Order and symmetry of simple games

Abstract

The aim of the paper is to use some known results of the theory of boolean functions and of the theory of finite groups for the classification and construction of simple games.Simple games can be seen as monotone boolean functions.In the introductory part the properties defining Post's classes are translated to game-theoretical properties.The second part gives a complete analysis of the non-Postian game-theoretical classes of half-hatf games, dual-equivalent games, ordered and weighted majority games with respect to set-inclusion and intersection.Symmetric boolean functions and symmetric games can be classified according to their symmetry group. In the third part the conditions of the existence of multiple-transitive games that are not fully symmetric are discussed.The last part gives a construction for those exceptional games that are sharply multiple-transitive without being trivial.The paper closes with an explicit construction of the family of the 13 most complex games; the corresponding automorphism, groups are Mathieu groups.