Bandit Max-Min Fair Allocation

TL;DR

Bandit max-min fair allocation (BMMFA) studies online fair division of indivisible items under semi-bandit feedback, where agent valuations are unknown and must be learned. The authors develop a UCB-based allocation algorithm that blends regret analysis with competitive-analysis ideas, achieving a regret bound of $R_T = O\big( \frac{m}{n}\sqrt{T}\ln T + m\sqrt{T\ln(mnT)} \big)$ and proving a matching lower bound of $\Omega\big( \frac{m}{n}\sqrt{T} \big)$ up to a logarithmic factor for large $T$. They introduce a surrogate regret framework to facilitate analysis and extend the results to matroid constraints, demonstrating robustness to broader feasibility structures. By addressing the interplay between learning valuations and fairness, this work advances online fair allocation with partial feedback and provides implications for subscription-style services and related online decision problems.

Abstract

In this paper, we study a new decision-making problem called the bandit max-min fair allocation (BMMFA) problem. The goal of this problem is to maximize the minimum utility among agents with additive valuations by repeatedly assigning indivisible goods to them. One key feature of this problem is that each agent's valuation for each item can only be observed through the semi-bandit feedback, while existing work supposes that the item values are provided at the beginning of each round. Another key feature is that the algorithm's reward function is not additive with respect to rounds, unlike most bandit-setting problems. Our first contribution is to propose an algorithm that has an asymptotic regret bound of $O(m\sqrt{T}\ln T/n + m\sqrt{T \ln(mnT)})$, where $n$ is the number of agents, $m$ is the number of items, and $T$ is the time horizon. This is based on a novel combination of bandit techniques and a resource allocation algorithm studied in the literature on competitive analysis. Our second contribution is to provide the regret lower bound of $Ω(m\sqrt{T}/n)$. When $T$ is sufficiently larger than $n$, the gap between the upper and lower bounds is a logarithmic factor of $T$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Claim 1
• proof
• Theorem 1: BKS2018
• Theorem 2
• Lemma 2
• proof
• Claim 2: BKS2018
• Lemma 3