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Now in its second edition, this popular textbook on game theory is unrivalled in the breadth of its coverage, the thoroughness of technical explanations and the number of worked examples included. Covering non-cooperative and cooperative games, this introduction to game theory includes advanced chapters on auctions, games with incomplete information, games with vector payoffs, stable matchings and the bargaining set. This edition contains new material on stochastic games, rationalizability, and the continuity of the set of equilibrium points with respect to the data of the game. The material is presented clearly and every concept is illustrated with concrete examples from a range of disciplines. With numerous exercises, and the addition of a solution manual for instructors with this edition, the book is an extensive guide to game theory for undergraduate through graduate courses in economics, mathematics, computer science, engineering and life sciences, and will also serve as useful reference for researchers.

**Author(s):** Michael Maschler, Eilon Solan, Shmuel Zamir

**Edition:** 2

**Publisher:** Cambridge University Press

**Year:** 2020

**Language:** English

**Pages:** 1050

Front Cover

CONTENTS

Acknowledgments

Notations

Introduction

1 The game of chess

1.1 Schematic description of the game

1.2 Analysis and results

1.3 Remarks

1.4 Exercises

2 Utility theory

2.1 Preference relations and their representation

2.2 Preference relations over uncertain outcomes: the model

2.3 The axioms of utility theory

2.4 The characterization theorem for utility functions

2.5 Utility functions and afï¬ne transformations

2.6 Infinite outcome set

2.7 Attitude towards risk

2.8 Subjective probability

2.9 Discussion

2.10 Remarks

2.11 Exercises

3 Extensive-form games

3.1 An example

3.2 Graphs and trees

3.3 Game trees

3.4 Chomp: David Gale's game

3.5 Games with chance moves

3.6 Games with imperfect information

3.7 Exercises

4 Strategic-form games

4.1 Examples and definition of strategic-form games

4.2 The relationship between the extensive form and the strategic form

4.3 Strategic-form games: solution concepts

4.4 Notation

4.5 Domination

4.6 Second-price auctions

4.7 The order of elimination of dominated strategies

4.8 Stability: Nash equilibrium

4.9 Properties of the Nash equilibrium

4.10 Security: the maxmin concept

4.11 The effect of elimination of dominated strategies

4.12 Two-player zero-sum games

4.13 Games with perfect information

4.14 Games on the unit square

4.15 Remarks

4.16 Exercises

5 Mixed strategies

5.1 The mixed extension of a strategic-form game

5.2 Computing equilibria in mixed strategies

5.3 The proof of Nash's Theorem

5.4 Generalizing Nash's Theorem

5.5 Utility theory and mixed strategies

5.6 The maxmin and the minmax in n-player games

5.7 Imperfect information: the value of information

5.8 Rationalizability

5.9 Evolutionarily stable strategies

5.10 The dependence of Nash equilibria on the payoffs of the game

5.11 Remarks

5.12 Exercises

6 Behavior strategies and Kuhn's Theorem

6.1 Behavior strategies

6.2 Kuhn's Theorem

6.3 Equilibria in behavior strategies

6.4 Kuhn's Theorem for infinite games

6.5 Remarks

6.6 Exercises

7 Equilibrium reï¬nements

7.1 Subgame perfect equilibrium

7.2 Rationality, backward induction, and forward induction

7.3 Perfect equilibrium

7.4 Sequential equilibrium

7.5 Remarks

7.6 Exercises

8 Correlated equilibria

8.1 Examples

8.2 Definition and properties of correlated equilibrium

8.3 Remarks

8.4 Exercises

9 Games with incomplete information and common priors

9.1 The Aumann model of incomplete information and the concept of knowledge

9.2 The Aumann model of incomplete information with beliefs

