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Improved Regret Bounds for Online Fair Division with Bandit Learning

Benjamin Schiffer, Shirley Zhang

TL;DR

This work addresses online fair division with $n$ players and $m$ item types where player-item values are drawn from unknown distributions and item types arrive online. It introduces a UCB-based algorithm that uses two rounds of linear programming to enforce proportionality in expectation while ensuring sufficient exploration, leveraging a high-probability confidence set over the unknown means. Under normalized valuations, the algorithm achieves a near-optimal $ ilde{O}(\,sqrt{T} olinebreak[0] )$ regret with high probability and improves upon the previous $ ilde{O}(T^{2/3})$ rate, illustrating a fundamental advantage of proportionality constraints over envy-freeness in learning settings. The paper also proves a matching lower bound for envy-freeness and discusses extensions to deterministic item types and broader limitations and future directions, including runtime considerations and applying the framework to other fairness notions.

Abstract

We study online fair division when there are a finite number of item types and the player values for the items are drawn randomly from distributions with unknown means. In this setting, a sequence of indivisible items arrives according to a random online process, and each item must be allocated to a single player. The goal is to maximize expected social welfare while maintaining that the allocation satisfies proportionality in expectation. When player values are normalized, we show that it is possible to with high probability guarantee proportionality constraint satisfaction and achieve $\tilde{O}(\sqrt{T})$ regret. To achieve this result, we present an upper confidence bound (UCB) algorithm that uses two rounds of linear optimization. This algorithm highlights fundamental aspects of proportionality constraints that allow for a UCB algorithm despite the presence of many (potentially tight) constraints. This result improves upon the previous best regret rate of $\tilde{O}(T^{2/3})$.

Improved Regret Bounds for Online Fair Division with Bandit Learning

TL;DR

This work addresses online fair division with players and item types where player-item values are drawn from unknown distributions and item types arrive online. It introduces a UCB-based algorithm that uses two rounds of linear programming to enforce proportionality in expectation while ensuring sufficient exploration, leveraging a high-probability confidence set over the unknown means. Under normalized valuations, the algorithm achieves a near-optimal regret with high probability and improves upon the previous rate, illustrating a fundamental advantage of proportionality constraints over envy-freeness in learning settings. The paper also proves a matching lower bound for envy-freeness and discusses extensions to deterministic item types and broader limitations and future directions, including runtime considerations and applying the framework to other fairness notions.

Abstract

We study online fair division when there are a finite number of item types and the player values for the items are drawn randomly from distributions with unknown means. In this setting, a sequence of indivisible items arrives according to a random online process, and each item must be allocated to a single player. The goal is to maximize expected social welfare while maintaining that the allocation satisfies proportionality in expectation. When player values are normalized, we show that it is possible to with high probability guarantee proportionality constraint satisfaction and achieve regret. To achieve this result, we present an upper confidence bound (UCB) algorithm that uses two rounds of linear optimization. This algorithm highlights fundamental aspects of proportionality constraints that allow for a UCB algorithm despite the presence of many (potentially tight) constraints. This result improves upon the previous best regret rate of .
Paper Structure (25 sections, 11 theorems, 98 equations, 2 algorithms)

This paper contains 25 sections, 11 theorems, 98 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Model
  5. Regret and Problem Formulation
  6. Notation
  7. Main Results
  8. Algorithm Overview
  9. Algorithm Intuition
  10. Properties of Proportionality Constraints
  11. Proof Sketch of Theorem \ref{['thm:sqrtT_regret']}
  12. Lower Bounds
  13. Discussion
  14. Non-random Item Types
  15. Limitations and Future Directions
  16. ...and 10 more sections

Key Result

Theorem 3.1

Suppose $n,m, T, a,b$ are known and that the true mean values satisfy $0 < a \le \mu^*_{ik} \le b$ for all $i \in [n]$, $k \in [m]$. With probability $1-1/T$, Algorithm algo:UCB will both satisfy the proportionality in expectation constraints for $\mu^*$ and have regret for $\mu^*$ of $\tilde{O}(n^5

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Theorem 3.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof : Proof sketch
  • Definition 5.1
  • ...and 14 more