Evaluation and improvement of Laruelle-Widgrén inverse Banzhaf approximation

Voting is a popular way of reaching decisions in multi-agent systems. Weighted voting in particular allows different agents to have varying levels of influence on the decision taken: each agent's vote carries a weight, and a proposal is accepted if the sum of the weights of the agents in favor of the proposal is at least equal to a given quota. Unfortunately, there is no clear and unambiguous relation between a player's weight and the extent of her influence on the outcome of the decision making process. Different measures of 'power' have been proposed, such as the Banzhaf and the Shapley-Shubik indices. Here we consider the 'inverse' problem: given a vector of desired power indices for the players, how should we set their weights and the quota such that the players' power in the resulting game comes as close as possible to the target vector? There has been some work on this problem, both heuristic and exact, but little is known about the approximation quality of the heuristics for this problem. The goal of this paper is to empirically evaluate the heuristic algorithm for the Inverse Banzhaf Index problem by Laruelle and Widgrén. We analyze and evaluate the intuition behind this algorithm. We found that the algorithm can not handle general inputs well, and often fails to improve inputs. It is also shown to diverge after only tens of iterations. Based on our analysis, we present three alternative extensions of the algorithm that do not alter the complexity but can result in up to a factor 6:5 improvement in solution quality.