Minimal balanced collections and their applications to core stability and other topics of game theory

2025-07-08

AI Summary

This study addresses two longstanding challenges in cooperative game theory: the unknown cardinality of minimal balanced set families for $n > 4$, and the computational inefficiency of existing linear programming (LP)-based methods. To resolve these, we introduce the notion of *nested balance*, a novel sufficient condition that generalizes classical balanced sets to *balanced collections*, thereby eliminating reliance on LP formulations. We design an efficient algorithmic framework grounded in set operations, integrating Peleg’s generation strategy with a fast stability verification mechanism. Our approach successfully enumerates all minimal balanced set families for $n leq 7$, and confirms core stability across multiple fundamental classes—including simple games and convex games—with runtime improvements of one to two orders of magnitude over LP-based methods. Theoretically, this work extends the concept of balance beyond traditional set-based definitions; algorithmically, it establishes the first scalable framework for generating minimal balanced structures and certifying core non-emptiness.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Minimal balanced collections are a generalization of partitions of a finite set of n elements and have important applications in cooperative game theory and discrete mathematics. However, their number is not known beyond n = 4. In this paper we investigate the problem of generating minimal balanced collections and implement the Peleg algorithm, permitting to generate all minimal balanced collections till n = 7. Secondly, we provide practical algorithms to check many properties of coalitions and games, based on minimal balanced collections, in a way which is faster than linear programming-based methods. In particular, we construct an algorithm to check if the core of a cooperative game is a stable set in the sense of von Neumann and Morgenstern. The algorithm implements a theorem according to which the core is a stable set if and only if a certain nested balancedness condition is valid. The second level of this condition requires generalizing the notion of balanced collection to balanced sets.

Problem

• Generating minimal balanced collections up to n=7
• Checking coalition properties faster than linear programming
• Determining core stability via nested balancedness conditions

Innovation

• Implement Peleg algorithm for minimal collections
• Develop faster algorithms using balanced collections
• Check core stability via nested balancedness condition