How to Gamble If You Must: Inequalities for Stochastic Processes

How to Gamble If You Must

Inequalities for Stochastic Processes

Lester E. Dubins

Leonard J. Savage

Edited and Updated by

William D. Sudderth

School of Statistics

University of Minnesota

David Gilat

Department of Statistics and Operations Research

School of Mathematical Sciences

Tel Aviv University

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 1965 by Lester E. Dubins and Leonard J. Savage

Copyright © 1976 by Lester E. Dubins

Copyright © 2014 by Dover Publications, Inc.

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2014, is an unabridged republication of the corrected 1976 Dover edition entitled Inequalities for Stochastic Processes which contained a new Bibliographic Supplement and a new Preface by Lester E. Dubins. The book was originally published as How to Gamble If You Must: Inequalities for Stochastic Processes in 1965 by the McGraw-Hill Book Company, Inc., New York. This 2014 edition contains a new Preface, an additional bibliographic supplement, Remarks on Open Problems, an Appendix on Finite Additivity, and an About the Authors page. All new material was provided by William Sudderth and David Gilat.

Library of Congress Cataloging-in-Publication Data

Dubins, Lester E., 1920–

How to gamble if you must: inequalities for stochastic processes / Lester E. Dubins and Leonard J. Savage; edited with a new preface by William D. Sudderth and David Gilat.—Dover edition.

      p. cm.

Originally published in 1965 as: How to gamble if you must. Republished in 1976 as: Inequalities for stochastic processes.

Includes bibliographical references and index.

eISBN-13: 978-0-486-79608-6

   1. Games of chance (Mathematics) 2. Gambling. 3. Stochastic inequalities. I. Savage, Leonard J. II. Sudderth, William D. III. Gilat, David (Mathematics professor) IV. Dubins, Lester E., 1920– Inequalities for stochastic processes. V Title. VI. Title: Inequalities for stochastic processes.

QA273.D77 2014

519.2′3—dc23

2014008632

Manufactured in the United States by Courier Corporation

78064301 2014

www.doverpublications.com

TO OUR FATHERS

About the Authors

Lester Eli Dubins

(April 27, 1920 – February 11, 2010)

Born and raised in New York. His college education was interrupted by World War II, in which he served as an Air Force officer stationed in Iceland. After the war he continued to work for several years as a civilian on radar research and development for the Air Force, but then resumed his studies at the University of Chicago as a graduate student of mathematics, obtaining his Ph.D in 1955 under the guidance of Irving Segal. After a year at the Institute for Advanced Study in Princeton and a few years at the Carnegie Institute of Technology in Pittsburgh, he joined the University of California at Berkeley in 1960, where he remained for the rest of his career as Professor of Mathematics and Statistics. Although noted primarily for his research in probability theory, scattered among his nearly one hundred scholarly publications are numerous papers on a wide range of mathematical themes, such as curves of minimal length under constraints on curvature and initial and final tangents (to be known as Dubin paths), Tarski circle squaring problem, convex analysis, and a variety of topics in topology and geometry.

Leonard Jimmie Savage

(November 20, 1917 – November 1, 1971)

Born and raised in Detroit. His severely poor eyesight from early childhood did not deter him from rising to substantial intellectual and scientific accomplishments. After obtaining a Ph.D degree in Mathematics from the University of Michigan in 1941, he spent a year at the Institute for Advanced Study in Princeton solving a problem in the Calculus of Variations suggested to him by John von Neumann. At von Neumann’s recommendation, Savage joined the Statistical Research Group at Columbia University in New York, engaged in those days to work for the war effort. After the war, in 1946, he joined the University of Chicago and in 1949 became one of the founders of its Statistics Department, serving as chairman of the department from 1956 to 1959. His most noted (and controversial) work was the 1954 book Foundations of Statistics which lays out his theory of utility, subjective probability, and Bayesian statistics. His staunch insistence on Bayesianism as the only sound approach to statistics alienated him from large circles in the statistical community. In 1960 he left Chicago to take up a professorship at the University of Michigan where he remained for four years before moving to Yale University to become the Eugene Higgins Professor of Statistics. Sadly, he had only seven years at Yale before his sudden premature death.

