An Application of Fixed-point Theory to Probabilistic Social Choice

TL;DR

This paper proves the existence of a novel randomized social decision scheme with superior fairness, efficiency, and strategyproofness properties using fixed-point theory, specifically Brouwer's theorem, marking a first in social choice mechanism proofs.

Contribution

It introduces a new application of fixed-point theory to establish the existence of advanced social choice mechanisms with desirable properties.

Findings

Disproves a conjecture by Aziz et al. (2013).

Establishes a strong existence result for random assignment.

Uses Brouwer's fixed-point theorem in social choice context.

Abstract

The purpose of this note is to prove the existence of a randomized mechanism, a social decision scheme (SDS), with desirable fairness, efficiency, and strategyproofness properties unmatched by all known SDSs. In particular, we disprove a conjecture by Aziz et al. (2013). Additionally, we obtain a strong existence result for the domain of random assignment. Both, the notion of efficiency and strategyproofness are based on stochastic dominance and have been studied extensively for random assignment. The proof makes crucial use of Brouwer's fixed-point theorem and is hence non-constructive. To the best of our knowledge, this is the first application of a fixed-point theorem to show the existence of a social choice function or mechanism.