Inverse power index problem: algorithms and complexity
Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of the votes exceeds a prespecified threshold. As the influence of a player over the voting outcome is not in general proportional to her assigned weight, various power indices have been proposed to measure each player’s influence. The inverse power index problem is the problem of designing a weighted voting game that achieves a set of target influences according to a predefined power index. In the first part of this thesis, we study the computational complexity of the inverse problem when the power index belongs to the class of semivalues. We prove that the inverse problem is computationally intractable for a broad family of semivalues, including all regular semivalues. As a special case of our general result, we establish computational hardness of the inverse problem for the Banzhaf indices and the Shapley values, arguably the most popular power indices. In the second part, we design efficient approximation algorithms for the inverse semi-value problem. We develop a unified methodology that leads to computationally efficient algorithms that solve the inverse semivalue problem to any desired accuracy. We perform an extensive experimental evaluation of our algorithms on both synthetic and real inputs. Our experiments show that our algorithms are scalable and achieve higher accuracy compared to previous methods in the literature.