Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration

We study the power index voting game design 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 consider 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 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 power index voting game design 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 in∗Algorithms, Combinatorics and Optimization; Centrum Wiskunde & Informatica; The Netherlands; Email: [email protected]. †Algorithmics; Delft University of Technology; The Netherlands; Email: [email protected]. ‡Department of Econometrics; Erasmus University Rotterdam; The Netherlands; Email: