A note about games-composition dimension

A minimum dimensional class of simple games

“…(iv) Theorem 4.3 in Freixas and Puente (2008) shows the existence of complete games with M-dimension m for every m ≥ 1. By considering the dual games of these complete games we obtain complete games of every codimension m. Further, Theorem 1.7.5 in Taylor and Zwicker (1999) and also Theorem 2.1 in Freixas and Puente (2001) show games with exponential M-dimension (or simply, dimension). By Remark 3.3(i), these games have U 1 -dimension, and by Lemma 3.1(ii), dim M (W ) ≤ dim U 1 (W ) for all W ∈ S. Hence, those games have exponential U 1 -dimension.…”

Cooperation through social influence

“…Our model is based on the linear threshold model for influence spread. In such a context we use [28], but this game admits a polynomial unweighted influence graph (G, f ) with respect to n for the corresponding unweighted influence game (G, f, n + 1, N).…”

“…In particular it remains open to know whether there are games with exponential dimension that also require an exponential number of players in any representation as influence games. In this line, we know that the simple game with exponential dimension with respect to the players of Section 2 in [28] can be represented by an unweighted influence game in polynomial time with respect to the number of players (see Figure 12). Another candidate is the simple game with exponential dimension of Theorem 8 in [20] for which we have been unable to show whether it can be represented by an (unweighted) influence game with polynomial number of agents.…”

“…Thus, as any simple game can be represented as an intersection or union of a finite number of weighted games, we have an alternative way to show the completeness of the family of weighted influence games with respect to the class of simple games (Theorem 1). However, as the dimension, the codimension, and the representation as a boolean weighted game of a simple game might be exponential in the number of players (but bounded by the number of maximal losing, minimal winning coalitions, or both, respectively) [28,25,22], we cannot conclude that any simple game can be represented by a weighted influence game whose number of agents is polynomial in the number of players. For the particular case of unweighted influence game we know the following.…”

Dimension and Codimension of Simple Games

“…As Γ * is a composition of n unanimity games, Γ * has dimension 2 n−1 [5] and Γ has codimension 2 n−1 (by Lemma 2). Proposition 1.…”