# Fair Exploration via Axiomatic Bargaining

arXiv: https://arxiv.org/abs/2106.02553 · HTML: https://mlsanity.com/paper/2106.02553/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 69 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Fairness in Learning: Classic and Contextual Bandits](https://mlsanity.com/paper/1605.07139/) (2016)
- [Mostly Exploration-Free Algorithms for Contextual Bandits](https://mlsanity.com/paper/1704.09011/) (2017)
- [Achieving Fairness in the Stochastic Multi-armed Bandit Problem](https://mlsanity.com/paper/1907.10516/) (2019)
- [Online Learning with an Unknown Fairness Metric](https://mlsanity.com/paper/1802.06936/) (2018)
- [Bayesian Incentive-Compatible Bandit Exploration](https://mlsanity.com/paper/1502.04147/) (2015)
- [The End of Optimism? An Asymptotic Analysis of Finite-Armed Linear Bandits](https://mlsanity.com/paper/1610.04491/) (2016)
- [Calibrated Fairness in Bandits](https://mlsanity.com/paper/1707.01875/) (2017)
- [Adaptive Exploration in Linear Contextual Bandit](https://mlsanity.com/paper/1910.06996/) (2019)
- [Minimal Exploration in Structured Stochastic Bandits](https://mlsanity.com/paper/1711.00400/) (2017)
- [Incentivizing Exploration with Selective Data Disclosure](https://mlsanity.com/paper/1811.06026/) (2018)
- [The Externalities of Exploration and How Data Diversity Helps Exploitation](https://mlsanity.com/paper/1806.00543/) (2018)
- [Fairness Incentives for Myopic Agents](https://mlsanity.com/paper/1705.02321/) (2017)
- [Quantifying the Burden of Exploration and the Unfairness of Free Riding](https://mlsanity.com/paper/1810.08743/) (2018)

## Built on by (most cited first, 22 of 22)

- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
- [Socially Fair Reinforcement Learning](https://mlsanity.com/paper/2208.12584/) (2022)
- [No-regret Algorithms for Fair Resource Allocation](https://mlsanity.com/paper/2303.06396/) (2023)
- [Multi-disciplinary fairness considerations in machine learning for clinical trials](https://mlsanity.com/paper/2205.08875/) (2022)
- [No-Regret Learning for Fair Multi-Agent Social Welfare Optimization](https://mlsanity.com/paper/2405.20678/) (2024)
- [Fairness-aware Online Price Discrimination with Nonparametric Demand Models](https://mlsanity.com/paper/2111.08221/) (2021)
- [Fairness-Aware Meta-Learning via Nash Bargaining](https://mlsanity.com/paper/2406.07029/) (2024)
- [Fair Resource Allocation in Virtualized O-RAN Platforms](https://mlsanity.com/paper/2402.11285/) (2024)
- [Network Revenue Management with Demand Learning and Fair Resource-Consumption Balancing](https://mlsanity.com/paper/2207.11159/) (2022)
- [Time Fairness in Online Knapsack Problems](https://mlsanity.com/paper/2305.13293/) (2023)
- [Interpolating Item and User Fairness in Multi-Sided Recommendations](https://mlsanity.com/paper/2306.10050/) (2023)
- [Fair Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/2408.17313/) (2024)
- [Towards Regulatory-Confirmed Adaptive Clinical Trials: Machine Learning Opportunities and Solutions](https://mlsanity.com/paper/2503.09226/) (2025)
- [Expert Merging in Sparse Mixture of Experts with Nash Bargaining](https://mlsanity.com/paper/2510.16138/) (2025)
- [Creator Incentives in Recommender Systems: A Cooperative Game-Theoretic Approach for Stable and Fair Collaboration in Multi-Agent Bandits](https://mlsanity.com/paper/2604.08643/) (2026)
- [Risk Preferences of Learning Algorithms](https://mlsanity.com/paper/2205.04619/) (2022)
- [Intuitions of Compromise: Utilitarianism vs. Contractualism](https://mlsanity.com/paper/2410.05496/) (2024)
- [Incentivized Collaboration in Active Learning](https://mlsanity.com/paper/2311.00260/) (2023)
- [Adversarial Attacks on Cooperative Multi-agent Bandits](https://mlsanity.com/paper/2311.01698/) (2023)
- [Learning with Limited Shared Information in Multi-agent Multi-armed Bandit](https://mlsanity.com/paper/2502.15338/) (2025)
- [Collaborative Min-Max Regret in Grouped Multi-Armed Bandits](https://mlsanity.com/paper/2506.10313/) (2025)
- [Exact Regret Frontiers and Externality Scheduling in Centralized Serial-Dictatorship Bandits](https://mlsanity.com/paper/2609.19963/) (2026)

## Abstract (arXiv)

Exploration is often necessary in online learning to maximize long-term reward, but it comes at the cost of short-term 'regret'. We study how this cost of exploration is shared across multiple groups. For example, in a clinical trial setting, patients who are assigned a sub-optimal treatment effectively incur the cost of exploration. When patients are associated with natural groups on the basis of, say, race or age, it is natural to ask whether the cost of exploration borne by any single group is 'fair'. So motivated, we introduce the 'grouped' bandit model. We leverage the theory of axiomatic bargaining, and the Nash bargaining solution in particular, to formalize what might constitute a fair division of the cost of exploration across groups. On the one hand, we show that any regret-optimal policy strikingly results in the least fair outcome: such policies will perversely leverage the most 'disadvantaged' groups when they can. More constructively, we derive policies that are optimally fair and simultaneously enjoy a small 'price of fairness'. We illustrate the relative merits of our algorithmic framework with a case study on contextual bandits for warfarin dosing where we are concerned with the cost of exploration across multiple races and age groups.