# Neural Payoff Machines: Predicting Fair and Stable Payoff Allocations Among Team Members

arXiv: https://arxiv.org/abs/2208.08798 · HTML: https://mlsanity.com/paper/2208.08798/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 65 references that are in the map.
- First posted on arXiv 18 Aug 2022.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Multi-Agent Actor-Critic for Mixed Cooperative-Competitive Environments](https://mlsanity.com/paper/1706.02275/) (2017)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Emergent Tool Use From Multi-Agent Autocurricula](https://mlsanity.com/paper/1909.07528/) (2019)
- [Human-level performance in first-person multiplayer games with population-based deep reinforcement learning](https://mlsanity.com/paper/1807.01281/) (2018)
- [Opportunities and Challenges in Explainable Artificial Intelligence (XAI): A Survey](https://mlsanity.com/paper/2006.11371/) (2020)
- [Neuron Shapley: Discovering the Responsible Neurons](https://mlsanity.com/paper/2002.09815/) (2020)
- [Gradients are Not All You Need](https://mlsanity.com/paper/2111.05803/) (2021)
- [A Marketplace for Data: An Algorithmic Solution](https://mlsanity.com/paper/1805.08125/) (2018)
- [Collaborative Machine Learning with Incentive-Aware Model Rewards](https://mlsanity.com/paper/2010.12797/) (2020)
- [Bounding the Estimation Error of Sampling-based Shapley Value Approximation](https://mlsanity.com/paper/1306.4265/) (2013)
- [Statistical Cost Sharing](https://mlsanity.com/paper/1703.03111/) (2017)
- [Lyapunov Exponents for Diversity in Differentiable Games](https://mlsanity.com/paper/2112.14570/) (2021)

## Built on by (most cited first, 6 of 6)

- [Explainability Techniques for Chemical Language Models](https://mlsanity.com/paper/2305.16192/) (2023)
- [Game-theoretic Counterfactual Explanation for Graph Neural Networks](https://mlsanity.com/paper/2402.06030/) (2024)
- [Goal Recognition Design for General Behavioral Agents using Machine Learning](https://mlsanity.com/paper/2404.03054/) (2024)
- [Using Cooperative Game Theory to Prune Neural Networks](https://mlsanity.com/paper/2311.10468/) (2023)
- [varepsilon-fractional Core Stability in Hedonic Games](https://mlsanity.com/paper/2311.11101/) (2023)
- [InfluenceNet: AI Models for Banzhaf and Shapley Value Prediction](https://mlsanity.com/paper/2503.08381/) (2025)

## Abstract (arXiv)

In many multi-agent settings, participants can form teams to achieve collective outcomes that may far surpass their individual capabilities. Measuring the relative contributions of agents and allocating them shares of the reward that promote long-lasting cooperation are difficult tasks. Cooperative game theory offers solution concepts identifying distribution schemes, such as the Shapley value, that fairly reflect the contribution of individuals to the performance of the team or the Core, which reduces the incentive of agents to abandon their team. Applications of such methods include identifying influential features and sharing the costs of joint ventures or team formation. Unfortunately, using these solutions requires tackling a computational barrier as they are hard to compute, even in restricted settings. In this work, we show how cooperative game-theoretic solutions can be distilled into a learned model by training neural networks to propose fair and stable payoff allocations. We show that our approach creates models that can generalize to games far from the training distribution and can predict solutions for more players than observed during training. An important application of our framework is Explainable AI: our approach can be used to speed-up Shapley value computations on many instances.