Directed and weighted majority games

In this paper we deal with several classes of simple games; the first class is the one of ordered simple games (i.e. they admit of a complete desirability relation). The second class consists of all zero-sum games in the first one. First of all we introduce a "natural" partial order on both classes respectively and prove that this order relation admits a rank function. Also the first class turns out to be a rank symmetric lattice. These order relations induce fast algorithms to generate both classes of ordered games. Next we focus on the class of weighted majority games with n persons, which can be mapped onto the class of weighted majority zero-sum games with n + 1 persons. To this end, we use in addition methods of linear programming, styling them for the special structure of ordered games. Thus, finally, we obtain algorithms, by combining LP-methods and the partial order relation structure. These fast algorithms serve to test any ordered game for the Weighted majority property. They provide a (frequently minimal) representation in case the answer to the test is affirmative.