Generating Functions for Computing the Myerson Value

The complexity of a computational problem is the order of computational resources which are necessary and sufficient to solve the problem. The algorithm complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce polynomial time algorithms based in generating functions for computing the Myerson value in weighted voting games restricted by a tree. Moreover, we apply the new generating algorithm for computing the Myerson value in the Council of Ministers of the European Union restricted by a communication structure. 1. The Myerson value for voting games In general, it is difficult to define the idea of power, but for the special case of voting power there are mathematical power indices that have been used. The first such power index was proposed by Shapley and Shubik [15] who apply the Shapley value [14] to the case of simple games. Another concept for measuring voting power was introduced by Banzhaf [1], a lawyer, whose work has appeared mainly in law journals, and whose index has been used in arguments in various legal proceedings. A simple game is a function v : 2 N → {0, 1}, such that v(N) = 1 and v is nonde-creasing, i.e., v(S) v(T) whenever S ⊆ T ⊆ N. A coalition is winning if v(S) = 1, and losing if v(S) = 0. The collection of all winning coalitions is denoted by W. We introduce a class of games called weighted voting games. The symbol [q; w 1 ,. .. , w n ] will be used, where the quota q and the weights w 1 ,. .. , w n are positive integers with 0 < q n i=1 w i. Here there are n players, w i is the number of votes of player i, and q is the quota needed for a coalition to win. Then, the above symbol represents the simple game v : 2 N → {0, 1} defined for all S ⊆ N by v(S) = 1 if w(S) q, 0 if w(S) < q, where w(S) = i∈S w i .