Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley-Shubik and the (normalized) Banzhaf power index.
 A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243–260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation. The number of threshold functions of eight variables is counted by ILLIAC II, the computer of the University of Illinois. Sets of optimum weights of majority elements realizing these functions also are investigated. Actually, canonical positive self-dual threshold functions of nine variables are investigated instead of directly investigating threshold functions of eight variables because it is easier to deal with them. The number and optimum weights of threshold functions of eight variables are easily obtained from these functions of nine variables and their realization.  In this paper we deal with several classes of simple games; the first class is the one of ordered simple games (i.e. they admit of a complete desirability relation). The second class consists of all zero-sum games in the first one.  In many multiagent settings, situations arise in which agents must collectively make decisions while not every agent is supposed to have an equal amount of influence in the outcome of such a decision. Weighted voting games are often used to deal with these situations. The amount of influence that an agent has in a weighted voting game can be measured by means of various power indices. This paper studies the problem of finding a weighted voting game in which the distribution of the influence among the agents is as close as possible to a given target value. We propose a method to exactly solve this problem. This method relies on a new efficient procedure for enumerating weighted voting games of a fixed number of agents. The enumeration algorithm we propose works by exploiting the properties of a specific partial order over the class of weighted voting games. The algorithm enumerates weighted voting games of a fixed number of agents in time exponential in the number of agents, and polynomial in the number of games output. As a consequence we obtain an exact anytime algorithm for designing weighted voting games. This paper defines the canonical representative of each equivalence class in the classification of the majority decision functions by complementing and permuting variables and by complementing the output. Also, a method is proposed to obtain all the representatives with their optimum structures, and a table of the representatives of the majority decision functions of up to six variables is provided. The reader should be familiar with the content of a previous paper by the authors, included as reference [1].