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

Author

Listed:
  • Dylan Laplace Mermoud

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Peter Sudhölter

    (SDU - University of Southern Denmark)

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.

Suggested Citation

  • Dylan Laplace Mermoud & Michel Grabisch & Peter Sudhölter, 2023. "Minimal balanced collections and their application to core stability and other topics of game theory," Post-Print halshs-04356803, HAL.
    • Handle: RePEc:hal:journl:halshs-04356803
    DOI: 10.1016/j.dam.2023.07.025
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-04356803
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