Karlin / Birnbaum Mathematical Methods and Theory in Games, Programming, and Economics

Volume 2: The Theory of Infinite Games
1. Auflage 2014
ISBN: 978-1-4832-2400-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 398 Seiten, Web PDF

Mathematical Methods and Theory in Games, Programming, and Economics, Volume II provides information pertinent to the mathematical theory of games of strategy. This book presents the mathematical tools for manipulating and analyzing large sets of strategies. Organized into nine chapters, this volume begins with an overview of the fundamental concepts in game theory, namely, strategy and pay-off. This text then examines the identification of strategies with points in Euclidean n-space, which is a convenience that simplifies the mathematical analysis. Other chapters provide a discussion of the theory of finite convex games. This book discusses as well the extension of the theory of convex continuous games to generalized convex games, which leads to the characterization that such games possess optimal strategies of finite type. The final chapter deals with the components of a simple two-person poker game. This book is a valuable resource for mathematicians, statisticians, economists, social scientists, and research workers.

Autoren/Hrsg.

Karlin, Samuel

Birnbaum, Z. W.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Methods and Theory in Games, Programming, and Economics;3
3;Copyright Page;5
4;Table of Contents;8
5;CHAPTER 1. THE DEFINITION OF A GAME AND THE MIN-MAX THEOREM;14
5.1;1.1 Introduction. Games in normal form;14
5.2;1.2 Examples;18
5.3;1.3 Choice of strategies;20
5.4;1.4 The min-max theorem for finite matrix games;23
5.5;1.5 General min-max theorem;27
5.6;1.6 Problems;29
5.7;Notes and references;32
6;CHAPTER 2. THE NATURE AND STRUCTURE OF INFINITE GAMES;35
6.1;2.1 Introduction;35
6.2;2.2 Games on the unit square;38
6.3;2.3 Classes of games on the unit square;41
6.4;2.4 Infinite games whose strategy spaces are known function spaces;48
6.5;2.5 How to solve infinite games;52
6.6;2.6 Problems;55
6.7;Notes and references;58
7;CHAPTER 3. SEPARABLE AND POLYNOMIAL GAMES;61
7.1;3.1 General finite convex games;61
7.2;3.2 The fixed-point method for finite convex games;64
7.3;3.3 Dimension relations for solutions of finite convex games;68
7.4;3.4 The method of dual cones;71
7.5;3.5 Structure of solution sets of separable games;74
7.6;3.6 General remarks on convex sets in En;76
7.7;3.7 The reduced moment spaces;79
7.8;3.8 Polynomial games;83
7.9;3.9 Problems;89
7.10;Notes and references;91
8;CHAPTER 4. GAMES WITH CONVEX KERNELS AND GENERALIZED CONVEX KERNELS;93
8.1;4.1 Introduction;93
8.2;4.2 Convex continuous games;94
8.3;4.3 Generalized convex games;97
8.4;4.4 Games with convex pay-off in En;102
8.5;4.5 A theorem on convex functions;108
8.6;4.6 Problems;111
8.7;Notes and references;113
9;CHAPTER 5. GAMES OF TIMING OF ONE ACTION FOR EACH PLAYER;114
9.1;5.1 Examples of games of timing;115
9.2;5.2 The integral equations of games of timing and their solutions;123
9.3;5.3 Integral equations with positive kernels;132
9.4;5.4 Existence proofs;137
9.5;5.5 The silent duel with general accuracy functions;149
9.6;5.6 Problems;153
9.7;Notes and references;156
10;CHAPTER 6. GAMES OF TIMING (CONTINUED);158
10.1;6.1 Games of timing of class I;158
10.2;6.2 Examples;159
10.3;6.3 Proof of Theorem 6.1.1;162
10.4;6.4 Games of timing involving several actions;166
10.5;6.5 Butterfly-shaped kernels;174
10.6;6.6 Problems;184
10.7;Notes and references;186
11;CHAPTER 7. MISCELLANEOUS GAMES;188
11.1;7.1 Games with analytic kernels;188
11.2;7.2 Bell-shaped kernels T;194
11.3;7.3 Bell-shaped games;197
11.4;7.4 Other types of continuous games;204
11.5;7.5 Invariant games;207
11.6;7.6 Problems;212
11.7;Notes and references;217
12;CHAPTER 8. INFINITE CLASSICAL GAMES NOT PLAYED OVER THE UNIT SQUARE ;219
12.1;8.1 Preliminary results (the Neyman-Pearson lemma);220
12.2;8.2 Application of the Neyman-Pearson lemma to a variational problem;223
12.3;8.3 The fighter-bomber duel;227
12.4;8.4 Solution of the fighter-bomber duel;231
12.5;8.5 The two-machine-gun duel;238
12.6;8.6 Problems;246
12.7;Notes and references;250
13;CHAPTER 9. POKE R AND GENERAL PARLOR GAMES;251
13.1;9.1 A simplified blackjack game;254
13.2;9.2 A poker model with one round of betting and one size of bet;258
13.3;9.3 A poker model with several sizes of bet;261
13.4;9.4 Poker model with two rounds of betting;267
13.5;9.5 Poker model with kraises;273
13.6;9.6 Poker with simultaneous moves;276
13.7;9.7 The Le Her Game;279
13.8;9.8 "High hand wins";282
13.9;9.9 Problems;286
13.10;Notes and references;290
14;SOLUTIONS TO PROBLEMS;292
15;APPENDIX A. VECTOR SPACES AND MATRICES;333
15.1;A.1 Euclidean and unitary spaces;333
15.2;A.2 Subspaces, linear independence, basis, direct sums, orthogonal complements;334
15.3;A.3 Linear transformations, matrices, and linear equations;337
15.4;A.4 Eigenvalues, eigenvectors, and the Jordan canonical form;342
15.5;A.5 Transposed, normal, and hermitian matrices; orthogonal complement;349
15.6;A.6 Quadratic form;352
15.7;A.7 Matrix-valued functions;354
15.8;A.8 Determinants; minors, cofactors;356
15.9;A.9 Some identities;359
15.10;A.10 Compound matrices;365
16;APPENDIX B. CONVEX SETS AND CONVEX FUNCTIONS;368
16.1;B.1 Convex sets in En;368
16.2;B.2 Convex hulls of sets and extreme points of convex sets;370
16.3;B.3 Convex cones;373
16.4;B.4 Convex and concave functions;375
17;APPENDIX C. MISCELLANEOUS TOPICS;378
17.1;C.1 Semicontinuous and equicontinuous functions;378
17.2;C.2 Fixed-point theorems;379
17.3;C.3 Set functions and probability distributions;380
18;BIBLIOGRAPHY;384
19;INDEX;398