# Cooperative Multi-Agent Fairness and Equivariant Policies

arXiv: https://arxiv.org/abs/2106.05727 · HTML: https://mlsanity.com/paper/2106.05727/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 57 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 29 of 29)

- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [Multi-Agent Actor-Critic for Mixed Cooperative-Competitive Environments](https://mlsanity.com/paper/1706.02275/) (2017)
- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [Inherent Trade-Offs in the Fair Determination of Risk Scores](https://mlsanity.com/paper/1609.05807/) (2016)
- [Learning to Communicate with Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/1605.06676/) (2016)
- [Algorithmic decision making and the cost of fairness](https://mlsanity.com/paper/1701.08230/) (2017)
- [Emergent Tool Use From Multi-Agent Autocurricula](https://mlsanity.com/paper/1909.07528/) (2019)
- [Emergence of Grounded Compositional Language in Multi-Agent Populations](https://mlsanity.com/paper/1703.04908/) (2017)
- [Multi-agent Reinforcement Learning in Sequential Social Dilemmas](https://mlsanity.com/paper/1702.03037/) (2017)
- [Delayed Impact of Fair Machine Learning](https://mlsanity.com/paper/1803.04383/) (2018)
- [Inherent Tradeoffs in Learning Fair Representations](https://mlsanity.com/paper/1906.08386/) (2019)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [The Frontiers of Fairness in Machine Learning](https://mlsanity.com/paper/1810.08810/) (2018)
- [Inequity aversion improves cooperation in intertemporal social dilemmas](https://mlsanity.com/paper/1803.08884/) (2018)
- [Fairness in Reinforcement Learning](https://mlsanity.com/paper/1611.03071/) (2016)
- [The AI Economist: Improving Equality and Productivity with AI-Driven Tax Policies](https://mlsanity.com/paper/2004.13332/) (2020)
- [Shapley Q-value: A Local Reward Approach to Solve Global Reward Games](https://mlsanity.com/paper/1907.05707/) (2019)
- [Operationalizing Individual Fairness with Pairwise Fair Representations](https://mlsanity.com/paper/1907.01439/) (2019)
- [Learning Fair Policies in Multiobjective (Deep) Reinforcement Learning with Average and Discounted Rewards](https://mlsanity.com/paper/2008.07773/) (2020)
- [Algorithms for Fairness in Sequential Decision Making](https://mlsanity.com/paper/1901.08568/) (2019)
- [Prosocial learning agents solve generalized Stag Hunts better than selfish ones](https://mlsanity.com/paper/1709.02865/) (2017)
- [Learning Fairness in Multi-Agent Systems](https://mlsanity.com/paper/1910.14472/) (2019)
- [Learning Fair Policies in Decentralized Cooperative Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2012.09421/) (2020)
- [Shapley Counterfactual Credits for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2106.00285/) (2021)
- [Evolving intrinsic motivations for altruistic behavior](https://mlsanity.com/paper/1811.05931/) (2018)
- [Fair Algorithms for Learning in Allocation Problems](https://mlsanity.com/paper/1808.10549/) (2018)
- [Investigating Trade-offs in Utility, Fairness and Differential Privacy in Neural Networks](https://mlsanity.com/paper/2102.05975/) (2021)
- [Low-Bandwidth Communication Emerges Naturally in Multi-Agent Learning Systems](https://mlsanity.com/paper/2011.14890/) (2020)

## Built on by (most cited first, 8 of 8)

- [Fairness in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2405.06909/) (2024)
- [Cooperation and Fairness in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2410.14916/) (2024)
- [Fairness in Multi-Agent Planning](https://mlsanity.com/paper/2212.00506/) (2022)
- [Temporal Fairness in Decision Making Problems](https://mlsanity.com/paper/2408.13208/) (2024)
- [Operationalising Rawlsian Ethics for Fairness in Norm-Learning Agents](https://mlsanity.com/paper/2412.15163/) (2024)
- [Learning in Multi-Objective Public Goods Games with Non-Linear Utilities](https://mlsanity.com/paper/2408.00682/) (2024)
- [MAFE: Enabling Equitable Algorithm Design in Multi-Agent Multi-Stage Decision-Making Systems](https://mlsanity.com/paper/2502.18534/) (2025)
- [ACPO: Agent-Chained Policy Optimization for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2606.30072/) (2026)

## Abstract (arXiv)

We study fairness through the lens of cooperative multi-agent learning. Our work is motivated by empirical evidence that naive maximization of team reward yields unfair outcomes for individual team members. To address fairness in multi-agent contexts, we introduce team fairness, a group-based fairness measure for multi-agent learning. We then prove that it is possible to enforce team fairness during policy optimization by transforming the team's joint policy into an equivariant map. We refer to our multi-agent learning strategy as Fairness through Equivariance (Fair-E) and demonstrate its effectiveness empirically. We then introduce Fairness through Equivariance Regularization (Fair-ER) as a soft-constraint version of Fair-E and show that it reaches higher levels of utility than Fair-E and fairer outcomes than non-equivariant policies. Finally, we present novel findings regarding the fairness-utility trade-off in multi-agent settings; showing that the magnitude of the trade-off is dependent on agent skill.