Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximating the Core via Iterative Coalition Sampling

Ian Gemp, Marc Lanctot, Luke Marris, Yiran Mao, Edgar Duéñez-Guzmán, Sarah Perrin, Andras Gyorgy, Romuald Elie, Georgios Piliouras, Michael Kaisers, Daniel Hennes, Kalesha Bullard, Kate Larson, Yoram Bachrach

TL;DR

The paper addresses the computational hardness of the core and least-core in transferable utility games and introduces scalable iterative algorithms that avoid solving large linear programs. It develops three approaches—Iterative Projections, Stochastic Subgradient Descent, and a Core Lagrangian saddle-point method—each with convergence guarantees and applicability to large, structured games. Empirical results demonstrate the methods’ scalability across weighted voting, graph, and marginal contribution network games, and they enable novel explainable AI (XAI) applications, including global/local feature explainability and data valuation. The findings show that core-based explanations can complement or surpass Shapley-value approaches in several XAI tasks while providing stability insights across diverse game representations, offering a practical pathway to integrate cooperative-game concepts in AI systems.

Abstract

The core is a central solution concept in cooperative game theory, defined as the set of feasible allocations or payments such that no subset of agents has incentive to break away and form their own subgroup or coalition. However, it has long been known that the core (and approximations, such as the least-core) are hard to compute. This limits our ability to analyze cooperative games in general, and to fully embrace cooperative game theory contributions in domains such as explainable AI (XAI), where the core can complement the Shapley values to identify influential features or instances supporting predictions by black-box models. We propose novel iterative algorithms for computing variants of the core, which avoid the computational bottleneck of many other approaches; namely solving large linear programs. As such, they scale better to very large problems as we demonstrate across different classes of cooperative games, including weighted voting games, induced subgraph games, and marginal contribution networks. We also explore our algorithms in the context of XAI, providing further evidence of the power of the core for such applications.

Approximating the Core via Iterative Coalition Sampling

TL;DR

The paper addresses the computational hardness of the core and least-core in transferable utility games and introduces scalable iterative algorithms that avoid solving large linear programs. It develops three approaches—Iterative Projections, Stochastic Subgradient Descent, and a Core Lagrangian saddle-point method—each with convergence guarantees and applicability to large, structured games. Empirical results demonstrate the methods’ scalability across weighted voting, graph, and marginal contribution network games, and they enable novel explainable AI (XAI) applications, including global/local feature explainability and data valuation. The findings show that core-based explanations can complement or surpass Shapley-value approaches in several XAI tasks while providing stability insights across diverse game representations, offering a practical pathway to integrate cooperative-game concepts in AI systems.

Abstract

The core is a central solution concept in cooperative game theory, defined as the set of feasible allocations or payments such that no subset of agents has incentive to break away and form their own subgroup or coalition. However, it has long been known that the core (and approximations, such as the least-core) are hard to compute. This limits our ability to analyze cooperative games in general, and to fully embrace cooperative game theory contributions in domains such as explainable AI (XAI), where the core can complement the Shapley values to identify influential features or instances supporting predictions by black-box models. We propose novel iterative algorithms for computing variants of the core, which avoid the computational bottleneck of many other approaches; namely solving large linear programs. As such, they scale better to very large problems as we demonstrate across different classes of cooperative games, including weighted voting games, induced subgraph games, and marginal contribution networks. We also explore our algorithms in the context of XAI, providing further evidence of the power of the core for such applications.
Paper Structure (30 sections, 7 theorems, 14 equations, 10 figures, 3 tables, 4 algorithms)

This paper contains 30 sections, 7 theorems, 14 equations, 10 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Approximating the Core through Iterative Payoff Adjustments via Coalition Sampling
  4. The $\epsilon$-Core via Iterative Projections
  5. The $\epsilon$-Core via Stochastic (Sub)Gradient Descent
  6. The Least-Core as a Saddle Point Problem
  7. Empirical Analysis
  8. Timing Sampled LPs vs. Core Lagrangian
  9. Analyzing Stability in Compact Game Representations
  10. Weighted Voting Games
  11. Graph Games
  12. Explainable AI
  13. Global Explainability
  14. Local Explainability
  15. Data Valuation
  16. ...and 15 more sections

Key Result

Theorem 1

Let $v : 2^I \rightarrow \mathbb{R}$ be a characteristic function. Given an $\epsilon$ and assuming the $\epsilon$-core exists, Algorithm alg:iterative_proj converges to an $\epsilon$-core imputation asymptotically, i.e.: $\lim_{T \rightarrow \infty} p_T \rightarrow \epsilon$-core$(v)$.

Figures (10)

  • Figure 1: (\ref{['fig:wvg:gauss']})-(\ref{['fig:wvg:beta']}) Heatmaps illustrating the impact of parameter axes on the least-core value (in color) in three weighted voting games. (\ref{['fig:wgg:erdosrenyi']})-(\ref{['fig:wgg:nws']}) Mean least-core value for Erdős-Rényi and Newman Watts Strogatz when sweeping over two hyperparameters. Constant hyperparameters are shown in parenthesis.
  • Figure 2: Approximate $\epsilon$ of the linear program (LP) versus the core Lagrangian (CL) method as a function of computation time (seconds). The $x$-axis corresponds to wall-clock time taken by both algorithms run side-by-side given $k \in \{500, 1000, 2000, 4000, 8000, 16000\}$ sampled coalitions for the LP method, $t_k$. A better core approximation quality is reflected in having a lower $\epsilon$ for the same runtime. The $y$-axis represents the approximate $\epsilon$ of the least-core solution found, $\hat{\epsilon}(p_{LP}, \hat{C})$ and $\hat{\epsilon}(p_{CL}, \hat{C})$, computed over the same set of $50,000$ coalitions, $\hat{C}$. Each data point $(t_k, \hat{\epsilon}(p_x, \hat{C}))$ represents an average over the same set of $2,500$ random weighted voting games with shading indicating standard error of the mean.
  • Figure 3: Correlation of Shapley and Core importance measures of features at both global dataset and individual instance levels.
  • Figure 4: Data Valuation on the Boston Housing, Diabetes, and Chat Bot Arena. Error bars correspond to 95% confidence intervals (1000 repeats).
  • Figure 5: Approximation quality of the linear program (LP), core Lagrangian (CL), and Shapley method as a function of computation time (seconds). The $x$-axis corresponds to wall-clock time taken by all algorithms run side-by-side given $k \in \{500, 1000, 2000, 4000, 8000, 16000\}$ sampled coalitions for the LP method, $t_k$. A better core approximation quality is reflected in having a lower $\epsilon$ for the same runtime. The left $y$-axis represents the approximate $\epsilon$ of the least-core solution found, $\hat{\epsilon}(p_{LP}, \hat{C})$ and $\hat{\epsilon}(p_{CL}, \hat{C})$, computed over the same set of $2^{25}$ coalitions, $\hat{C}$. As the Shapley solution is not motivated by stability, we measure the approximate solution's quality by euclidean distance to the Shapley value estimated with $10$ million Monte-Carlo samples. Each Shapley approximation is estimated using $125 \times k$ samples. Each data point $(t_k, \hat{\epsilon}(p_x, \hat{C}))$ represents an average over the same set of $2,500$ random weighted voting games with shading indicating standard error of the mean.
  • ...and 5 more figures

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Definition 4
  • Theorem
  • ...and 1 more