9.3 An infinite set of states of the world

9.4 The Harsanyi model of games with incomplete information

9.5 Incomplete information as a possible interpretation of mixed strategies

9.6 The common prior assumption: inconsistent beliefs

9.7 Remarks

9.8 Exercises

10 Games with incomplete information: the general model

10.1 Belief spaces

10.2 Belief and knowledge

10.3 Examples of belief spaces

10.4 Belief subspaces

10.5 Games with incomplete information

10.6 The concept of consistency

10.7 Remarks

10.8 Exercises

11 The universal belief space

11.1 Belief hierarchies

11.2 Types

11.3 Definition of the universal belief space

11.4 Remarks

11.5 Exercises

12 Auctions

12.1 Notation

12.2 Common auction methods

12.3 Definition of a sealed-bid auction with private values

12.4 Equilibrium

12.5 The symmetric model with independent private values

12.6 The Envelope Theorem

12.7 Risk aversion

12.8 Mechanism design

12.9 Individually rational mechanisms

12.10 Finding the optimal mechanism

12.11 Remarks

12.12 Exercises

13 Repeated games

13.1 The model

13.2 Examples

13.3 The T-stage repeated game

13.4 Characterization of the set of equilibrium payoffs of the T-stage repeated game

13.5 Inï¬nitely repeated games

13.6 The discounted game

13.7 Subgame perfectness

13.8 Uniform equilibrium

13.9 Discussion

13.10 Remarks

13.11 Exercises

14 Repeated games with vector payoffs

14.1 Notation

14.2 The model

14.3 Examples

14.4 Connections between approachable and excludable sets

14.5 A geometric condition for the approachability of a set

14.6 Characterizations of convex approachable sets

14.7 Application 1: Repeated games with incomplete information

14.8 Application 2: Challenge the expert

14.9 Discussion

14.10 Remarks

14.11 Exercises

15 Stochastic games

15.1 The model

15.2 The T-stage stochastic game

15.3 The inïfinite discounted game

15.4 Remarks

15.5 Exercises

16 Bargaining games

16.1 Notation

16.2 The model

16.3 Properties of the Nash solution

16.4 Existence and uniqueness of the Nash solution

16.5 Another characterization of the Nash solution

16.6 The minimality of the properties of the Nash solution

16.7 Critiques of the properties of the Nash solution

16.8 Monotonicity properties

16.9 Bargaining games with more than two players

16.10 Remarks

16.11 Exercises

17 Coalitional games with transferable utility

17.1 Examples

17.2 Strategic equivalence

17.3 A game as a vector in a Euclidean space

17.4 Special families of games

17.5 Solution concepts

17.6 Geometric representation of the set of imputations

17.7 Remarks

17.8 Exercises

18 The core

18.1 Definition of the core

18.2 Balanced collections of coalitions

18.3 The Bondareva-Shapley Theorem

18.4 Market games

18.5 Additive games

18.6 The consistency property of the core

18.7 Convex games

18.8 Spanning tree games

18.9 Flow games

18.10 The core for general coalitional structures

18.11 Remarks

18.12 Exercises

19 The Shapley value

19.1 The Shapley properties

19.2 Solutions satisfying some of the Shapley properties

19.3 The definition and characterization of the Shapley value

19.4 Examples

19.5 An alternative characterization of the Shapley value

19.6 Application: the Shapley-Shubik power index

19.7 Convex games

19.8 The consistency of the Shapley value

19.9 Remarks

19.10 Exercises

20 The bargaining set

20.1 Definition of the bargaining set

20.2 The bargaining set in two-player games

20.3 The bargaining set in three-player games

20.4 The bargaining set in convex games

20.5 Discussion

20.6 Remarks

20.7 Exercises

21 The nucleolus

21.1 Definition of the nucleolus

21.2 Nonemptiness and uniqueness of the nucleolus

21.3 Properties of the nucleolus

21.4 Computing the nucleolus

21.5 Characterizing the prenucleolus

21.6 The consistency of the nucleolus

