Desirability and social rankings

TL;DR

This paper characterizes five social ranking solutions in coalitional games based on desirability relations, providing axiomatic foundations and practical insights into their similarities and differences.

Contribution

It introduces an axiomatic framework for social ranking solutions based on desirability, characterizing five existing solutions and analyzing their properties and relations.

Findings

Axiomatic characterizations of five social ranking solutions.

Identification of similarities among the solutions.

Application to a bicameral legislature scenario.

Abstract

In coalitional games, a player $i$ is regarded as strictly more desirable than player $j$ if substituting $j$ with $i$ within any coalition leads to a strict augmentation in the value of certain coalitions, while preserving the value of the others. We adopt a property-driven approach to 'integrate' the notion of the desirability relation into a total relation by establishing sets of independent axioms leading to the characterization of solutionconcepts from the related literature. We focus on social ranking solutions consistent with the desirability relation and propose complementary sets of properties for the axiomatic characterization of five existing solutions: Ceteris Paribus (CP-)majority, lexicographic excellence (lex-cel), dual-lex, $L^{(1)}$ solution and its dual version $L^{(1)}_{*}$ . These characterizations reveal additional similarities among the five solutions and emphasize the essential characteristics that should be taken into account when selecting a social ranking. A practical scenario involving a bicameral legislature is studied.