  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 this work, 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.
