Least Core Computation of Weighted Voting Games

We introduce a pseudo-polynomial numerical method for computing an element of the least core in three distinct variations of weighted voting games: simple, multiple, and the disjunction of simple weighted voting games. The least core intended to find a stable sharing, i.e. no coalition is tempted to reject it, particularly by taxing the coalitions that reject this sharing with a minimum tax. We establish a compelling equivalence between computing the least core of a weighted voting game and solving the linear relaxation of the Gilmore and Gomory model for bin packing. We use our approach to solve a number of real-life problems, such as Electoral College of the United States of America, the Voting system of European Union and the qualified majority of the Council of EU. For example, the linear system, associated with the qualified majority vote, containing more than 4000 constraints with almost 19000 variables, was solved in less than 35 s.