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

TL;DR

This paper advances the understanding of minimal balanced collections by developing algorithms to generate and analyze them, with applications to core stability in cooperative game theory, extending computational capabilities up to n=7.

Contribution

It introduces practical algorithms for generating minimal balanced collections and checking core stability, surpassing previous linear programming methods, and extends the concept to balanced sets.

Findings

Generated all minimal balanced collections for n up to 7.

Developed faster algorithms for core stability verification.

Extended the notion of balanced collections to balanced sets.

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.