Dimension of complete simple games with minimum

Weighted majority games have the property that players are totally ordered by the desirability relation (introduced by Isbell in [J.R. Isbell, A class of majority games, Quarterly Journal of Mathematics, 7 (1956) 183-187]) because weights induce it. Games for which this relation is total are called complete simple games. Taylor and Zwicker proved in [A.D. Taylor, W.S. Zwicker, Weighted voting, multicameral representation, and power, Games and Economic Behavior 5 (1993) 170-181] that every simple game (or monotonic finite hypergraph) can be represented by an intersection of weighted majority games and consider the dimension of a game as the needed minimum number of them to get it. They provide the existence of non-complete simple games of every dimension and left open the problem for complete simple games. In this paper we prove that their result can be extended for these games and give a constructive procedure to get complete simple games of every dimension. The curious fact is that to obtain our result it is enough to consider the simplest kind of complete simple games, i.e. those that admit a minimum lattice representative. As a consequence, it is proved that there is no connection between 'having small dimension' and 'being totally ordered by the desirability relation'. (c) 2007 Elsevier B.V. All rights reserved.