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An application of power indices for the family of weighted majority games in partition function form

José M. Alonso-Meijide, Livino M. Armijos-Toro, Balbina V. Casas-Méndez, Manuel A. Mosquera

TL;DR

This paper tackles the problem of measuring power in weighted majority games under partition function form by generalizing two established indices to this broader framework. It derives $CM_i(q;\boldsymbol{w})$ and $HCM_i(q;\boldsymbol{w})$ and contrasts them with $DP_i$ and $PG_i$—extending the prior Deegan-Packel and Public Good approaches and building on Colomer-Martínez and Holler-Colomer-Martínez foundations. The authors then apply these indices to the National Assembly of Ecuador during 2021 under a plurality rule, showing how power distributions evolve as legislative benches form and reconfigure; weight-aware indices (CM/HCM) capture shifts that purely coalition-counting indices (DP/PG) may overlook. The work demonstrates the practical relevance of partition-form power indices for analyzing complex, changing political bodies and suggests avenues for axiomatic and computational development in this setting.

Abstract

Based on Holler (1982) and Armijos-Toro et al. (2021) we propose two power indices to measure the influence of the players in the class of weighted majority games in partition function form. We compare these new power indices with their original versions on the class of games in characteristic function form. Finally, we use both pairs of power indices for games in partition function form to study the distribution of power in the National Assembly of Ecuador that emerged after the elections of February 7, 2021.

An application of power indices for the family of weighted majority games in partition function form

TL;DR

This paper tackles the problem of measuring power in weighted majority games under partition function form by generalizing two established indices to this broader framework. It derives and and contrasts them with and —extending the prior Deegan-Packel and Public Good approaches and building on Colomer-Martínez and Holler-Colomer-Martínez foundations. The authors then apply these indices to the National Assembly of Ecuador during 2021 under a plurality rule, showing how power distributions evolve as legislative benches form and reconfigure; weight-aware indices (CM/HCM) capture shifts that purely coalition-counting indices (DP/PG) may overlook. The work demonstrates the practical relevance of partition-form power indices for analyzing complex, changing political bodies and suggests avenues for axiomatic and computational development in this setting.

Abstract

Based on Holler (1982) and Armijos-Toro et al. (2021) we propose two power indices to measure the influence of the players in the class of weighted majority games in partition function form. We compare these new power indices with their original versions on the class of games in characteristic function form. Finally, we use both pairs of power indices for games in partition function form to study the distribution of power in the National Assembly of Ecuador that emerged after the elections of February 7, 2021.
Paper Structure (9 sections, 1 theorem, 19 equations, 3 figures, 9 tables)

This paper contains 9 sections, 1 theorem, 19 equations, 3 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Simple games in characteristic and partition function form
  3. Simple games in characteristic function form
  4. Simple games in partition function form
  5. Power indices for weighted majority games in partition function form
  6. A political example: The National Assembly of Ecuador
  7. Original formation of the National Assembly of Ecuador: May 2021
  8. Formation of the National Assembly of Ecuador: from Jun 2021 to Dec 2021
  9. Conclusions

Key Result

Proposition 2.1

Let $\mathcal{C}\subseteq EC^N$ be such that there is no relation with respect to $\sqsubseteq$ between any pair $(S;\mathsf{P}),(T;\mathsf{Q})\in \mathcal{C}$. Then, there exists a unique simple game in partition function form, $(N,v)$, such that $\mathcal{M}(v)=\mathcal{C}$.

Figures (3)

  • Figure 1: National Assembly of Ecuador in May 2021.
  • Figure 2: Composition of the legislative benches - National Assembly of Ecuador in June 2021.
  • Figure 3: National Assembly of Ecuador from June to December 2021.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.1
  • Definition 3.1
  • ...and 4 more