Online Fair Division with Budget Constraints

2026-07-25

AI Summary

This study addresses the online discrete fair allocation problem under generalized budget constraints, where items arrive sequentially and must be irrevocably assigned to agents or a charity, with envy-freeness evaluated only over feasible subsets for each recipient. The work identifies “bounded density extension” as a key structural condition, establishing the first optimal deterministic approximation bounds. It proposes a learning-augmented framework based on joint value–size-type predictions that achieves both consistency and robustness. Theoretical analysis shows the algorithm guarantees feasible envy-free approximations for arbitrary item sizes and attains optimality under homogeneous valuations and small-item regimes. Moreover, resource augmentation significantly strengthens fairness guarantees, and joint prediction strictly outperforms marginal prediction in enhancing allocation quality.

Abstract

We study an online variant of discrete fair division under generalized assignment budget constraints. Goods arrive one at a time and must be assigned irrevocably to a feasible agent or to charity, which holds all unallocated goods, while fairness is evaluated only against budget-feasible subsets of every recipient's bundle. We first show that, without additional structure, no deterministic online algorithm can guarantee any fixed approximation to feasible envy-freeness, even in highly symmetric instances. We then identify bounded density spread as a structural condition that restores meaningful guarantees, obtaining approximation algorithms for arbitrary item sizes and showing that, under common valuations and sufficiently small goods, these guarantees can be strengthened to an optimal deterministic frontier. We further study resource augmentation, where the online algorithm is allowed slightly larger budgets than the fairness benchmark, and characterize the resulting improvement in the achievable guarantees. Finally, we develop a learning-augmented framework based on predicting joint value-size types, proving consistency under perfect predictions, robustness to prediction error, and showing that separate predictions of value and size marginals are insufficient to recover strong fairness guarantees.

Problem

• online fair division
• budget constraints
• envy-freeness
• resource augmentation
• learning-augmented algorithms

Innovation

• online fair division
• budget constraints
• bounded density spread
• resource augmentation
• learning-augmented algorithms