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

Ferenc Illés

TL;DR

It is shown that the Shapley value calculation is pseudo-polynomial for linearly representable games, which is a generalization of many classical and recent results in the literature.

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.

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

TL;DR

It is shown that the Shapley value calculation is pseudo-polynomial for linearly representable games, which is a generalization of many classical and recent results in the literature.

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.
Paper Structure (15 sections, 13 theorems, 60 equations, 4 tables)

This paper contains 15 sections, 13 theorems, 60 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integral representation
  4. Expected value representation
  5. Linearly representable games
  6. Shapley value of a single player
  7. Notations and basic properties
  8. Algorithm and complexity
  9. Shapley value of the whole game
  10. Reverse recursion
  11. Algorithm and complexity
  12. Summary and possible extensions
  13. Optimizing the algorithm
  14. Application for other game types
  15. Limitations

Key Result

Lemma 2.1

For any $x\in[0, 1]$ and any finite set $\mathcal{H}$, $\mathbb{P}^{\mathcal{H}}_x:2^\mathcal{H}\to [0,1]$ is a probability distribution.

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Remark 2.1
  • Definition 3.1
  • Remark 3.1
  • Definition 3.2
  • Definition 3.3
  • Example 3.1: weighted voting games
  • Example 3.2: bankruptcy games
  • Example 3.3: liability games
  • ...and 29 more