Книга по ТИ (&

Книга по ТИ (&

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Unformatted text preview: Contents 1 Introduction and Examples 1 1.1 What is Game Theory? . . . . . . . . . . . . . . . . . . . . 1 1.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Noncooperative Games . . . . . . . . . . . . . . . . . 2 1.2.2 Cooperative Games . . . . . . . . . . . . . . . . . . 6 1.3 Short Outline of the Course . . . . . . . . . . . . . . . . . . 11 1.4 Exercises to Chapter 1 . . . . . . . . . . . . . . . . . . . . . 12 I Cooperative Game Theory 15 2 Domination: Stable Sets and the Core 17 2.1 Imputations and Domination . . . . . . . . . . . . . . . . . 17 2.2 The Core and the D-Core . . . . . . . . . . . . . . . . . . . 19 2.3 Simple Games . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Stable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Exercises to Chapter 2 . . . . . . . . . . . . . . . . . . . . . 24 3 Balancedness and the Core 27 3.1 Balanced Games and the Core . . . . . . . . . . . . . . . . 27 3.2 Farkas’ Lemma and the Duality Theorem . . . . . . . . . . 29 3.3 Nonnegative Balanced Games . . . . . . . . . . . . . . . . . 32 3.4 Exercises to Chapter 3 . . . . . . . . . . . . . . . . . . . . . 35 4 The Shapley Value 37 4.1 The Shapley Value . . . . . . . . . . . . . . . . . . . . . . . 37 4.2 Other Characterizations . . . . . . . . . . . . . . . . . . . . 43 4.2.1 Dividends . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.2 Strong Monotonicity . . . . . . . . . . . . . . . . . . 44 4.2.3 Multilinear Extension . . . . . . . . . . . . . . . . . 45 i ii CONTENTS 4.3 Potential and Reduced Game . . . . . . . . . . . . . . . . . 47 4.3.1 The Potential Approach to the Shapley Value . . . . 47 4.3.2 Reduced Games . . . . . . . . . . . . . . . . . . . . 50 4.4 Exercises to Chapter 4 . . . . . . . . . . . . . . . . . . . . . 54 5 Core, Shapley Value and Weber Set 57 5.1 The Weber set . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Convex Games . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Random Order Values . . . . . . . . . . . . . . . . . . . . . 61 5.4 Weighted Shapley Values . . . . . . . . . . . . . . . . . . . 65 5.5 Exercises to Chapter 5 . . . . . . . . . . . . . . . . . . . . . 67 6 Combinatorial and Voting Games 71 6.1 Assignment and Permutation Games . . . . . . . . . . . . . 71 6.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 71 6.1.2 The Birkhoff-von Neumann Theorem . . . . . . . . . 74 6.1.3 Total Balancedness of Permutation Games . . . . . . 76 6.2 Flow Situations and Flow Games . . . . . . . . . . . . . . . 78 6.2.1 Networks and Flows . . . . . . . . . . . . . . . . . . 78 6.2.2 Flow Games . . . . . . . . . . . . . . . . . . . . . . . 81 6.3 Voting Games: The Banzhaf Value . . . . . . . . . . . . . . 83 6.4 Exercises to Chapter 6 . . . . . . . . . . . . . . . . . . . . . 86 7 The (Pre)Nucleolus 91 7.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 7.2 Definitions and Preliminary Results . . . . . . . . . . . . . 93 7.3 The Kohlberg Criterion . . . . . . . . . . . . . . . . . . . .7....
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Книга по ТИ (&

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