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Optimally Interpolating between Ex-Ante Fairness and Welfare

Mikael Møller Høgsgaard, Panagiotis Karras, Wenyue Ma, Nidhi Rathi, Chris Schwiegelshohn

TL;DR

This work addresses balancing ex-ante fairness and welfare in resource allocation by formalizing fair-to-welfare interpolation (FWI). It introduces two efficient algorithms, $\varepsilon$-Mix and $\textsc{Simple-Mix}$, that fuse a given ex-ante fair prior with a welfare-maximizing mechanism to produce allocations that are $\alpha$-fair while achieving near-optimal welfare; $\varepsilon$-Mix achieves an optimal multi-criteria approximation with $O(1/\varepsilon^{2})$ samples, and Simple-Mix offers similar guarantees with zero computational overhead beyond the underlying mechanisms. The authors prove the theoretical properties, including tightness results for the Simple-Mix strategy, and demonstrate empirical behavior across resource allocation, reviewer matching, and sortition tasks on real and synthetic data. The framework is mechanism-agnostic, preserving approximation factors of the underlying fair and welfare mechanisms, and supports practical deployment via straightforward sampling from the fair prior. Overall, the paper provides a principled, scalable approach to interpolate between fairness and welfare that is both theoretically grounded and empirically validated.

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.

Optimally Interpolating between Ex-Ante Fairness and Welfare

TL;DR

This work addresses balancing ex-ante fairness and welfare in resource allocation by formalizing fair-to-welfare interpolation (FWI). It introduces two efficient algorithms, -Mix and , that fuse a given ex-ante fair prior with a welfare-maximizing mechanism to produce allocations that are -fair while achieving near-optimal welfare; -Mix achieves an optimal multi-criteria approximation with samples, and Simple-Mix offers similar guarantees with zero computational overhead beyond the underlying mechanisms. The authors prove the theoretical properties, including tightness results for the Simple-Mix strategy, and demonstrate empirical behavior across resource allocation, reviewer matching, and sortition tasks on real and synthetic data. The framework is mechanism-agnostic, preserving approximation factors of the underlying fair and welfare mechanisms, and supports practical deployment via straightforward sampling from the fair prior. Overall, the paper provides a principled, scalable approach to interpolate between fairness and welfare that is both theoretically grounded and empirically validated.

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 ( and ) that achieve different trade-off guarantees with respect to fairness and welfare. achieves an optimal multi-criteria approximation with respect to fairness and welfare, while 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.
Paper Structure (12 sections, 4 theorems, 38 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 12 sections, 4 theorems, 38 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Framework
  3. Interpolating Between Fairness and Welfare
  4. Simple Mixing to achieve Fairness and Welfare guarantees
  5. Welfare guarantees of Lg are tight
  6. Ancillary fairness properties of Lg
  7. Experiments
  8. Data sets
  9. Evaluation Details
  10. FWI Instance: Resource allocation
  11. FWI Instance: Reviewers Matching
  12. FWI Instance: Sortition

Key Result

Theorem 3.1

Consider an instance $\mathcal{I}(V,p^{ f}, \mathcal{M}_\lambda, \alpha)$ of FWI along with an approximation factor $\varepsilon>0$ and let $p^{\varepsilon}$ be the distribution of the output of Algorithm alg:cap. Then, $p^{\varepsilon}$ is $\alpha$-fair and obtains the following welfare guarantees using $O(1/\varepsilon^2)$-many samples drawn from the fair prior.

Figures (3)

  • Figure 1: The mean score for different scenarios of FWI with $\alpha\in(0,1]$
  • Figure 2: Empirical standard deviations for the corresponding plots in Figure \ref{['fig:means']}
  • Figure 3: Number of the center $K$ affects Likelihood $\mathcal{L}$

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof of Lemma \ref{['lem:fair']}
  • Claim 3.4
  • proof : Proof of Claim \ref{['cor:valuepx']}
  • proof : Proof of Lemma \ref{['lem:aproxemix']}
  • Theorem 4.1
  • Remark 4.2
  • proof : Proof of Theorem \ref{['thm:smix']}