---
title: "Approximating the Shapley Value Using Stratified Empirical Bernstein Sampling"
authors:
  - "Mark A. Burgess"
  - "Archie C. Chapman"
doi: "10.24963/ijcai.2021/11"
url: "https://ijcai.org/proceedings/2021/11"
pdf: "https://ijcai.org/proceedings/2021/0011.pdf"
venue: "Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence"
year: 2021
pages: "73-81"
track: "Main Track"
keywords:
  - "Agent-based and Multi-agent Systems: Cooperative Games"
  - "Uncertainty in AI: Uncertainty Representations"
---

# Approximating the Shapley Value Using Stratified Empirical Bernstein Sampling

Mark A. Burgess, Archie C. Chapman

*Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence*, pages 73-81. DOI: [10.24963/ijcai.2021/11](https://doi.org/10.24963/ijcai.2021/11)

## Abstract

The Shapley value is a well recognised method for dividing the value of joint effort in cooperative games. However, computing the Shapley value is known to be computationally hard, so stratified sample-based estimation is sometimes used. For this task, we provide two contributions to the state of the art. First, we derive a novel concentration inequality that is tailored to stratified Shapley value estimation using sample variance information. Second, by sequentially choosing samples to minimize our inequality, we develop a new and more efficient method of sampling to estimate the Shapley value. We evaluate our sampling method on a suite of test cooperative games, and our results demonstrate that it outperforms or is competitive with existing stratified sample-based estimation approaches to computing the Shapley value.