Lexicographic Ranking based on Minimal Winning Coalitions

TL;DR

This paper introduces the Lexicographic Ranking based on Minimal Winning Coalitions (LRM), a new player ranking method in simple games that aligns with desirability relations and relates to existing indices like the Public Good Index.

Contribution

The paper proposes the LRM solution, providing its characterization and analyzing its properties and relation to other ranking methods in simple games.

Findings

LRM is monotonic with respect to desirability when applicable.

LRM is rooted in minimal winning coalitions.

The relation between LRM and criticality-based ranking is explored.

Abstract

In this paper, we consider the consistency of the desirability relation with the ranking of the players in a simple game provided by some well-known solutions, in particular the Public Good Index [14] and the criticality-based ranking [1]. We define a new ranking solution, the Lexicographic Ranking based on Minimal winning coalitions (LRM), strongly related to the Public Good Index being rooted in the minimal winning coalitions of the simple game, proving that it is monotonic with respect to the desirability relation [17], when it holds. A suitable characterization of the LRM solution is provided. Finally, we investigate the relation among the LRM solution and the criticality-based ranking, referring to the dual game.