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Game Theory Maschler,Solan,Zamir.pdf

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    Contents
    Acknowledgments
    Notations
    Introduction
    What is game theory?
    How to use this book
    1 The game of chess
    Chapter summary
    1.1 Schematic description of the game
    1.2 Analysis and results
    1.3 Remarks
    1.4 Exercises
    2 Utility theory
    Chapter summary
    2.1 Preference relations and their representation
    2.2 Preference relations over uncertain outcomes
    2.3 The axioms of utility theory
    2.3.1 Continuity
    2.3.2 Monotonicity
    2.3.3 Simplification of lotteries
    2.3.4 Independence
    2.4 The characterization theorem
    2.5 Utility functions and affine transformations
    2.6 Infinite outcome set
    2.7 Attitude towards risk
    2.8 Subjective probability
    2.9 Discussion
    2.9.1 The assumption of completeness
    2.9.2 The assumption of transitivity
    2.9.3 Perceptions of probability
    2.9.4 The Axiom of Simplification
    2.9.5 Other aspects that can influence preferences
    2.10 Remarks
    2.11 Exercises
    3 Extensive-form games
    Chapter summary
    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
    3.6.1 Strategies in games with imperfect information
    3.7 Exercises
    4 Strategic-form games
    Chapter summary
    4.1 Examples and definition of strategic-form games
    4.2 The relationship between extensive and strategic forms
    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
    4.9.1 Stability
    4.9.2 A self-fulfilling agreement
    4.9.3 Equilibrium and evolution
    4.9.4 Equilibrium from the normative perspective
    4.10 Security: the maxmin concept
    4.11 Elimination of dominated strategies
    4.12 Two-player zero-sum games
    4.13 Games with perfect information
    4.14 Games on the unit square
    4.14.1 A two-player zero-sum game on the unit square
    4.14.2 A two-player non-zero-sum game on the unit square
    4.15 Remarks
    4.16 Exercises
    5 Mixed strategies
    Chapter summary
    5.1 The mixed extension of a strategic-form game
    5.2 Computing equilibria in mixed strategies
    5.2.1 The direct approach
    5.2.2 Computing equilibrium points
    5.2.3 The indifference principle
    5.2.4 Dominance and equilibrium
    5.2.5 Two-player zero-sum games and linear programming
    5.2.6 Two-player games that are not zero sum
    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 Evolutionarily stable strategies
    5.9 Remarks
    5.10 Exercises
    6 Behavior strategies and Kuhn's Theorem
    Chapter summary
    6.1 Behavior strategies
    6.2 Kuhn's Theorem
    6.2.1 Conditions for the existence of an equivalent mixed strategy to any behavior strategy
    6.2.2 Representing (x;) as a product of probabilities
    6.2.3 Proof of the second direction of Theorem 6.11: sufficiency
    6.2.4 Conditions guaranteeing the existence of a behavior strategy equivalent to a mixed strategy
    6.3 Equilibria in behavior strategies
    6.4 Kuhn's Theorem for infinite games
    6.4.1 Definitions of pure strategy, mixed strategy, and behavior strategy
    6.4.2 Equivalence between mixed strategies and behavior strategies
    6.4.3 Statement of Kuhn's Theorem for infinite games and its proof
    6.5 Remarks
    6.6 Exercises
    7 Equilibrium refinements
    Chapter summary
    7.1 Subgame perfect equilibrium
    7.2 Rationality, and backward and forward induction
    7.3 Perfect equilibrium
    7.3.1 Perfect equilibrium in strategic-form games
    7.3.2 Perfect equilibrium in extensive-form games
    7.4 Sequential equilibrium
    7.5 Remarks
    7.6 Exercises
    8 Correlated equilibria
    Chapter summary
    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
    Chapter summary
    9.1 The Aumann model and the concept of knowledge
    9.2 The Aumann model with beliefs
    9.3 An infinite set of states of the world
    9.4 The Harsanyi model
    9.4.1 Belief hierarchies
    9.4.2 Strategies and payoffs
    9.4.3 Equilibrium in games with incomplete information
    9.5 A possible interpretation of mixed strategies
    9.6 The common prior assumption
    9.7 Remarks
    9.8 Exercises
    10 Games with incomplete information: the general model
    Chapter summary
    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
    Chapter summary
    11.1 Belief hierarchies
    11.2 Types
    11.3 Definition of the universal belief space
