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

Zineb Abidi Perier and Vincent Merlin

Mathematical Social Sciences, 2025, vol. 137, issue C

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.

Keywords: Power indices; Penrose Law; Banzhaf index; Voting; Local governance (search for similar items in EconPapers)
JEL-codes: C71 D72 H77 (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matsoc:v:137:y:2025:i:c:s0165489625000332

DOI: 10.1016/j.mathsocsci.2025.102418

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