# The Shapley Value of Classifiers in Ensemble Games

arXiv: https://arxiv.org/abs/2101.02153 · HTML: https://mlsanity.com/paper/2101.02153/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 55 references that are in the map.
- First posted on arXiv 6 Jan 2021.

## Builds on (most cited first, 19 of 19)

- [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)
- [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)
- [graph2vec: Learning Distributed Representations of Graphs](https://mlsanity.com/paper/1707.05005/) (2017)
- [Improving KernelSHAP: Practical Shapley Value Estimation via Linear Regression](https://mlsanity.com/paper/2012.01536/) (2020)
- [Characteristic Functions on Graphs: Birds of a Feather, from Statistical Descriptors to Parametric Models](https://mlsanity.com/paper/2005.07959/) (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)
- [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)
- [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)
- [NetLSD: Hearing the Shape of a Graph](https://mlsanity.com/paper/1805.10712/) (2018)
- [Geometric Scattering for Graph Data Analysis](https://mlsanity.com/paper/1810.03068/) (2018)
- [A simple yet effective baseline for non-attributed graph classification](https://mlsanity.com/paper/1811.03508/) (2018)
- [A Simple Baseline Algorithm for Graph Classification](https://mlsanity.com/paper/1810.09155/) (2018)
- [Ensemble Pruning based on Objection Maximization with a General Distributed Framework](https://mlsanity.com/paper/1806.04899/) (2018)
- [Analysing Neural Network Topologies: a Game Theoretic Approach](https://mlsanity.com/paper/1904.08166/) (2019)

## Built on by (most cited first, 23 of 23)

- [Beta Shapley: a Unified and Noise-reduced Data Valuation Framework for Machine Learning](https://mlsanity.com/paper/2110.14049/) (2021)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [shapiq: Shapley Interactions for Machine Learning](https://mlsanity.com/paper/2410.01649/) (2024)
- [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)
- [Energy-Based Learning for Cooperative Games, with Applications to Valuation Problems in Machine Learning](https://mlsanity.com/paper/2106.02938/) (2021)
- [Ensembles of Random SHAPs](https://mlsanity.com/paper/2103.03302/) (2021)
- [Differentially Private Shapley Values for Data Evaluation](https://mlsanity.com/paper/2206.00511/) (2022)
- [Fair yet Asymptotically Equal Collaborative Learning](https://mlsanity.com/paper/2306.05764/) (2023)
- [Statistical Significance of Feature Importance Rankings](https://mlsanity.com/paper/2401.15800/) (2024)
- [An Imprecise SHAP as a Tool for Explaining the Class Probability Distributions under Limited Training Data](https://mlsanity.com/paper/2106.09111/) (2021)
- [Interpretable Ensembles of Hyper-Rectangles as Base Models](https://mlsanity.com/paper/2303.08625/) (2023)
- [Generating Effective Ensembles for Sentiment Analysis](https://mlsanity.com/paper/2402.16700/) (2024)
- [On-device Content-based Recommendation with Single-shot Embedding Pruning: A Cooperative Game Perspective](https://mlsanity.com/paper/2411.13052/) (2024)
- [A Comprehensive Study of Shapley Value in Data Analytics](https://mlsanity.com/paper/2412.01460/) (2024)
- [Shapley Value Approximation Based on k-Additive Games](https://mlsanity.com/paper/2502.04763/) (2025)
- [FW-Shapley: Real-time Estimation of Weighted Shapley Values](https://mlsanity.com/paper/2503.06602/) (2025)
- [An Explainable Stacked Ensemble Model for Static Route-Free Estimation of Time of Arrival](https://mlsanity.com/paper/2203.09438/) (2022)
- [Shapley Chains: Extending Shapley Values to Classifier Chains](https://mlsanity.com/paper/2303.17243/) (2023)
- [SIM-Shapley: A Stable and Computationally Efficient Approach to Shapley Value Approximation](https://mlsanity.com/paper/2505.08198/) (2025)
- [Accelerated Shapley Value Approximation for Data Evaluation](https://mlsanity.com/paper/2311.05346/) (2023)
- [Antithetic Sampling for Top-k Shapley Identification](https://mlsanity.com/paper/2504.02019/) (2025)
- [Shapley-Value-Based Feature Attribution for Data Masking](https://mlsanity.com/paper/2607.28946/) (2026)

## Abstract (arXiv)

What is the value of an individual model in an ensemble of binary classifiers? We answer this question by introducing a class of transferable utility cooperative games called textitensemble games. In machine learning ensembles, pre-trained models cooperate to make classification decisions. To quantify the importance of models in these ensemble games, we define textitTroupe -- an efficient algorithm which allocates payoffs based on approximate Shapley values of the classifiers. We argue that the Shapley value of models in these games is an effective decision metric for choosing a high performing subset of models from the ensemble. Our analytical findings prove that our Shapley value estimation scheme is precise and scalable; its performance increases with size of the dataset and ensemble. Empirical results on real world graph classification tasks demonstrate that our algorithm produces high quality estimates of the Shapley value. We find that Shapley values can be utilized for ensemble pruning, and that adversarial models receive a low valuation. Complex classifiers are frequently found to be responsible for both correct and incorrect classification decisions.