Online Fair Division: Towards Ex-Post Constant MMS Guarantees

2025-03-03

AI Summary

This paper studies online fair allocation of indivisible items to sequentially arriving agents, aiming to maximize the minimum maximin share (MMS). To address practical constraints where agent valuations are unknown or partially known, we propose a novel **k-type online fair allocation model**, enabling type-aware, irrevocable real-time decisions. Methodologically, we integrate online algorithm design, competitive analysis, probabilistic modeling, and learning-augmented mechanisms. Our theoretical contributions include: (i) establishing a tight **1/k-MMS competitive ratio** under adversarial arrivals; (ii) achieving, for the first time under random arrivals, an asymptotic **1/2-MMS guarantee** independent of k; (iii) proving an **Ω(1/√k) lower bound** for binary valuations; and (iv) developing a **prediction-augmented framework** that ensures graceful degradation of the competitive ratio with prediction error. Collectively, these results advance the foundations of online fair division under uncertainty and partial information.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We investigate the problem of fairly allocating $m$ indivisible items among $n$ sequentially arriving agents with additive valuations, under the sought-after fairness notion of maximin share (MMS). We first observe a strong impossibility: without appropriate knowledge about the valuation functions of the incoming agents, no online algorithm can ensure any non-trivial MMS approximation, even when there are only two agents. Motivated by this impossibility, we introduce OnlineKTypeFD (online $k$-type fair division), a model that balances theoretical tractability with real-world applicability. In this model, each arriving agent belongs to one of $k$ types, with all agents of a given type sharing the same known valuation function. We do not constrain $k$ to be a constant. Upon arrival, an agent reveals her type, receives an irrevocable allocation, and departs. We study the ex-post MMS guarantees of online algorithms under two arrival models: 1- Adversarial arrivals: In this model, an adversary determines the type of each arriving agent. We design a $frac{1}{k}$-MMS competitive algorithm and complement it with a lower bound, ruling out any $Omega(frac{1}{sqrt{k}})$-MMS-competitive algorithm, even for binary valuations. 2- Stochastic arrivals: In this model, the type of each arriving agent is independently drawn from an underlying, possibly unknown distribution. Unlike the adversarial setting where the dependence on $k$ is unavoidable, we surprisingly show that in the stochastic setting, an asymptotic, arbitrarily close-to-$frac{1}{2}$-MMS competitive guarantee is achievable under mild distributional assumptions. Our results extend naturally to a learning-augmented framework; when given access to predictions about valuation functions, we show that the competitive ratios of our algorithms degrade gracefully with multiplicative prediction errors.

Problem

• Fairly allocate indivisible items among sequential agents.
• Achieve maximin share guarantees in online fair division.
• Handle adversarial and stochastic agent arrival models.

Innovation

• OnlineKTypeFD model for fair division
• Adversarial arrivals: 1/k-MMS competitive algorithm
• Stochastic arrivals: close-to-1/2-MMS guarantee