text: We study the inverse power index problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we study various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions (also known as shift-maximal losing coalitions) of a game from its list of roof coalitions (also known as shift-minimal winning coalitions), and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the synthesis problem, we then improve on this naive algorithm, and we obtain an enumeration algorithm that runs in quadratic exponential time (that is, O(2^(n^2) p(n)) for a polynomial p). Moreover, we show that this algorithm runs in output-polynomial time, making it the best possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime algorithm for the inverse power index problem that runs in exponential time. This algorithm is straightforward and general: it computes the error for each game enumerated, and outputs the game that minimizes this error. By the genericity of our approach, our algorithm can be used to find a weighted voting game that optimizes any exponential time computable function. We implement our algorithm for the case of the normalized Banzhaf index, and we perform experiments in order to study performance and error convergence.
  @cite_11: For f a weighted voting scheme used by n voters to choose between two candidates, the nShapley-Shubik Indices (or Shapley values) of f provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 [SS54] and are widely studied in social choice theory as a measure of the "influence" of voters. The Inverse Shapley Value Problem is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant e>0 our algorithm runs in fixed poly(n) time (the degree of the polynomial is independent of e) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error e. If there is a "reasonable" voting scheme in which all voting weights are integers at most poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error e=n−1 8.
summary: The other recent work is @cite_11 , by This paper gives provides as a main result an algorithm for the inverse power index problem for the case of the Shapley-Shubik index, and has a certain approximation guarantee: in addition to a target power index, the algorithm takes a precision parameter @math and guarantees to output a weighted voting game of which the power index is @math -close to it, on the precondition that there an @math -close weighted voting game with the property that the quota is not too skewed, in a particular sense. This is, to our knowledge, the only polynomial time algorithm for a power index voting game design problem that provides an approximation guarantee in any sense.