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Autre Publication Scientifique Documents de travail du Centre d'Économie de la Sorbonne Année : 2023

Minimal balanced collections: generation, applications and generalization

Résumé

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 pratical 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 to generalize the notion of balanced collection to balanced sets.
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halshs-03972833 , version 1 (03-02-2023)

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  • HAL Id : halshs-03972833 , version 1

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Dylan Laplace Mermoud, Michel Grabisch, Peter Sudhölter. Minimal balanced collections: generation, applications and generalization. 2023. ⟨halshs-03972833⟩
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