# On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games

## TL;DR

This paper investigates the computational difficulty of designing weighted voting games to match specified influence levels, proving that the inverse problem is generally intractable for many semivalues including Banzhaf and Shapley indices.

## Contribution

It establishes the intractability of the inverse semivalue problem for a broad class of power indices, including the most popular ones.

## Key findings

- Inverse problem is computationally intractable for many semivalues.
- Hardness results include Banzhaf and Shapley indices.
- Results highlight fundamental computational limits in voting game design.

## Abstract

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.