{"ID":2897004,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.05898","arxiv_id":"2507.05898","title":"Minimal balanced collections and their applications to core stability and other topics of game theory","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.","short_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 impl...","url_abs":"https://arxiv.org/abs/2507.05898","url_pdf":"https://arxiv.org/pdf/2507.05898v1","authors":"[\"Dylan Laplace Mermoud\",\"Michel Grabisch\",\"Peter Sudhölter\"]","published":"2025-07-08T11:39:03Z","proceeding":"cs.GT","tasks":"[\"cs.GT\",\"econ.TH\",\"math.CO\"]","methods":"[]","has_code":false}