The collaboration of the authors which led to this book is presented in the preface to the first Dover edition.

Preface to the Second Dover Edition

The previous Dover edition appeared in 1976, five years after the untimely death of Leonard J. Savage at the age of 54. In his impressive preface to the 1976 edition of the book, Lester Dubins pays tribute to his friend Jimmie Savage - to his remarkable intellect and scholarship as well as to their mutual friendship and mathematical collaboration. Having also praised Savage’s outstanding writing skills, Dubins - alluding to the bibliographic supplement appended to the new edition - concludes his preface by saying: .. reading some of the references will, I hope, enable the reader to visualize, at least in outline form, the kind of revision that might be possible were there alive my much missed friend, Leonard Jimmie Savage.

Lester Eli Dubins died in 2010 and it was his wish, stated explicitly in his will, that in any further edition of the book, the text - as originally composed by him and Savage - be left intact. In adherence to this wish, the present edition is identical with the preceding one, except for the following, hopefully useful, additions:

1. A Bibliographic Supplement (2014) listing, to the best of our ability, the relevant literature that has appeared during the 38 years since the publication of the previous edition.

2. Remarks on Open Problems: Dubins and Savage leave many open problems scattered throughout the book which they list at the end. Some of these problems have been partially or completely solved over the years. We briefly discuss some of the solutions and in each case point to the relevant references in the bibliographic supplement.

3. Appendix on Finite Additivity. Readers who are used to the conventional, countably additive approach to probability may have difficulty with the (more general) finitely additive framework used in the book. The main purpose of this appendix is to provide a simple introduction to the finitely additive probability theory needed to understand the statements and proofs in the book. The appendix was written in collaboration with Roger Purves.

We feel that the title, How to Gamble if You Must, of the original 1965 edition of the book, better reflects its content and its authors’ intention than the subtitle Inequalities for Stochastic Processes. The order of the titles is therefore restored to its original.

William D. Sudderth, University of Minnesota

David Gilat, Tel Aviv University

IT IS ALMOST ALWAYS GAMBLING THAT ENABLES ONE TO FORM A FAIRLY CLEAR IDEA OF A MANIFESTATION OF CHANCE; IT IS GAMBLING THAT GAVE BIRTH TO THE CALCULUS OF PROBABILITY; IT IS TO GAMBLING THAT THIS CALCULUS OWES ITS FIRST FALTERING UTTERANCES AND ITS MOST RECENT DEVELOPMENTS; IT IS GAMBLING THAT ENABLES US TO CONCEIVE OF THIS CALCULUS IN THE MOST GENERAL WAY; IT IS, THEREFORE, GAMBLING THAT ONE MUST STRIVE TO UNDERSTAND, BUT ONE SHOULD UNDERSTAND IT IN A PHILOSOPHIC SENSE, FREE FROM ALL VULGAR IDEAS.

LOUIS BACHELIER

PREFACE TO THE DOVER EDITION

The occasion of this edition provides an opportunity to honor the memory of Leonard J. Savage, who suddenly died of a heart attack on November 1, 1971.

Much will certainly be written about Jimmie Savage as a man and about Jimmie Savage as a scientist. In this preface, it seems appropriate to confine myself to a few of my many recollections of our collaboration in the writing of this book.

Jimmie had already made a wonderful impression on me when I was a student at the University of Chicago in the early 1950’s. He was different from the other mathematicians I had heretofore known. He often tested his understanding of the most abstract mathematics with colorful, concrete examples that seemed to leap instantly and effortlessly into his mind. He was often searching for answers, and he shared his thought processes with you. In this way, you were his equal, for the search was a joint venture—you searched too. But one could not fail to be impressed with his quickness of mind as well as with the breadth and depth of his knowledge.

