Mergeable weighted majority games and characterizations of some power indices
Livino M. Armijos-Toro, José M. Alonso-Meijide, Manuel A. Mosquera
TL;DR
The paper addresses the problem of characterizing power allocation in weighted majority games by developing a mergeability framework and providing an axiomatic characterization of the Colomer‑Martínez power index (CM) on weighted majority games. It introduces WM‑mergeable weighted majority games and proves that the unique index on $\mathcal{SIW}$ satisfying efficiency, null-player, weighted symmetry, and weighted Deegan‑Packel‑like mergeability is $CM_i(q;\boldsymbol{w}) = \frac{1}{|M(q;\boldsymbol{w})|} \sum_{S \in M_i(q;\boldsymbol{w})} \frac{w_i}{w_S}$. The paper also defines the Holler‑Colomer‑Martínez index (HCM) as $HCM_i(q;\boldsymbol{w}) = \frac{|M_i(q;\boldsymbol{w})|\,w_i}{\sum_{j} |M_j(q;\boldsymbol{w})|\,w_j}$, combining Public Good ideas with weights, and provides a weighted mergeability characterization. An empirical application to Ecuador’s National Assembly demonstrates how CM and HCM align with weight structure and differ from SS, DP, and PG, illustrating practical impact for parliamentary power analysis. These results yield a principled basis for selecting power indices in weighted voting contexts and pave the way for generalizations to unions and consortia in future work.
Abstract
In this paper, we introduce a notion of mergeable weighted majority games with the aim of providing the first characterization of the Colomer-Martínez power index (Colomer and Martínez in J Theor Polit 7(1):41-63, 1995). Furthermore, we define and characterize a new power index for the family of weighted majority games that combines ideas of the Public Good (Holler in Polit Stud 30(2):262-271, 1982) and Colomer-Martínez power indices. Finally, we analyze the National Assembly of Ecuador using these and some other well-known power indices.