A Lexicographic Public Good Ranking
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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 [12] and the criticality-based ranking [1]. We define a new ranking solution, the lexicographic Public Good ranking (LPGR), 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 [15], when it holds. A suitable characterization of the LPGR solution is provided. Finally, we investigate the relation among the LPGR solution and the criticality-based ranking, referring to the dual game.
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