# The Shapley Value in Machine Learning

arXiv: https://arxiv.org/abs/2202.05594 · HTML: https://mlsanity.com/paper/2202.05594/

## Key facts

- Cited by 98 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 83 references that are in the map.
- First posted on arXiv 11 Feb 2022.

## Builds on (most cited first, 30 of 32)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Data Shapley: Equitable Valuation of Data for Machine Learning](https://mlsanity.com/paper/1904.02868/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Towards Efficient Data Valuation Based on the Shapley Value](https://mlsanity.com/paper/1902.10275/) (2019)
- [On Explainability of Graph Neural Networks via Subgraph Explorations](https://mlsanity.com/paper/2102.05152/) (2021)
- [Problems with Shapley-value-based explanations as feature importance measures](https://mlsanity.com/paper/2002.11097/) (2020)
- [Beta Shapley: a Unified and Noise-reduced Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2110.14049/) (2021)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [The Shapley Taylor Interaction Index](https://mlsanity.com/paper/1902.05622/) (2019)
- [A Principled Approach to Data Valuation for Federated Learning](https://mlsanity.com/paper/2009.06192/) (2020)
- [Explaining Deep Neural Networks with a Polynomial Time Algorithm for Shapley Values Approximation](https://mlsanity.com/paper/1903.10992/) (2019)
- [Asymmetric Shapley values: incorporating causal knowledge into model-agnostic explainability](https://mlsanity.com/paper/1910.06358/) (2019)
- [Causal Shapley Values: Exploiting Causal Knowledge to Explain Individual Predictions of Complex Models](https://mlsanity.com/paper/2011.01625/) (2020)
- [Sampling Permutations for Shapley Value Estimation](https://mlsanity.com/paper/2104.12199/) (2021)
- [L-Shapley and C-Shapley: Efficient Model Interpretation for Structured Data](https://mlsanity.com/paper/1808.02610/) (2018)
- [Neuron Shapley: Discovering the Responsible Neurons](https://mlsanity.com/paper/2002.09815/) (2020)
- [Shapley explainability on the data manifold](https://mlsanity.com/paper/2006.01272/) (2020)
- [A Distributional Framework for Data Valuation](https://mlsanity.com/paper/2002.12334/) (2020)
- [GTG-Shapley: Efficient and Accurate Participant Contribution Evaluation in Federated Learning](https://mlsanity.com/paper/2109.02053/) (2021)
- [GraphSVX: Shapley Value Explanations for Graph Neural Networks](https://mlsanity.com/paper/2104.10482/) (2021)
- [Efficient nonparametric statistical inference on population feature importance using Shapley values](https://mlsanity.com/paper/2006.09481/) (2020)
- [Bounding the Estimation Error of Sampling-based Shapley Value Approximation](https://mlsanity.com/paper/1306.4265/) (2013)
- [Shapley values for feature selection: The good, the bad, and the axioms](https://mlsanity.com/paper/2102.10936/) (2021)
- [Shapley Flow: A Graph-based Approach to Interpreting Model Predictions](https://mlsanity.com/paper/2010.14592/) (2020)
- [Efficient computation and analysis of distributional Shapley values](https://mlsanity.com/paper/2007.01357/) (2020)
- [Shapley Counterfactual Credits for Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2106.00285/) (2021)
- [A Multilinear Sampling Algorithm to Estimate Shapley Values](https://mlsanity.com/paper/2010.12082/) (2020)
- [The Shapley Value of Classifiers in Ensemble Games](https://mlsanity.com/paper/2101.02153/) (2021)
- [Finding Influential Training Samples for Gradient Boosted Decision Trees](https://mlsanity.com/paper/1802.06640/) (2018)

