Geodesic
Hierarchies in Voting Simple Games
Contributions to game theory and management, Tome 2 (2009), pp. 72-80.

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A work by Friedman, McGrath and Parker introduced the concept of a hierarchy of a simple voting game and characterized which hierarchies, induced by the desirability relation, are achievable in weighted games. They proved that no more hierarchies are obtainable if weighted games are replaced by the larger class of linear games. In a subsequent paper by Freixas and Pons, it was proved that only four hierarchies, conserving the ordinal equivalence between the Shapley–Shubik and the Penrose–Banzhaf–Coleman power indices, are non–achievable in simple games. It was also proved that all achievable hierarchies are obtainable in the class of weakly linear games. In this paper, we define a new class of totally pre–ordered games, the almost linear games, smaller than the class of weakly linear games, and prove that all hierarchies achievable in simple games are already achievable in almost linear games.
Keywords: simple game, power index, desirability, weak desirability, almost desirability, linear game, weakly linear game, almost linear game. 
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Josep Freixas; Montserrat Pons. Hierarchies in Voting Simple Games. Contributions to game theory and management, Tome 2 (2009), pp. 72-80. http://geodesic.mathdoc.fr/item/CGTM_2009_2_a6/

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