Axiomatic and Dynamic Results for Power Indexes under Symmetry

Ruey-Rong Huang,Yu-Hsien Liao

doi:10.3390/axioms10040345

Abstract

Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based on the difference between “participation processes” and “allocating results”, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification.

Highlights

Axioms 2021, 10, 345. https://doi.Three power indexes are considered in this article, the equal allocation non-separable costs (EANSC), the efficient Banzhaf–Coleman index (EBCI) and the efficient Banzhaf–Owen index (EBOI)
Based on the results of Theorem 2, in a legislative institution, by continuously adjusting the difference between any two legislative representatives as to their respective participation in all professional committee bills, power allocation among all legislative representatives will become gradually closer to the result of power allocation derived from the EANSC
The efficient Banzhaf–Owen index (EBOI) is the only power allocation that conforms to the two natures of “complete and proper allocation of power” and “balancing the total difference between any two legislative representatives as to their respective participation in all professional committee bills”

Summary

Introduction

Three power indexes are considered in this article, the equal allocation non-separable costs (EANSC), the efficient Banzhaf–Coleman index (EBCI) and the efficient Banzhaf–. Maschler and Owen [13], Moulin [6] and Peleg [14] conducted axiomatic characterizations of the Shapley value, the EANSC and the Prekernal respectively by considering this notion of symmetry. Can unique symmetry axioms be derived from different behavioral models and be applied to conduct axiomatic characterizations of these power indexes?. The allocative efficiency of resources would combined with different symmetry axioms to axiomatically characterize three power indexes. This article would used the power indexes, each with a different set of symmetry axioms, to produce three different modification functions and related dynamic processes. According to the axiomatic results, an efficient power index that approximates one of the symmetrical states implies that it approximates all three power indexes

Preliminaries

Symmetry and Axiomatic Results

Dynamic Processes

Discussions

Conclusions