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Distributed Agreement in the Arrovian Framework

Kenan Wood, Hammurabi Mendes, Jonad Pulaj

TL;DR

This work studies a weaker distributed task in which crash faults are introduced, IIA is not required, and the consensus property is relaxed to either $k$-set agreement or $\epsilon$-approximate agreement using any metric on the set of preferences.

Abstract

Preference aggregation is a fundamental problem in voting theory, in which public input rankings of a set of alternatives (called preferences) must be aggregated into a single preference that satisfies certain soundness properties. The celebrated Arrow Impossibility Theorem is equivalent to a distributed task in a synchronous fault-free system that satisfies properties such as respecting unanimous preferences, maintaining independence of irrelevant alternatives (IIA), and non-dictatorship, along with consensus since only one preference can be decided. In this work, we study a weaker distributed task in which crash faults are introduced, IIA is not required, and the consensus property is relaxed to either $k$-set agreement or $ε$-approximate agreement using any metric on the set of preferences. In particular, we prove several novel impossibility results for both of these tasks in both synchronous and asynchronous distributed systems. We additionally show that the impossibility for our $ε$-approximate agreement task using the Kendall tau or Spearman footrule metrics holds under extremely weak assumptions.

Distributed Agreement in the Arrovian Framework

TL;DR

This work studies a weaker distributed task in which crash faults are introduced, IIA is not required, and the consensus property is relaxed to either -set agreement or -approximate agreement using any metric on the set of preferences.

Abstract

Preference aggregation is a fundamental problem in voting theory, in which public input rankings of a set of alternatives (called preferences) must be aggregated into a single preference that satisfies certain soundness properties. The celebrated Arrow Impossibility Theorem is equivalent to a distributed task in a synchronous fault-free system that satisfies properties such as respecting unanimous preferences, maintaining independence of irrelevant alternatives (IIA), and non-dictatorship, along with consensus since only one preference can be decided. In this work, we study a weaker distributed task in which crash faults are introduced, IIA is not required, and the consensus property is relaxed to either -set agreement or -approximate agreement using any metric on the set of preferences. In particular, we prove several novel impossibility results for both of these tasks in both synchronous and asynchronous distributed systems. We additionally show that the impossibility for our -approximate agreement task using the Kendall tau or Spearman footrule metrics holds under extremely weak assumptions.
Paper Structure (9 sections, 15 theorems, 18 equations, 1 figure)

This paper contains 9 sections, 15 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Relations
  4. Distributed Aggregation Maps
  5. Initial Observations
  6. Distributed Set and Approximate Preference Aggregation
  7. Arrovian Impossibilities in Synchronous and Asynchronous Systems
  8. Related Work
  9. Conclusion

Key Result

Proposition 2.11

If $|X| = m$, then $\mathrm{diam}_{\mathtt{KT}}(L(X)) = \frac{m^2-m}{2}$ and $\mathrm{diam}_{\mathtt{SF}}(L(X)) = \left\lfloor \frac{m^2}{2} \right\rfloor$.

Figures (1)

  • Figure 1: Visualization of $R_1'$ and $R_{\left\lfloor j/2 \right\rfloor+1}'$ in the proofs of Corollaries \ref{['Cor:general-kendall-tau']} and \ref{['Cor:general-spearman-footrule']}.

Theorems & Definitions (49)

  • Definition 2.1: Distributed Aggregation Map
  • Definition 2.2: Unanimity
  • Definition 2.3: IIA
  • Definition 2.4: Decisive
  • Definition 2.5: Dictatorship
  • Definition 2.6: Set Agreement
  • Definition 2.7: Metric Diameter
  • Definition 2.8: Approximate Agreement
  • Definition 2.9: Kendall tau
  • Definition 2.10: Rank and Spearman's footrule
  • ...and 39 more