Linearly representable games and pseudo-polynomial calculation of the Shapley value

TL;DR

This paper introduces linearly representable games, a class of TU games describable by parameters equal to the number of players, and demonstrates that their Shapley value can be computed in pseudo-polynomial time, generalizing existing results.

Contribution

The paper defines linearly representable games and proves that the Shapley value can be computed in pseudo-polynomial time for this class, extending previous work.

Findings

Shapley value calculation is pseudo-polynomial for linearly representable games.

The method becomes polynomial when parameters are polynomial in the number of players.

Includes classical and recent game classes like weighted voting and bankruptcy games.

Abstract

We introduce the notion of linearly representable games. Broadly speaking, these are TU games that can be described by as many parameters as the number of players, like weighted voting games, airport games, or bankruptcy games. We show that the Shapley value calculation is pseudo-polynomial for linearly representable games. This is a generalization of many classical and recent results in the literature. Our method naturally turns into a strictly polynomial algorithm when the parameters are polynomial in the number of players.