Discrete gambling and stochastic games
The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of chance. However, it was not until the middle of the twentieth century that mathematicians developed general techniques for maximizing the chances of beating a casino or winni...
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| Format: | Buch |
| Sprache: | German |
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New York [u.a.]
Springer
1996
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| Schriftenreihe: | Applications of mathematics
32 |
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| Online-Zugang: | Inhaltsverzeichnis |
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| 100 | 1 | |a Maitra, Ashok P. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Discrete gambling and stochastic games |c Ashok P. Maitra ; William D. Sudderth |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1996 | |
| 300 | |a XI, 244 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applications of mathematics |v 32 | |
| 500 | |a Literaturverz. S. 228 - 237 | ||
| 520 | 3 | |a The theory of probability began in the seventeenth century with attempts to calculate the odds of winning in certain games of chance. However, it was not until the middle of the twentieth century that mathematicians developed general techniques for maximizing the chances of beating a casino or winning against an intelligent opponent. These methods of finding optimal strategies are at the heart of the modern theory of stochastic control and stochastic games | |
| 520 | |a This monograph provides an introduction to the ideas of gambling theory and stochastic games. The first chapters introduce the ideas and notation of gambling theory. Chapters 3 and 4 consider "leavable" and "nonleavable" problems that form the core theory of this subject. Chapters 5, 6, and 7 cover stationary strategies, approximation results, and two-person zero-sum stochastic games, respectively. Throughout, the authors have included examples, and there are problem sets at the end of each chapter | ||
| 650 | 7 | |a Jeux de hasard (Mathématiques) |2 ram | |
| 650 | 7 | |a Speltheorie |2 gtt | |
| 650 | 7 | |a Stochastische processen |2 gtt | |
| 650 | 4 | |a Gambling | |
| 650 | 4 | |a Games of chance (Mathematics) | |
| 650 | 4 | |a Stochastic inequalities | |
| 650 | 0 | 7 | |a Stochastisches Spiel |0 (DE-588)4129289-3 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Optimales Stoppen |0 (DE-588)4230259-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Spieltheorie |0 (DE-588)4056243-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diskreter stochastischer Prozess |0 (DE-588)4150187-1 |2 gnd |9 rswk-swf |
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| 689 | 1 | 2 | |a Spieltheorie |0 (DE-588)4056243-8 |D s |
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| 700 | 1 | |a Sudderth, William D. |e Verfasser |4 aut | |
| 830 | 0 | |a Applications of mathematics |v 32 |w (DE-604)BV000895226 |9 32 | |
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Datensatz im Suchindex
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|---|---|
| DE-BY-TUM_katkey | 777279 |
| DE-BY-TUM_location | Mag |
| DE-BY-TUM_media_number | 040001901691 |
| _version_ | 1820804925093838848 |
| adam_text | Contents
1 Introduction 1
1.1 Preview 2
1.2 Prerequisites 2
1.3 Numbering 2
2 Gambling Houses and the Conservation of Fairness 5
2.1 Introduction 5
2.2 Gambles, Gambling Houses, and Strategies 6
2.3 Stopping Times and Stop Rules 9
2.4 An Optional Sampling Theorem 11
2.5 Martingale Convergence Theorems 15
2.6 The Ordinals and Transfinite Induction 15
2.7 Uncountable State Spaces and Continuous Time 18
2.8 Problems for Chapter 2 19
3 Leavable Gambling Problems 23
3.1 The Fundamental Theorem 24
3.2 The One Day Operator and the Optimality Equation .... 26
3.3 The Utility of a Strategy 27
3.4 Some Examples 30
3.5 Optimal Strategies 42
3.6 Backward Induction: An Algorithm for U 48
3.7 Problems for Chapter 3 52
x Contents
4 Nonleavable Gambling Problems 59
4.1 Introduction 59
4.2 Understanding u(cr) 60
4.3 A Characterization of V 68
4.4 The Optimality Equation for V 69
4.5 Proving Optimality 70
4.6 Some Examples 70
4.7 Optimal Strategies 75
4.8 Another Characterization of V 78
4.9 An Algorithm for V 82
4.10 Problems for Chapter 4 84
5 Stationary Families of Strategies 89
5.1 Introduction 89
5.2 Comparing Strategies 90
5.3 Finite Gambling Problems 94
5.4 Nonnegative Stop or Go Problems 96
5.5 Leavable Houses 101
5.6 An Example of Blackwell and Ramakrishnan 106