    11.4 Remarks
    11.5 Exercises
    12 Auctions
    Chapter summary
    12.1 Notation
    12.2 Common auction methods
    12.3 Definition of a sealed-bid auction
    12.4 Equilibrium
    12.5 The symmetric model
    12.5.1 Analyzing auctions: an example
    12.5.2 Equilibrium strategies
    12.5.3 The Revenue Equivalence Theorem
    12.5.4 Entry fees
    12.6 The Envelope Theorem
    12.7 Risk aversion
    12.8 Mechanism design
    12.8.1 The revelation principle
    12.8.2 The Revenue Equivalence Theorem
    12.9 Individually rational mechanisms
    12.10 Finding the optimal mechanism
    12.11 Remarks
    12.12 Exercises
    13 Repeated games
    Chapter summary
    13.1 The model
    13.2 Examples
    13.3 The T-stage repeated game
    13.3.1 Histories and strategies
    13.3.2 Payoffs and equilibria
    13.3.3 The minmax value
    13.4 Equilibrium payoffs of the T-stage repeated game
    13.4.1 Proof of the Folk Theorem: example
    13.4.2 Detailed proof of the Folk Theorem
    13.5 Infinitely repeated games
    13.6 The discounted game
    13.7 Uniform equilibrium
    13.8 Discussion
    13.9 Remarks
    13.10 Exercises
    14 Repeated games with vector payoffs
    Chapter summary
    14.1 Notation
    14.2 The model
    14.3 Examples
    14.4 Approachable and excludable sets
    14.5 The approachability of a set
    14.6 Characterizations of convex approachable sets
    14.7 Application 1: Repeated games
    14.8 Application 2: Challenge the expert
    14.8.1 The model
    14.8.2 Existence of a no-regret strategy: a special case
    14.8.3 Existence of a no-regret strategy: the general case
    14.9 Discussion
    14.10 Remarks
    14.11 Exercises
    15 Bargaining games
    Chapter summary
    15.1 Notation
    15.2 The model
    15.3 Properties of the Nash solution
    15.3.1 Symmetry
    15.3.2 Efficiency
    15.3.3 Covariance under positive affine transformations
    15.3.4 Independence of irrelevant alternatives (IIA)
    15.4 Existence and uniqueness of the Nash solution
    15.5 Another characterization of the Nash solution
    15.5.1 Interpersonal comparison of utilities
    15.5.2 The status quo region
    15.6 The minimality of the Nash solution
    15.7 Critiques of the properties of the Nash solution
    15.8 Monotonicity properties
    15.9 Bargaining games with more than two players
    15.10 Remarks
    15.11 Exercises
    16 Coalitional games with transferable utility
    Chapter summary
    16.1 Examples
    16.1.1 Profit games
    16.1.2 Cost games
    16.1.3 Simple games
    16.1.4 Weighted majority games
    16.1.5 Market games
    16.1.6 Sequencing games
    16.1.7 Spanning tree games
    16.1.8 Cost-sharing games
    16.2 Strategic equivalence
    16.3 A game as a vector in a Euclidean space
    16.4 Special families of games
    16.5 Solution concepts
    16.6 Geometric representation of the set of imputations
    16.7 Remarks
    16.8 Exercises
    17 The core
    Chapter summary
    17.1 Definition of the core
    17.2 Balanced collections of coalitions
    17.3 The Bondareva--Shapley Theorem
    17.3.1 The Bondareva--Shapley condition is a necessary condition for the nonemptiness of the core
    17.3.2 The Bondareva--Shapley condition is a sufficient condition for the nonemptiness of the core
    17.3.3 A proof of the Bondareva--Shapley Theorem using linear programming
    17.4 Market games
    17.4.1 The balanced cover of a coalitional game
    17.4.2 Every totally balanced game is a market game
    17.5 Additive games
    17.6 The consistency property of the core
    17.7 Convex games
    17.8 Spanning tree games
    17.9 Flow games
    17.10 The core for general coalitional structures
    17.11 Remarks
    17.12 Exercises
    18 The Shapley value
    Chapter summary
    18.1 The Shapley properties
    18.1.1 Efficiency
    18.1.2 Symmetry
    18.1.3 Covariance under strategic equivalence
    18.1.4 The null player property
    18.1.5 Additivity property
    18.2 Solutions of some of the Shapley properties
    18.3 The definition of the Shapley value
    18.4 Examples
    18.5 An alternative characterization
    18.5.1 The marginality property
    18.5.2 The second characterization of the Shapley value
    18.6 Application: the Shapley-Shubik power index
    18.6.1 The power index of the United Nations Security Council
    18.7 Convex games
    18.8 The consistency of the Shapley value
    18.9 Remarks
    18.10 Exercises
    19 The bargaining set
