Cover

Half-title

Title

Copyright

Contents

Preface

1 Introduction to the Shapley value

1 The Shapley value

2 The other papers in this volume

3 Some remarks on the work of Lloyd Shapley

NOTES

REFERENCES

I Ancestral papers

2 A value for n-person games

1 Introduction

2 Definitions

3 Determination of the value function

4 Elementary properties of the value

5 Examples

6 Derivation of the value from a bargaining model

NOTES

REFERENCES

3 A method for evaluating the distribution of power in a committee system

Appendix

NOTES

II Reformulations and generalizations

4 The expected utility of playing a game

1 Introduction

2 Utility theory

3 Comparing positions in games

4 Risk posture

5 Simple games

6 Discussion

NOTES

REFERENCES

5 The Shapley-Shubik and Banzhaf power indices as probabilities

1 The Shapley - Shubik and Banzhaf indices

2 Comparison of the indices

3 A probability model for the power indices

4 Examples and calculations

5 Implications for using the power indices

6 Partial homogeneity assumptions: a political example

7 Other approaches to comparing the power indices

REFERENCES

6 Weighted Shapley values

1 Background and summary

2 Weighted Shapley values

3 Probabilistic definition of weighted Shapley values

4 An axiomatic characterization of the family of weighted Shapley values

5 Duality

6 Other formulas for [phi][sub(w)] and [phi][sub(w)]*

7 Reduction of partnerships and families

REFERENCES

7 Probabilistic values for games

1 Introduction

2 Definitions and notation

3 The linearity and dummy axioms

4 The monotonicity axiom

5 Values for superadditive games

6 Values for simple games

7 Symmetric values

8 Efficiency without symmetry: Random-order values

9 The Shapley value

REFERENCES

8 Combinatorial representations of the Shapley value based on average relative payoffs

Abstract

1 Introduction

2 The main result

3 Extensions

REFERENCES

9 The potential of the Shapley value

1 Introduction

2 The potential

3 Consistency

NOTES

REFERENCES

10 Multilinear extensions of games

1 Definition

2 Interpretation

3 Relation to the power index

4 Applications to measure games

5 Composite games

6 Generalizations

7 Example: The bilateral oligopoly

REFERENCES

III Coalitions

11 Coalitional value

1 Solutions of games with coalition structures

2 The value of a coalition structure

3 Stability of coalition structures

NOTES

REFERENCES

12 Endogenous formation of links between players and of coalitions: an application of the Shapley value

1 Introduction

2 Looking ahead with the Myerson value

3 The formal model

4 An illustration

5 Some weighted majority games

6 A natural structure that is not internally complete

7 Natural structures that depend on the rule of order

8 Discussion

Appendix

NOTES

REFERENCES

IV Large games

13 Values of large finite games

1 Introduction

2 Games

3 Technologies

4 Results

5 Example

NOTES

REFERENCES

14 Payoffs in nonatomic economies: an axiomatic approach

1 Introduction

2 Nonatomic economies with transferable, differentiate utilities

3 Statement of the theorem

NOTES

REFERENCES

15 Values of smooth nonatomic games: the method of multilinear approximation

1 Introduction

2 Preliminaries

3 Multilinear nonatomic games and finite games

4 The approximation theorem

5 Internality

6 The uniqueness of the value

7 The asymptotic value on pNA

8 Characterization of the Aumann-Shapley value without the linearity axiom

9 Characterizations of the value on pNA∞

10 Application to cost allocation

11 Bernstein's polynomials

REFERENCES

16 Nondifferentiable TU markets: the value

Abstract

1 Introduction

2 The model

3 The characteristic function

4 The core

5 The value

Appendix

REFERENCES

V Cost allocation and fair division

17 Individual contribution and just compensation

Abstract

1 Introduction

2 Sharing as a cooperative game

3 Aumann-Shapley pricing

REFERENCES

18 The Aumann-Shapley prices: a survey

1 Introduction

2 Other characterizations of the Aumann-Shapley price mechanism

3 Price mechanisms without the break-even assumption

4 The Aumann-Shapley price mechanism on classes of nondifferentiable cost functions

5 The extension of the Aumann-Shapley prices to cost functions with fixed-cost component

6 The Aumann-Shapley price mechanism as an incentive compatible scheme

7 The Aumann-Shapley price mechanism as a demand-compatible scheme

8 Aumann-Shapley prices and contestable market theory

9 Aumann-Shapley prices and Ramsey prices

REFERENCES

VI NTU games

19 Utility comparison and the theory of games

Appendix: the existence theorem

Discussion

NOTES

20 Paths leading to the Nash set

Abstract

1 Introduction

2 Notation, definitions, and preliminary results

3 The dynamic system

4 Asymptotically stable Nash points

NOTES

REFERENCES