Given n players, a simple game is a 0/1 valued TU cooperative game verifying unanimity and monotonicity. In such a game, a coalition, i.e., a subset of players, always wins (winning coalition) or loses (losing coalition). It is known that each simple simple can be expressed as the intersection (or the union) of weighted voting games. A simple game is called weighted voting (majority) game if there is a quote q and an assignment of a non-negative integer weight to each player in such a way tha...
Given n players, a simple game is a 0/1 valued TU cooperative game verifying unanimity and monotonicity. In such a game, a coalition, i.e., a subset of players, always wins (winning coalition) or loses (losing coalition). It is known that each simple simple can be expressed as the intersection (or the union) of weighted voting games. A simple game is called weighted voting (majority) game if there is a quote q and an assignment of a non-negative integer weight to each player in such a way that a coalition is winning if and only if the sum of the weights of its players is greater than or equal to the integer quota q.
Two important and well studied concepts relating simple games with weighted games are the dimension and the codimension.
The dimension is the minimum number of weighted games such that their intersections generate the considered simple game. In the same vein, the codimension is the minimum number of weighted games such that their unions generate the considered simple game. There are some previous studies about the dimension and the codimension of simple games.
Nevertheless no complete classification of dimension or codimension of simple games is known even for small number of players. In this work we initiate a systematic such classification with respect to the dimension and the codimension parameters.
We introduce the concept of multidimension of a simple game, the minimum number of intersections and unions of weighted games to generate the considered simple game.
A similar concept was introduced in another topic (Boolean functions) by Goldberg.
We also classify some simple games with respect to their multidimension.
Moreover, we present some particular results of the dimension, the codimension and the multidimension for specific simple games depending on properties of the (minimal) winning coalitions.
Finally, we study possible relations between k-trade robustness and k-invarinat-trade robustness (concepts that characterize the subclass of simple games called complete simple games) with respect to the dimension, the codimension and the multidimension.