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Testing Penrose Limit Theorem: A case study of French local data

Author

Listed:
  • Zineb Abidi Perier

    (ERUDITE - Equipe de Recherche sur l’Utilisation des Données Individuelles en lien avec la Théorie Economique - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12 - Université Gustave Eiffel, TEPP - Théorie et évaluation des politiques publiques - CNRS - Centre National de la Recherche Scientifique)

  • Vincent Merlin

    (TEPP - Travail, Emploi et Politiques Publiques - UPEM - Université Paris-Est Marne-la-Vallée - CNRS - Centre National de la Recherche Scientifique, CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR - Université de Rennes - CNRS - Centre National de la Recherche Scientifique)

Abstract

In 1946, Penrose argued that in a weighted quota game, if the number of players is sufficiently large and the weight associated with the largest player is bounded, the Banzhaf/Penrose power of a player is approximately proportional to its weight. This conjecture is now known as the Penrose Limit Theorem (PLT). However, as the weight of the largest player increases and/or when it is surrounded by an ocean of small players, its Banzhaf/Penrose power approaches one, even if its own weight is far less than 50% of the total weight. This paper aims to empirically determine the conditions under which this assertion holds. Can we identify the threshold weight (as a percentage of the total sum of weights) below which the Penrose Limit Theorem applies? To address this question, we analyze a panel of 1,251 French intercommunal councils, where each town is represented by a given number of delegates. In particular, we compare the normalized Banzhaf index of the largest city to its weight in the council. As a consequence, we propose an alternative allocation rule that French law might adopt, considering factors such as the weight of the largest city in the council and the number of its members.

Suggested Citation

  • Zineb Abidi Perier & Vincent Merlin, 2025. "Testing Penrose Limit Theorem: A case study of French local data," Post-Print halshs-05090071, HAL.
    • Handle: RePEc:hal:journl:halshs-05090071
    DOI: 10.1016/j.mathsocsci.2025.102418
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    References listed on IDEAS

    as
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    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior
    • H77 - Public Economics - - State and Local Government; Intergovernmental Relations - - - Intergovernmental Relations; Federalism

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