21.7 Weighted majority games

21.8 The bankruptcy problem

21.9 Discussion

21.10 Remarks

21.11 Exercises

22 Social choice

22.1 Social welfare functions

22.2 Social choice functions

22.3 Non-manipulability

22.4 Discussion

22.5 Remarks

22.6 Exercises

23 Stable matching

23.1 The model

23.2 Existence of stable matching: the men's courtship algorithm

23.3 The women's courtship algorithm

23.4 Comparing matchings

23.5 Extensions

23.6 Remarks

23.7 Exercises

24 Appendices

24.1 Fixed point theorems

24.2 The Separating Hyperplane Theorem

24.3 Linear programming

24.4 Remarks

24.5 Exercises

References

Index

Back Cover

CONTENTS

Acknowledgments

Notations

Introduction

1 The game of chess

1.1 Schematic description of the game

1.2 Analysis and results

1.3 Remarks

1.4 Exercises

2 Utility theory

2.1 Preference relations and their representation

2.2 Preference relations over uncertain outcomes: the model

2.3 The axioms of utility theory

2.4 The characterization theorem for utility functions

2.5 Utility functions and afï¬ne transformations

2.6 Infinite outcome set

2.7 Attitude towards risk

2.8 Subjective probability

2.9 Discussion

2.10 Remarks

2.11 Exercises

3 Extensive-form games

3.1 An example

3.2 Graphs and trees

3.3 Game trees

3.4 Chomp: David Gale's game

3.5 Games with chance moves

3.6 Games with imperfect information

3.7 Exercises

4 Strategic-form games

4.1 Examples and definition of strategic-form games

4.2 The relationship between the extensive form and the strategic form

4.3 Strategic-form games: solution concepts

4.4 Notation

4.5 Domination

4.6 Second-price auctions

4.7 The order of elimination of dominated strategies

4.8 Stability: Nash equilibrium

4.9 Properties of the Nash equilibrium

4.10 Security: the maxmin concept

4.11 The effect of elimination of dominated strategies

4.12 Two-player zero-sum games

4.13 Games with perfect information

4.14 Games on the unit square

4.15 Remarks

4.16 Exercises

5 Mixed strategies

5.1 The mixed extension of a strategic-form game

5.2 Computing equilibria in mixed strategies

5.3 The proof of Nash's Theorem

5.4 Generalizing Nash's Theorem

5.5 Utility theory and mixed strategies

5.6 The maxmin and the minmax in n-player games

5.7 Imperfect information: the value of information

5.8 Rationalizability

5.9 Evolutionarily stable strategies

5.10 The dependence of Nash equilibria on the payoffs of the game

5.11 Remarks

5.12 Exercises

6 Behavior strategies and Kuhn's Theorem

6.1 Behavior strategies

6.2 Kuhn's Theorem

6.3 Equilibria in behavior strategies

6.4 Kuhn's Theorem for infinite games

6.5 Remarks

6.6 Exercises

7 Equilibrium reï¬nements

7.1 Subgame perfect equilibrium

7.2 Rationality, backward induction, and forward induction

7.3 Perfect equilibrium

7.4 Sequential equilibrium

7.5 Remarks

7.6 Exercises

8 Correlated equilibria

8.1 Examples

8.2 Definition and properties of correlated equilibrium

8.3 Remarks

8.4 Exercises

9 Games with incomplete information and common priors

9.1 The Aumann model of incomplete information and the concept of knowledge

9.2 The Aumann model of incomplete information with beliefs

9.3 An infinite set of states of the world

9.4 The Harsanyi model of games with incomplete information

9.5 Incomplete information as a possible interpretation of mixed strategies

9.6 The common prior assumption: inconsistent beliefs

9.7 Remarks

9.8 Exercises

10 Games with incomplete information: the general model

10.1 Belief spaces

10.2 Belief and knowledge

10.3 Examples of belief spaces

10.4 Belief subspaces

10.5 Games with incomplete information

10.6 The concept of consistency

10.7 Remarks

10.8 Exercises

11 The universal belief space

11.1 Belief hierarchies

11.2 Types

11.3 Definition of the universal belief space

11.4 Remarks