    Chapter summary
    19.1 Definition of the bargaining set
    19.2 The bargaining set in two-player games
    19.3 The bargaining set in three-player games
    19.4 The bargaining set in convex games
    19.5 Discussion
    19.6 Remarks
    19.7 Exercises
    20 The nucleolus
    Chapter summary
    20.1 Definition of the nucleolus
    20.2 Nonemptiness and uniqueness of the nucleolus
    20.3 Properties of the nucleolus
    20.4 Computing the nucleolus
    20.5 Characterizing the prenucleolus
    20.6 The consistency of the nucleolus
    20.7 Weighted majority games
    20.8 The bankruptcy problem
    20.8.1 The model
    20.8.2 The case n=2
    20.8.3 The case n>2
    20.8.4 The nucleolus of a bankruptcy problem
    20.9 Discussion
    20.10 Remarks
    20.11 Exercises
    21 Social choice
    Chapter summary
    21.1 Social welfare functions
    21.2 Social choice functions
    21.3 Non-manipulability
    21.4 Discussion
    21.5 Remarks
    21.6 Exercises
    22 Stable matching
    Chapter summary
    22.1 The model
    22.2 The men's courtship algorithm
    22.3 The women's courtship algorithm
    22.4 Comparing matchings
    22.4.1 The lattice structure of the set of stable matchings
    22.5 Extensions
    22.5.1 When the number of men does not equal the number of women
    22.5.2 Strategic considerations
    22.5.3 The desire to remain single, or: getting married, but not at any price
    22.5.4 Polygamous matching: placement of students in universities
    22.5.5 Unisexual matchings
    22.6 Remarks
    22.7 Exercises
    23 Appendices
    Chapter summary
    23.1 Fixed point theorems
    23.1.1 Sperner's Lemma
    23.1.2 Brouwer's Fixed Point Theorem
    23.1.3 Kakutani's Fixed Point Theorem
    23.1.4 The KKM Theorem
    23.2 The Separating Hyperplane
    23.3 Linear programming
    23.4 Remarks
    23.5 Exercises
    References
    Index
    more information - www.cambridge.org/9781107005488
    Game Theory Covering both noncooperative and cooperative games, this comprehensive introduction to game theory also includes some advanced chapters on auctions, games with incomplete information, games with vector payoffs, stable matchings, and the bargaining set. Mathematically oriented, the book presents every theorem alongside a proof. The material is presented clearly and every concept is illustrated with concrete examples from a broad range of disciplines. With numerous exercises the book is a thorough and extensive guide to game theory from undergraduate through graduate courses in economics, mathematics, computer science, engineering, and life sciences to being an authoritative reference for researchers. Michael Maschler was a professor in the Einstein Institute of Mathematics and the Center for the Study of Rationality at the Hebrew University of Jerusalem in Israel. He greatly contributed to cooperative game theory and to repeated games with incomplete information. Eilon Solan is a professor in the School of Mathematical Sciences at Tel Aviv University in Israel. The main topic of his research is repeated games. He serves on the editorial board of several academic journals. Shmuel Zamir is a professor emeritus in the Department of Statistics and the Center for the Study of Rationality at the Hebrew University of Jerusalem in Israel. The main topics of his research are games with incomplete information and auction theory. He is the editor-in-chief of the International Journal of Game Theory.
    Game Theory MICHAEL MASCHLER EILON SOLAN SHMUEL ZAMIR Translated from Hebrew by Ziv Hellman English Editor Mike Borns
    cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107005488 C The Estate of the late Michael Maschler, Eilon Solan and Shmuel Zamir 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2013 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Zamir, Shmuel. [Torat ha-mishakim. English] Game theory / Michael Maschler, Eilon Solan, Shmuel Zamir ; translated from Hebrew by Ziv Hellman ; English editor, Mike Borns. pages cm Translation of: Torat ha-mishakim / Shemu’el Zamir, Mikha’el Mashler ve-Elon Solan. Includes bibliographical references and index. ISBN 978-1-107-00548-8 (hardback) 1. Game theory. QA269.Z3613 2013 519.3 – dc23 I. Maschler, Michael, 1927–2008. 2012050827 II. Solan, Eilon. III. Title. ISBN 978-1-107-00548-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
    To Michael Maschler
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