
  • Remark 3: Mutation rate vs. urn size
  • Remark 4: Majority voting
  • Remark 5: Static or growing urn
  • Example 2: Condorcet cycle
  • ...and 27 more