Bandit Max-Min Fair Allocation

TL;DR

This paper introduces the bandit max-min fair allocation problem, proposing an algorithm with regret bounds and establishing lower bounds, addressing the challenge of allocating indivisible goods with semi-bandit feedback.

Contribution

It presents a novel algorithm for BMMFA with proven regret bounds and establishes fundamental lower bounds, advancing understanding of fair allocation under bandit feedback.

Findings

Proposed an algorithm with $O(mrac{ ext{sqrt}(T) ext{ln} T}{n} + m ext{sqrt}(T ext{ln}(mnT)))$ regret.

Established a regret lower bound of $ ext{Omega}(mrac{ ext{sqrt}(T)}{n})$.

Identified a logarithmic gap between upper and lower bounds when $T$ is large.

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 $\Omega(m\sqrt{T}/n)$. When $T$ is sufficiently larger than $n$, the gap between the upper and lower bounds is a logarithmic factor of $T$.