11.5 Exercises

12 Auctions

12.1 Notation

12.2 Common auction methods

12.3 Definition of a sealed-bid auction with private values

12.4 Equilibrium

12.5 The symmetric model with independent private values

12.6 The Envelope Theorem

12.7 Risk aversion

12.8 Mechanism design

12.9 Individually rational mechanisms

12.10 Finding the optimal mechanism

12.11 Remarks

12.12 Exercises

13 Repeated games

13.1 The model

13.2 Examples

13.3 The T-stage repeated game

13.4 Characterization of the set of equilibrium payoffs of the T-stage repeated game

13.5 Inï¬nitely repeated games

13.6 The discounted game

13.7 Subgame perfectness

13.8 Uniform equilibrium

13.9 Discussion

13.10 Remarks

13.11 Exercises

14 Repeated games with vector payoffs

14.1 Notation

14.2 The model

14.3 Examples

14.4 Connections between approachable and excludable sets

14.5 A geometric condition for the approachability of a set

14.6 Characterizations of convex approachable sets

14.7 Application 1: Repeated games with incomplete information

14.8 Application 2: Challenge the expert

14.9 Discussion

14.10 Remarks

14.11 Exercises

15 Stochastic games

15.1 The model

15.2 The T-stage stochastic game

15.3 The inïfinite discounted game

15.4 Remarks

15.5 Exercises

16 Bargaining games

16.1 Notation

16.2 The model

16.3 Properties of the Nash solution

16.4 Existence and uniqueness of the Nash solution

16.5 Another characterization of the Nash solution

16.6 The minimality of the properties of the Nash solution

16.7 Critiques of the properties of the Nash solution

16.8 Monotonicity properties

16.9 Bargaining games with more than two players

16.10 Remarks

16.11 Exercises

17 Coalitional games with transferable utility

17.1 Examples

17.2 Strategic equivalence

17.3 A game as a vector in a Euclidean space

17.4 Special families of games

17.5 Solution concepts

17.6 Geometric representation of the set of imputations

17.7 Remarks

17.8 Exercises

18 The core

18.1 Definition of the core

18.2 Balanced collections of coalitions

18.3 The Bondareva-Shapley Theorem

18.4 Market games

18.5 Additive games

18.6 The consistency property of the core

18.7 Convex games

18.8 Spanning tree games

18.9 Flow games

18.10 The core for general coalitional structures

18.11 Remarks

18.12 Exercises

19 The Shapley value

19.1 The Shapley properties

19.2 Solutions satisfying some of the Shapley properties

19.3 The definition and characterization of the Shapley value

19.4 Examples

19.5 An alternative characterization of the Shapley value

19.6 Application: the Shapley-Shubik power index

19.7 Convex games

19.8 The consistency of the Shapley value

19.9 Remarks

19.10 Exercises

20 The bargaining set

20.1 Definition of the bargaining set

20.2 The bargaining set in two-player games

20.3 The bargaining set in three-player games

20.4 The bargaining set in convex games

20.5 Discussion

20.6 Remarks

20.7 Exercises

21 The nucleolus

21.1 Definition of the nucleolus

21.2 Nonemptiness and uniqueness of the nucleolus

21.3 Properties of the nucleolus

21.4 Computing the nucleolus

21.5 Characterizing the prenucleolus

21.6 The consistency of the nucleolus

21.7 Weighted majority games

21.8 The bankruptcy problem

21.9 Discussion

21.10 Remarks

21.11 Exercises

22 Social choice

22.1 Social welfare functions

22.2 Social choice functions

22.3 Non-manipulability

22.4 Discussion

22.5 Remarks

22.6 Exercises

23 Stable matching

23.1 The model

23.2 Existence of stable matching: the men's courtship algorithm

23.3 The women's courtship algorithm

23.4 Comparing matchings

23.5 Extensions

23.6 Remarks

23.7 Exercises

24 Appendices

24.1 Fixed point theorems

24.2 The Separating Hyperplane Theorem

24.3 Linear programming

24.4 Remarks

24.5 Exercises

References

Index

Back Cover