Directed and weighted majority games

Abstract

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 withn persons, which can be mapped onto the class of weighted majority zero-sum games withn+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 combiningLP-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.