Dimension of complete simple games with minimum

The main contribution of this paper is to calculate the dimension of complete simple games with minimum using their characteristic invariants. As a consequence of this result we can deduce that there are complete simple games of every dimension. In a weighted majority game we can line up the voters in the order determined by noncreasing weight. Speaking informally, it certainly seems as if this same line-up should be one of creasing or decreasing influence (the exact definition of player i has more influence than player j is given by the desirability relation introduced by Isbell (1956)) over outcomes in the voting system modeled by the game. Indeed, the phrase �carries more weight� is used colloquially to mean �has more influence�. Complete games can be described as those for which it is possible to line up the players in some linear ordering for which influence is decreasing. This informally justify that every weighted majority game is complete. Then, a natural extension of weighted majority games are complete games, for which the desirability order is total. Carreras and Freixas (1996) associate a system of quantities (characteristic invariants) with every complete simple game and state their basic properties. Then they show these quantities determine the game (uniqueness) and that every system (admissible system) is associated with some complete simple game. A partial motivation to introduce characteristic invariants is the issue of characterizations of weightedness and we notice that the assumability condition of Elgot (1960) and the trade robustness of Taylor and Zwicker (1992) are adapted to the representation given by characteristic invariants. A particular case of complete simple games are those with minimum that have been studied by Freixas (1997) and Freixas and Puente (1998). The first work provides necessary and sufficient conditions to determine when a game of this type is weighted. In the second one the characteristic invariants are used to facilitate the calculus of different kind of solutions like the nucleolus, the kernel and semivalues. Different situations described through rules of decision can be modeling using complete simple games with minimum; among them, the United Nations Security Council and the procedure to amend the Canadian Constitution. It is well known that every simple game can be represented as intersection of weighted majority games. It nevertheless becomes of interest to ask how efficiently this can be done for a given simple game. The question of efficiency leads to the definition of dimension. A simple game is said to be of dimension k if and only if it can be represented as the intersection of exactly k weighted majority games, but not as the intersection of k - 1 weighted majority games. Most naturally occurring voting systems in use are modeled by simple games of dimension one or two. Interesting examples of dimension 2 are the United States Federal System and the procedure to amend the Canadian Constitution. A central question to suggest itself is whether or not there are simple games of every dimension. This was answered by Taylor and Zwicker (1999). They proved that for all positive integer n, there is a simple game of dimension n. However, the Taylor and Zwicker�s proof is based in a class of games highly not complete. In this situation we raise the next question: is it possible to define families of complete simple games such that their dimension increases with the number of players?, i.e., for all positive integer n, there is a complete simple game of dimension n? The answer to this question is given in this paper by the determination of the dimension of complete simple games with minimum. This result shows that the complexity of the game is not closely connected with the fact that the desirability relation is total.

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