When in June of 1956 I returned to Chicago after a year’s absence, Jimmie amazed me utterly by telling me that he had been unable to settle the problem of how to play Red-and-Black wisely. How could so simple-sounding a problem escape his (and others’) immediate solution, I wondered. So I began to think about this enticing problem. Little did I then anticipate the delightful conversations and correspondence that were to develop from that beginning, nor did I then realize the influence that it was destined to have on my own career. But to return to June 1956—

Within a few days of Jimmie’s telling me of the problem of Red-and-Black, I happened to make this trivial observation: if bold play is optimal, it is not uniquely so. If I was amazed at learning that the problem of Red-and-Black was open, I was even more so at Jimmie’s reaction to my all but immediate observation. He was so incredulous that he checked, and double-checked—and even at the blackboard! He made me feel good about my tiny observation by saying that he had always taken the contrary for granted. It was his manifest appreciation of that infinitesimal advance that gave me the courage to accept his invitation that I work with him. His ability to recognize and to appreciate genuinely even the tiniest of contributions greatly enhanced my pleasure in working with him during the succeeding months and years.

By the end of the summer of 1956, Red-and-Black was no longer an enigma to us. We had shown that bold play was optimal (Section 5.11). What launched this book, however, was our mistaken confidence that we could, without difficulty, extend our understanding to the more general problem of primitive casinos (Chapter 6). A plausible conjecture was that bold play was again optimal (Section 6.1). So I (and perhaps Jimmie, too) was shocked when, after we had spent more than two years in seeking a proof, Aryeh Dvoretsky discovered that bold play was not optimal (Section 6.6), at least if playing time is limited. Our—or at least my—confidence in the conjecture shaken, we nevertheless persevered and, within a few additional months, settled a combinatorial inequality that had stumped us for more than two years, and which was all that had remained to complete the proof that bold play was indeed optimal for subfair, primitive casinos. This was the now innocent-looking inequality of Section 6.3 which asserts that certain sequences M are Cartesian.

Another concrete problem that we found easier to pose than to settle was to determine those casinos that are, in a certain sense, fair (Section 10.3). We made some progress, were pleased by certain mildly surprising observations, and then, were much surprised by a discovery of Donald Ornstein (Section 10.2). The main problem of giving necessary and sufficient conditions for a casino to be fair was finally solved, mainly by the prodigious efforts of Jimmie during the year 1958–59, which he spent in Rome.

At what point did we break with tradition and develop the theory without the countably additive hypothesis? It happened essentially thus. In our work on special problems such as Red-and-Black, we had, like many others before us (and I mention in particular Snell, 1952), used the idea of conservation of fairness, which in this book finds important expression in the basic Theorem 2.12.1. But it was not until I had heard Aryeh Dvoretsky speak about the problem of bold play for primitive casinos (with limited playing time) that I began to appreciate the power and generality of that idea. I immediately saw the possibilities of a general theorem, and established one, but one which applied only to countably additive problems, and not even to all such. What was most mystifying and puzzling, however, was that it should require, as it did, a preliminary technical result of moderate difficulty, namely, a measurable selection theorem. It was at this point that Jimmie had the daring idea, influenced no doubt by his association with Bruno de Finetti, that we aim for a general, finitely additive theory. Once Jimmie, with some assistance from de Finetti, had persuaded me that we were not embarking on totally unjustifiable heresy, we proceeded to develop the appropriate mathematics. The main difficulty which we encountered was to determine the proper class of functions which were to be integrated by strategies, which class turned out to be the inductively integrable functions (Theorem 2.8.1). Once that was accomplished, the fundamental ingredients to a theory of gambling were quickly developed, in particular, the basic Theorem 2.12.1 and its companion, Theorem 2.14.1. The finitely additive approach made it possible to understand the true, and far less central, role of the measurable selection theorem, which theorem now appears later in Section 2.16.

An abstract problem that we raised but did not settle was the extent to which stationary families of strategies are adequate. For us, it was already a challenge to settle this for various special cases, including that in which the fortune space, F, is finite (Section 3.9). Many further developments of this problem of stationarity have taken place, as a number of entries in the Bibliographic Supplement indicate, the most surprising of which is due to Ornstein (1969).

