Optimally Interpolating between Ex-Ante Fairness and Welfare

TL;DR

This paper introduces a flexible framework for balancing fairness and welfare in resource allocation, providing efficient algorithms with optimal trade-offs and validating them through experiments on real data.

Contribution

It proposes a general interpolation framework between fairness and welfare, with two algorithms achieving optimal or near-optimal trade-offs, applicable to any underlying mechanisms.

Findings

$oldsymbol{ ext{$ extit{ε}$-Mix}$ achieves optimal multi-criteria approximation.

$ extbf{Simple-Mix}$ achieves near-optimal trade-offs with minimal overhead.

Experimental results confirm theoretical guarantees on real datasets.

Abstract

For the fundamental problem of allocating a set of resources among individuals with varied preferences, the quality of an allocation relates to the degree of fairness and the collective welfare achieved. Unfortunately, in many resource-allocation settings, it is computationally hard to maximize welfare while achieving fairness goals. In this work, we consider ex-ante notions of fairness; popular examples include the \emph{randomized round-robin algorithm} and \emph{sortition mechanism}. We propose a general framework to systematically study the \emph{interpolation} between fairness and welfare goals in a multi-criteria setting. We develop two efficient algorithms ($\varepsilon-Mix$ and $Simple-Mix$) that achieve different trade-off guarantees with respect to fairness and welfare. $\varepsilon-Mix$ achieves an optimal multi-criteria approximation with respect to fairness and welfare, while $Simple-Mix$ achieves optimality up to a constant factor with zero computational overhead beyond the underlying \emph{welfare-maximizing mechanism} and the \emph{ex-ante fair mechanism}. Our framework makes no assumptions on either of the two underlying mechanisms, other than that the fair mechanism produces a distribution over the set of all allocations. Indeed, if these mechanisms are themselves approximation algorithms, our framework will retain the approximation factor, guaranteeing sensitivity to the quality of the underlying mechanisms, while being \emph{oblivious} to them. We also give an extensive experimental analysis for the aforementioned ex-ante fair mechanisms on real data sets, confirming our theoretical analysis.