Achievable hierarchies in voting games with abstention

It is well known that he in‡uence relation orders the voters the same way as the classical Banzhaf and Shapley-Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the in‡uence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of in‡uence relation. Keywords : Game theory (3; 2) Voting rules Abstention Decision Support Systems Weightedness and Completeness Hierarchies Mathematics Subject Classi…cation (2000) 91A12 05C65 94C10 “Departament de Matemàtica Aplicada 3 i Escola Politècnica Superior d’Enginyeria de Manresa, Universitat Politècnica de Catalunya". Research partially supported by “Ministerio de Economía y Competitividad proyecto MTM2012-34426/FEDER" and “Govern de la Generalitat SGR 2009–1029" ; [email protected]. yCorresponding author : [email protected], University of Yaounde I ; MASS laboratory University of Cergy Pontoise ; THEMA laboratory. [email protected], University of Yaounde I.