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Fair and Welfare-Efficient Constrained Multi-matchings under Uncertainty

Elita Lobo, Justin Payan, Cyrus Cousins, Yair Zick

TL;DR

The approaches presented enable scalable constrained resource allocation under uncertainty for many combinations of objectives and preference models and demonstrate the efficacy of the approaches on three publicly available conference reviewer assignment datasets.

Abstract

We study fair allocation of constrained resources, where a market designer optimizes overall welfare while maintaining group fairness. In many large-scale settings, utilities are not known in advance, but are instead observed after realizing the allocation. We therefore estimate agent utilities using machine learning. Optimizing over estimates requires trading-off between mean utilities and their predictive variances. We discuss these trade-offs under two paradigms for preference modeling -- in the stochastic optimization regime, the market designer has access to a probability distribution over utilities, and in the robust optimization regime they have access to an uncertainty set containing the true utilities with high probability. We discuss utilitarian and egalitarian welfare objectives, and we explore how to optimize for them under stochastic and robust paradigms. We demonstrate the efficacy of our approaches on three publicly available conference reviewer assignment datasets. The approaches presented enable scalable constrained resource allocation under uncertainty for many combinations of objectives and preference models.

Fair and Welfare-Efficient Constrained Multi-matchings under Uncertainty

TL;DR

The approaches presented enable scalable constrained resource allocation under uncertainty for many combinations of objectives and preference models and demonstrate the efficacy of the approaches on three publicly available conference reviewer assignment datasets.

Abstract

We study fair allocation of constrained resources, where a market designer optimizes overall welfare while maintaining group fairness. In many large-scale settings, utilities are not known in advance, but are instead observed after realizing the allocation. We therefore estimate agent utilities using machine learning. Optimizing over estimates requires trading-off between mean utilities and their predictive variances. We discuss these trade-offs under two paradigms for preference modeling -- in the stochastic optimization regime, the market designer has access to a probability distribution over utilities, and in the robust optimization regime they have access to an uncertainty set containing the true utilities with high probability. We discuss utilitarian and egalitarian welfare objectives, and we explore how to optimize for them under stochastic and robust paradigms. We demonstrate the efficacy of our approaches on three publicly available conference reviewer assignment datasets. The approaches presented enable scalable constrained resource allocation under uncertainty for many combinations of objectives and preference models.

Paper Structure

This paper contains 38 sections, 22 theorems, 77 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related work
  4. Fair Resource Allocation under Uncertainty
  5. Fair Resource Allocation
  6. Optimizing Allocations under Uncertainty
  7. Robust Welfare Optimization
  8. Scope:
  9. Robust Allocation for Utilitarian Welfare
  10. Robust Allocation for Group Egalitarian Welfare
  11. Robust Allocation for Monotonic Welfare Functions
  12. Stochastic Welfare Optimization
  13. CVaR Allocation for Utilitarian Welfare
  14. CVaR Allocation for Group Egalitarian Welfare
  15. Experiments
  16. ...and 23 more sections

Key Result

Proposition 3.0

The problem in eq:robust_utilitarian is equivalent to solving where $\mathbf{p} = \bm{\xi} - \bm{Q}^\intercal \bm{\beta}$, $\mathbf{q} = \sum_{i=1}^{\ell} \bm{\lambda}_i\mathbf{S}_i^{-1}\bar{\mathbf{v}}_i$, and $\mathbf{T}=\bigl(\sum_{i=1}^{\ell}\bm{\lambda}_{i} \mathbf{S}_{i}^{-1}\bigr)^{-1}$. Let $\bm{\xi}^*$ be the optimal $\bm{\xi}$ in eq:adversarial_utili

Figures (4)

  • Figure 1: Left: $\operatorname{CVaR}$ as noise increases for AAMAS 2015. Right: Convergence behavior of the Iterated Quadratic Program (Iterated QP) vs. Adversarial Projected Subgradient Ascent approach on AAMAS 2015.
  • Figure 2: $\operatorname{CVaR}_{0.01}$ as noise increases for AAMAS $2016$ and $2021$.
  • Figure 3: Convergence of the Iterated QP vs. adversarial projected subgradient ascent on AAMAS $2016$ dataset for the adversarial USW objective. The Iterated QP (in blue) converges much faster.
  • Figure 4: Relative loss (in GESW) of the maximum USW solution, compared to the optimal GESW solution. Results are reported for a synthetic $2$-group example, varying 1) the divisor applied to artificially scale the minority group's valuations, 2) the ratio of the minority group to the overall number of papers, and 3) the number of valuations per paper that are artifically set to $0$.

Theorems & Definitions (34)

  • Example 2.1: The Importance of Considering Uncertainty
  • Proposition 3.0: Robust Utilitarian Welfare Dual
  • Corollary 3.0: Utilitarian Welfare with Polyhedral Uncertainty
  • Corollary 3.0: Utilitarian Welfare with Ellipsoidal Uncertainty
  • Proposition 3.1: Robust Group Egalitarian Dual
  • Corollary 3.1: Group Egalitarian Welfare with Polyhedral Uncertainty
  • Corollary 3.1: Group Egalitarian Welfare with Ellipsoidal Uncertainty
  • Proposition 3.1: Decomposition for Monotonic Welfare Functions
  • Proposition 4.0: Approximate $\CVaR$ of USW
  • Proposition 4.0: Sample Complexity of Approximate $\CVaR$CVaR of $\USW$USW
  • ...and 24 more