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

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. • Empirical validation of Penrose’s Limit Theorem in French inter-communal councils reveals its dependence on municipality size and council composition. • The theorem holds as long as the size of the major player is bounded, and the number of players plays a role, too. • Using a toy theoretical model, an original apportionment method is proposed to reduce the power of the largest municipality and promote fairer representation.