ILP and CP models for normalized integer weighted voting game design problem

Abstract

Weighted voting games are a family of cooperative games, where each voter or player carries a weight. If the sum of the weights of the players in favor of a proposal is larger than or equal to a given quota, then this proposal is accepted. In the weighted game field, the problem of normalized integer weighted voting game representation (NIWVG), and the integer weighted voting game design problem (IWVGD) or inverse power index problem have been subjects of several studies. A solution to the last problem is an integer-weighted voting game with a specific power index, as close as possible (following a norm) to a given influence vector. Finding a normalized integer-weighted voting game solution for the inverse power index problem is a challenging task. The nature of the power index and the norm awarded in the problem can impact the approach used to solve this problem. In this paper, exact approaches based on integer linear programming (ILP) and constraint programming (CP) are proposed for this problem. The models proposed in this paper combine the Shapley–Shubik index or Banzhaf–Coleman index with Manhattan and Chebyshev norms respectively. Computational results show that ILP is an adequate approach when the power index and the norm can be linearized while CP appears to be suitable for the remaining cases of voting games with a small number of players.