# On the Convergence of the Shapley Value in Parametric Bayesian Learning Games

arXiv: https://arxiv.org/abs/2205.07428 · HTML: https://mlsanity.com/paper/2205.07428/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 26 references that are in the map.
- First posted on arXiv 16 May 2022.

## Builds on (most cited first, 7 of 7)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [Age Progression/Regression by Conditional Adversarial Autoencoder](https://mlsanity.com/paper/1702.08423/) (2017)
- [Collaborative Machine Learning with Incentive-Aware Model Rewards](https://mlsanity.com/paper/2010.12797/) (2020)
- [Improved Algorithms for Collaborative PAC Learning](https://mlsanity.com/paper/1805.08356/) (2018)
- [Incentivizing Collaboration in Machine Learning via Synthetic Data Rewards](https://mlsanity.com/paper/2112.09327/) (2021)
- [Tight Bounds for Collaborative PAC Learning via Multiplicative Weights](https://mlsanity.com/paper/1805.09217/) (2018)

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

- [Data Banzhaf: A Robust Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2205.15466/) (2022)
- [Probably Approximate Shapley Fairness with Applications in Machine Learning](https://mlsanity.com/paper/2212.00630/) (2022)
- [Data Distribution Valuation](https://mlsanity.com/paper/2410.04386/) (2024)
- [Goal-Oriented Communications in Federated Learning via Feedback on Risk-Averse Participation](https://mlsanity.com/paper/2305.11633/) (2023)
- [Bayesian Regression Markets](https://mlsanity.com/paper/2310.14992/) (2023)
- [Uncertainty-Aware Multimodal Learning via Conformal Shapley Intervals](https://mlsanity.com/paper/2602.00171/) (2026)

## Abstract (arXiv)

Measuring contributions is a classical problem in cooperative game theory where the Shapley value is the most well-known solution concept. In this paper, we establish the convergence property of the Shapley value in parametric Bayesian learning games where players perform a Bayesian inference using their combined data, and the posterior-prior KL divergence is used as the characteristic function. We show that for any two players, under some regularity conditions, their difference in Shapley value converges in probability to the difference in Shapley value of a limiting game whose characteristic function is proportional to the log-determinant of the joint Fisher information. As an application, we present an online collaborative learning framework that is asymptotically Shapley-fair. Our result enables this to be achieved without any costly computations of posterior-prior KL divergences. Only a consistent estimator of the Fisher information is needed. The effectiveness of our framework is demonstrated with experiments using real-world data.