## Built on by (most cited first, 30 of 98)

- [Training Data Influence Analysis and Estimation: A Survey](https://mlsanity.com/paper/2212.04612/) (2022)
- [Algorithms to estimate Shapley value feature attributions](https://mlsanity.com/paper/2207.07605/) (2022)
- [Unified Gradient-Based Machine Unlearning with Remain Geometry Enhancement](https://mlsanity.com/paper/2409.19732/) (2024)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [Data-OOB: Out-of-bag Estimate as a Simple and Efficient Data Value](https://mlsanity.com/paper/2304.07718/) (2023)
- [WeightedSHAP: analyzing and improving Shapley based feature attributions](https://mlsanity.com/paper/2209.13429/) (2022)
- [Approximating the Shapley Value without Marginal Contributions](https://mlsanity.com/paper/2302.00736/) (2023)
- [The Inadequacy of Shapley Values for Explainability](https://mlsanity.com/paper/2302.08160/) (2023)
- [Explaining Quantum Circuits with Shapley Values: Towards Explainable Quantum Machine Learning](https://mlsanity.com/paper/2301.09138/) (2023)
- [MultiModN- Multimodal, Multi-Task, Interpretable Modular Networks](https://mlsanity.com/paper/2309.14118/) (2023)
- [Lost in Moderation: How Commercial Content Moderation APIs Over- and Under-Moderate Group-Targeted Hate Speech and Linguistic Variations](https://mlsanity.com/paper/2503.01623/) (2025)
- [Interpretable Machine Learning for TabPFN](https://mlsanity.com/paper/2403.10923/) (2024)
- [Towards Explainable Artificial Intelligence (XAI): A Data Mining Perspective](https://mlsanity.com/paper/2401.04374/) (2024)
- [One Sample Fits All: Approximating All Probabilistic Values Simultaneously and Efficiently](https://mlsanity.com/paper/2410.23808/) (2024)
- [Differentially Private Shapley Values for Data Evaluation](https://mlsanity.com/paper/2206.00511/) (2022)
- [SHAP-XRT: The Shapley Value Meets Conditional Independence Testing](https://mlsanity.com/paper/2207.07038/) (2022)
- [Probably Approximate Shapley Fairness with Applications in Machine Learning](https://mlsanity.com/paper/2212.00630/) (2022)
- [A comparative assessment of deep learning models for day-ahead load forecasting: Investigating key accuracy drivers](https://mlsanity.com/paper/2302.12168/) (2023)
- [A Unified Gradient-based Framework for Task-agnostic Continual Learning-Unlearning](https://mlsanity.com/paper/2505.15178/) (2025)
- [Explaining Similarity in Vision-Language Encoders with Weighted Banzhaf Interactions](https://mlsanity.com/paper/2508.05430/) (2025)
- [Generating Explanations to Understand and Repair Embedding-based Entity Alignment](https://mlsanity.com/paper/2312.04877/) (2023)
- [A Survey on Contribution Evaluation in Vertical Federated Learning](https://mlsanity.com/paper/2405.02364/) (2024)
- [CAMs as Shapley Value-based Explainers](https://mlsanity.com/paper/2501.06261/) (2025)
- [Where on Earth? A Vision-Language Benchmark for Probing Model Geolocation Skills Across Scales](https://mlsanity.com/paper/2510.10880/) (2025)
- [An Odd Estimator for Shapley Values](https://mlsanity.com/paper/2602.01399/) (2026)
- [For Better or Worse: The Impact of Counterfactual Explanations' Directionality on User Behavior in xAI](https://mlsanity.com/paper/2306.07637/) (2023)
- [CHG Shapley: Efficient Data Valuation and Selection towards Trustworthy Machine Learning](https://mlsanity.com/paper/2406.11730/) (2024)
- [Explanatory Model Monitoring to Understand the Effects of Feature Shifts on Performance](https://mlsanity.com/paper/2408.13648/) (2024)
- [HyperSHAP: Shapley Values and Interactions for Explaining Hyperparameter Optimization](https://mlsanity.com/paper/2502.01276/) (2025)
- [Souper-Model: How Simple Arithmetic Unlocks State-of-the-Art LLM Performance](https://mlsanity.com/paper/2511.13254/) (2025)

## Abstract (arXiv)

Over the last few years, the Shapley value, a solution concept from cooperative game theory, has found numerous applications in machine learning. In this paper, we first discuss fundamental concepts of cooperative game theory and axiomatic properties of the Shapley value. Then we give an overview of the most important applications of the Shapley value in machine learning: feature selection, explainability, multi-agent reinforcement learning, ensemble pruning, and data valuation. We examine the most crucial limitations of the Shapley value and point out directions for future research.