Abstract
This paper studies the complexity of computing a representation of a simple game as the intersection (union) of weighted majority games, as well as, the dimension or the codimension. We also present some examples with linear dimension and exponential codimension with respect to the number of players.
Highlights
Introduction and preliminariesWe consider the so-called simple games and the computational complexity of representing them as unions or intersections of weighted majority games
Γ is said to be self-dual if Γ = Γ∗
Given two simple games Γ1 = (N1, W1) and Γ2 = (N2, W2), they are equivalent if N1 = N2 and W1 = W2
Summary
We consider the so-called simple games and the computational complexity of representing them as unions or intersections of weighted majority games. The complement of the family of winning coalitions is the family of losing coalitions L, i.e., L = P(N ) \ W Any of those set families determine uniquely the game Γ and constitute one of the usual forms of representation for simple games [12], the size of the representation is not, in general, polynomial in the number of players [10]. A simple game Γ is a weighted majority game (WMG) if it admits a representation by means of n + 1 nonnegative real numbers [q; w1, . Note that the converse statement of the last sentence is not true in general as there are weighted games which are not self-dual
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