Procedural Fairness in Multi-Agent Bandits

TL;DR

This work introduces procedural fairness as a legitimacy-driven objective for multi-agent multi-armed bandits, defining equal decision-making influence and associating it with agent-specific favourite arms. It develops three fairness notions—Procedural Fairness, Equality Fairness, and Utilitarian Fairness—each paired with learning algorithms and theoretical guarantees, including regret bounds and core membership results. Theoretical impossibility results show these fairness notions are incompatible in general, while procedurally fair policies can lie in the procedural core, offering stability against coalitions. Empirical evaluation, including a real-world PrefLib dataset, demonstrates that procedural fairness achieves a strong balance across fairness criteria, maintaining legitimacy while remaining robust across diverse settings. The work argues for prioritizing process-based fairness as a design principle with broad applicability to participatory budgeting, resource allocation, and governance of autonomous systems.

Abstract

In the context of multi-agent multi-armed bandits (MA-MAB), fairness is often reduced to outcomes: maximizing welfare, reducing inequality, or balancing utilities. However, evidence in psychology, economics, and Rawlsian theory suggests that fairness is also about process and who gets a say in the decisions being made. We introduce a new fairness objective, procedural fairness, which provides equal decision-making power for all agents, lies in the core, and provides for proportionality in outcomes. Empirical results confirm that fairness notions based on optimizing for outcomes sacrifice equal voice and representation, while the sacrifice in outcome-based fairness objectives (like equality and utilitarianism) is minimal under procedurally fair policies. We further prove that different fairness notions prioritize fundamentally different and incompatible values, highlighting that fairness requires explicit normative choices. This paper argues that procedural legitimacy deserves greater focus as a fairness objective, and provides a framework for putting procedural fairness into practice.

Figures (3)

• Figure 1: Average fairness metrics per policy type. Each column refers to a specific policy maximizing a certain fairness notion, such as procedural, equality, or utilitarian fairness, maximizing utility-based Nash social welfare, or the Generalized Gini Index. Each bar represents a score for a fairness type, such as procedural, equality, or utilitarian fairness, and the error bars represent one standard deviation from the mean.
• Figure 2: Shows each fairness notion's optimal policy scored on the three metrics. Each bar indicates each fairness metric, and the columns indicate each fairness notion or algorithm's optimal policy. This graph has no error bars at it is the result from a single bandit instance (the dataset).
• Figure 3: Comparison of procedural fairness algorithm’s performance over time: (a) fairness scores, (b) policy evolution.

Theorems & Definitions (44)

• Definition 1: Utility
• Definition 2: Decision Share
• Definition 3: Utility-Based Nash Welfare
• Definition 4: Decision-Share-Based Nash Welfare
• Definition 5: Procedural Fairness
• Definition 6: Equality Fairness
• Definition 7: Utilitarian Fairness
• Theorem 1
• Lemma 1
• Lemma 2