5.7 Markov Families of Strategies 109
5.8 Stationary Plans in Dynamic Programming 109
5.9 Problems for Chapter 5 110
6 Approximation Theorems 113
6.1 Introduction 113
6.2 Analytic Sets 114
6.3 Optimality Equations 124
6.4 Special Cases of Theorem 1.2 128
6.5 The Going Up Property of M 139
6.6 Dynamic Capacities and the Proof of Theorem 1.2 144
6.7 Approximating Functions 150
6.8 Composition Closure and Saturated House 158
6.9 Problems for Chapter 6 165
7 Stochastic Games 171
7.1 Introduction 171
7.2 Two Person, Zero Sum Games 172
7.3 The Dynamics of Stochastic Games 176
7.4 Stochastic Games with lim sup Payoff 179
7.5 Other Payoff Functions 180
7.6 The One Day Operator 181
7.7 Leavable Games 184
7.8 Families of Optimal Strategies for Leavable Games 189
7.9 Examples of Leavable Games 191
7.10 A Modification of Leavable Games and the Operator T . . . 196
Contents xi
7.11 An Algorithm for the Value of a Nonleavable Game 198
7.12 The Optimality Equation for V 201
7.13 Good Strategies in Nonleavable Games 203
7.14 Win, Lose, or Draw 207
7.15 Recursive Matrix Games 210
7.16 Games of Survival 212
7.17 The Big Match 216
7.18 Problems for Chapter 7 221
References 227
Symbol Index 239
Index 241
|
| any_adam_object | 1 |
| author | Maitra, Ashok P. Sudderth, William D. |
| author_facet | Maitra, Ashok P. Sudderth, William D. |
| author_role | aut aut |
| author_sort | Maitra, Ashok P. |
| author_variant | a p m ap apm w d s wd wds |
| building | Verbundindex |
| bvnumber | BV010840597 |
| callnumber-first | Q - Science |
| callnumber-label | QA271 |
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| callnumber-subject | QA - Mathematics |
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| ctrlnum | (OCoLC)33276855 (DE-599)BVBBV010840597 |
| dewey-full | 795/.01/5193 |
| dewey-hundreds | 700 - The arts |
| dewey-ones | 795 - Games of chance |
| dewey-raw | 795/.01/5193 |
| dewey-search | 795/.01/5193 |
| dewey-sort | 3795 11 45193 |
| dewey-tens | 790 - Recreational and performing arts |
| discipline | Sport Mathematik |
| format | Book |
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| id | DE-604.BV010840597 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-23T14:13:59Z |
| institution | BVB |
| isbn | 0387946284 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007246671 |
| oclc_num | 33276855 |
| open_access_boolean | |
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| owner_facet | DE-12 DE-739 DE-91 DE-BY-TUM DE-703 DE-824 DE-634 DE-83 DE-29T |
| physical | XI, 244 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Springer |
| record_format | marc |
| series | Applications of mathematics |
| series2 | Applications of mathematics |
| spellingShingle | Maitra, Ashok P. Sudderth, William D. Discrete gambling and stochastic games Applications of mathematics Jeux de hasard (Mathématiques) ram Speltheorie gtt Stochastische processen gtt Gambling Games of chance (Mathematics) Stochastic inequalities Stochastisches Spiel (DE-588)4129289-3 gnd Optimales Stoppen (DE-588)4230259-6 gnd Spieltheorie (DE-588)4056243-8 gnd Diskreter stochastischer Prozess (DE-588)4150187-1 gnd |
| subject_GND | (DE-588)4129289-3 (DE-588)4230259-6 (DE-588)4056243-8 (DE-588)4150187-1 |
| title | Discrete gambling and stochastic games |
| title_auth | Discrete gambling and stochastic games |
| title_exact_search | Discrete gambling and stochastic games |
| title_full | Discrete gambling and stochastic games Ashok P. Maitra ; William D. Sudderth |
| title_fullStr | Discrete gambling and stochastic games Ashok P. Maitra ; William D. Sudderth |
| title_full_unstemmed | Discrete gambling and stochastic games Ashok P. Maitra ; William D. Sudderth |
| title_short | Discrete gambling and stochastic games |
| title_sort | discrete gambling and stochastic games |
| topic | Jeux de hasard (Mathématiques) ram Speltheorie gtt Stochastische processen gtt Gambling Games of chance (Mathematics) Stochastic inequalities Stochastisches Spiel (DE-588)4129289-3 gnd Optimales Stoppen (DE-588)4230259-6 gnd Spieltheorie (DE-588)4056243-8 gnd Diskreter stochastischer Prozess (DE-588)4150187-1 gnd |
| topic_facet | Jeux de hasard (Mathématiques) Speltheorie Stochastische processen Gambling Games of chance (Mathematics) Stochastic inequalities Stochastisches Spiel Optimales Stoppen Spieltheorie Diskreter stochastischer Prozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007246671&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000895226 |
| work_keys_str_mv | AT maitraashokp discretegamblingandstochasticgames AT sudderthwilliamd discretegamblingandstochasticgames |