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Maximum Welfare Allocations under Quantile Valuations

Haris Aziz, Shivika Narang, Mashbat Suzuki

TL;DR

This work introduces quantile valuations, a non-additive, quantile-based model for aggregating preferences over bundles of indivisible items, parameterized by τ ∈ [0,1]. It analyzes the computational complexity and algorithmic guarantees for maximizing utilitarian (USW-USW) and egalitarian (ESW-ESW) welfare under both balanced and unbalanced allocations, for goods and chores. The results reveal a sharp balance-dependent divide: balanced USW-USW is hard to approximate while balanced ESW-ESW is tractable under heterogeneous τ, whereas unbalanced settings often allow near-exact or τ-specific polynomial-time algorithms but can also be NP-hard for many τ values. The paper also covers identical valuations, showing tractability gains, and provides reductions to classic problems (e.g., kDM, Exact3Cover, Set Cover) to establish hardness, along with practical greedy and scapegoat algorithms for near-optimal performance. Overall, the work refines our understanding of how quantile-based preferences shape the feasibility and design of fair and efficient allocations in both goods and chores contexts, offering near-optimal methods and clear computational trade-offs with potential impact on review assignment, fair division, and resource allocation problems.

Abstract

We propose a new model for aggregating preferences over a set of indivisible items based on a quantile value. In this model, each agent is endowed with a specific quantile, and the value of a given bundle is defined by the corresponding quantile of the individual values of the items within it. Our model captures the diverse ways in which agents may perceive a bundle, even when they agree on the values of individual items. It enables richer behavioral modeling that cannot be easily captured by additive valuation functions. We study the problem of maximizing utilitarian and egalitarian welfare within the quantile-based valuation setting. For each of the welfare functions, we analyze the complexity of the objectives. Interestingly, our results show that the complexity of both objectives varies significantly depending on whether the allocation is required to be balanced. We provide near-optimal approximation algorithms for utilitarian welfare, and for egalitarian welfare, we present exact algorithms whenever possible.

Maximum Welfare Allocations under Quantile Valuations

TL;DR

This work introduces quantile valuations, a non-additive, quantile-based model for aggregating preferences over bundles of indivisible items, parameterized by τ ∈ [0,1]. It analyzes the computational complexity and algorithmic guarantees for maximizing utilitarian (USW-USW) and egalitarian (ESW-ESW) welfare under both balanced and unbalanced allocations, for goods and chores. The results reveal a sharp balance-dependent divide: balanced USW-USW is hard to approximate while balanced ESW-ESW is tractable under heterogeneous τ, whereas unbalanced settings often allow near-exact or τ-specific polynomial-time algorithms but can also be NP-hard for many τ values. The paper also covers identical valuations, showing tractability gains, and provides reductions to classic problems (e.g., kDM, Exact3Cover, Set Cover) to establish hardness, along with practical greedy and scapegoat algorithms for near-optimal performance. Overall, the work refines our understanding of how quantile-based preferences shape the feasibility and design of fair and efficient allocations in both goods and chores contexts, offering near-optimal methods and clear computational trade-offs with potential impact on review assignment, fair division, and resource allocation problems.

Abstract

We propose a new model for aggregating preferences over a set of indivisible items based on a quantile value. In this model, each agent is endowed with a specific quantile, and the value of a given bundle is defined by the corresponding quantile of the individual values of the items within it. Our model captures the diverse ways in which agents may perceive a bundle, even when they agree on the values of individual items. It enables richer behavioral modeling that cannot be easily captured by additive valuation functions. We study the problem of maximizing utilitarian and egalitarian welfare within the quantile-based valuation setting. For each of the welfare functions, we analyze the complexity of the objectives. Interestingly, our results show that the complexity of both objectives varies significantly depending on whether the allocation is required to be balanced. We provide near-optimal approximation algorithms for utilitarian welfare, and for egalitarian welfare, we present exact algorithms whenever possible.

Paper Structure

This paper contains 50 sections, 29 theorems, 13 equations, 1 figure, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Utilitarian Welfare.
  4. Egalitarian Welfare.
  5. Identical Valuations.
  6. Chores.
  7. Related Work
  8. Quantile based preferences.
  9. Allocating Indivisible Items.
  10. Model
  11. Balanced Allocations
  12. Goods and Chores.
  13. Balanced Allocations
  14. Utilitarian Social Welfare
  15. Hardness of Approximation
  16. ...and 35 more sections

Key Result

theorem 1

Given instance $I=\langle N,M,v,\tau\rangle$ where $m< n^2$ and $m=kn$, it is NP-hard to find an $\ifstrempty{O\left(\frac{m/n}{\log (m/n)}\right)}{\textrm{USW}}{O\left(\frac{m/n}{\log (m/n)}\right)\textrm{-USW{}}}$ balanced allocation.

Figures (1)

  • Figure 1: Quantile-wise tractability or intractability of max ESW-ESW. Red dashed lines show values of $\tau$ for which maximizing ESW-ESW is NP-hard, black solid lines show a value for $\tau$ for which we have polytime algorithms.

Theorems & Definitions (51)

  • definition 1: Quantile Valuations
  • definition 2: Utilitarian Social Welfare (USW-USW)
  • definition 3: Egalitarian Social Welfare (ESW-ESW)
  • theorem 1
  • proof
  • theorem 2
  • proof
  • lemma 1
  • proof
  • proposition 1
  • ...and 41 more