Bandit Max-Min Fair Allocation (2505.05169v1)

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 $T$.

Analysis of Bandit Max-Min Fair Allocation

The paper under consideration focuses on a novel problem in sequential decision-making, specifically, the bandit max-min fair allocation (BMMFA) problem. This problem involves distributing indivisible items among multiple agents in a manner that maximizes the minimum utility across the agents. The challenge stems from the fact that each agent's valuation for an item is only known through a stochastic, semi-bandit feedback system and not disclosed upfront.

Problem Framework and Contributions

The BMMFA problem is fundamentally an online variation of the classical fair allocation problem commonly studied in algorithmic game theory. The goal in BMMFA is twofold: (1) to allocate resources fairly among agents by maximizing the minimum utility and (2) to adaptively learn the valuations of agents to make better allocation decisions over time. Distinctively, the problem diverges from other learning paradigms by having a non-additive reward function across rounds, posing unique challenges in formulating an effective algorithmic strategy.

The primary contributions of this paper are twofold:

1. The proposal of an algorithm achieving an upper regret bound of $O(m\sqrt{T}\ln T/n + m\sqrt{T \ln(mnT)})$ for large $T$, where $n$ represents the number of agents, $m$ signifies the number of items, and $T$ is the time horizon.
2. Establishing a lower regret bound of $\Omega(m\sqrt{T}/n)$. The gap between these upper and lower bounds is limited by a logarithmic factor related to the time horizon $T$.

Algorithmic Approach

The construction of the proposed algorithm is rooted in combining bandit algorithms with competitive analysis methods. Specifically, it leverages the multiplicative weight update method to dynamically adjust allocations based on estimated valuations. The algorithm is equipped to handle the semi-bandit feedback by utilizing upper confidence bounds (UCB) to refine valuation estimates, ensuring all agents receive a competitive utility relative to their unknown true preferences. The analysis employs surrogate metrics to indirectly evaluate regret, enabling robust performance even in uncertain environments.

Implications and Future Directions

The theoretical contributions are clear, with implications spanning both practical applications and theoretical advancements. In practical terms, the BMMFA problem framework has potential applications in managing subscription services where customer satisfaction is paramount, and resource allocation has direct business implications. Theoretically, this research outlines methods that optimize regret when dealing with bandit problems possessing non-additive reward structures, thus pushing boundaries in online learning algorithm design.

Future research directions may involve exploring variations of the problem setup, such as allowing correlations between agents’ valuations or incorporating other forms of feedback mechanisms. Additionally, deepening the analysis to close the gap between theoretical bounds or applying these insights into more generalized allocation problems could offer further breakthroughs.

By addressing the new challenges in BMMFA, this paper offers substantial groundwork upon which future explorations in fair online allocation might be scaffolded, fostering a more equitable approach to sequential decision problems in uncertain informational landscapes.