  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.
