Efficient Approximation Algorithms for the Inverse Semivalue Problem

Weighted voting games are typically used to model situations where a number of agents vote against or for a proposal. In such games, a proposal is accepted if a weighted sum of the votes exceeds a specified threshold. As the influence of a player over the 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 desirable influences as they are measured by a predefined power index. Recent work has shown that exactly solving the inverse power index problem is computationally intractable when the power index is in the class of semivalues-a broad class that includes the popular Shapley value and Banzhaf index. In this work, we design efficient approximation algorithms for the inverse semivalue 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.