It is almost no exaggeration to say that working with Jimmie, whether face to face or by correspondence, was always a joy for me. His letters were a delight. I chuckled over his humor and enjoyed his insights. Our work was accomplished mainly through correspondence punctuated by periods together. The letters grew into manuscripts which invariably went through several editions. The final drafts that Jimmie would write were, for me, a thing of beauty. At least once, I undertook to write the final draft of something we had done. And I took great pains, much thought and much time, until I was totally satisfied that it could not be improved upon, not even by Jimmie. But after Jimmie read my draft and then wrote his (I do not say rewrote, for there was barely any resemblance), my chagrin gradually changed to delight and admiration at his achievements, both mathematical and expository.

More than once, Jimmie told me that he considered our work to be an important contribution, and that it would be so judged by our peers. Perhaps that is why the constant flow of calls upon his time and talents did not prevent him from pouring into our joint work so much of his energy and genius. Working thus with Jimmie for more than three years, it would have been all but impossible for anyone not to produce something of interest.

So, upon coming to Berkeley in 1959—three years and three months after Jimmie had introduced me to the problem of Red-and-Black—I was eager to communicate our results and methods, especially those mentioned above. It was a pleasure to lecture about our work, and to have an interested audience, especially in the person of David Blackwell, who suggested that we publish our findings as a book. At every stage in the writing of this book, Jimmie was an artist interested in the quality of every aspect of the book, whether it be the elegance of an argument or the appearance of a page. Though his example was inimitable, it could not fail to make his coauthor a better collaborator. Our efforts were fully rewarded with the sense of completion which came with the circulation of the mimeographed edition, dated August 28, 1960.

Each of the subsequent editions, whether mimeographed or printed, and this one, in particular, differs but little from the preceding. In order to alleviate the present publisher’s concern about possible misunderstandings as to the nature of the book, its title and subtitle have been permuted for this edition. The score or so of minor mechanical errors that miraculously appeared in the last edition—and which were brought to light mainly by the participants in a seminar led by Richard Olshen and Jimmie Savage at Yale in 1967 have been corrected, of course. More important, this edition contains a Bibliographic Supplement. The inclusion of this Supplement, unannotated as it is, can merely call attention to recent advances. But reading some of the references will, I hope, enable the reader to visualize, at least in outline form, the kind of revision that might be possible were there alive today my much missed friend, Leonard Jimmie Savage.

L. E. DUBINS

University of California

Berkeley, California

August 1975

PREFACE TO THE FIRST EDITION

The immediate object of this book, a chapter in the pure mathematics of probability, is to explore and to share with other mathematicians certain problems that have excited our curiosity and that lead to somewhat novel methods and phenomena. A light introduction to these problems is in Chapter 1.

In spite of the probabilist’s tendency to invoke gambling imagery, works on probability have little influence, for good or bad, on the gambling man. But probability is not behind other parts of mathematics in its influence on practical affairs, and this book about theoretical problems of how to make the best of a bad and risky situation also may eventually lead to applications.

We have tried to write the book for mathematicians generally. Technical knowledge of probability is not a logical prerequisite, except for isolated passages, but it will ordinarily be a psychological one. No one may be disposed to read the whole book through, so we have tried to help skippers and postponers by providing them with signposts.

We began to ask each other how to play red-and-black in the summer of 1956, and this report of our research—in our minds first a note, then a paper, and at last a book—has been nearly finished ever since. Six written preliminary announcements are (Savage 1957), (Dubins and Savage 1960, 1960a), (Savage 1962), and (Dubins and Savage 1963, 1965). We also gave talks about the research, particularly at the following meetings: the Institute of Mathematical Statistics at Atlantic City in September, 1957, and at Albuquerque in April, 1962; the American Mathematical Society at Chicago in January, 1960.

The flexible system of making references to the Bibliography by such expressions as a book by Doob (1953), (Doob 1953), or (Doob 1953, p. 67) is familiar and almost self-explanatory. Other mechanical matters should also be unobtrusive, but some points may be mentioned.

Technical terms are defined with the aid of italics, which usually provide the only signal that a sentence is a definition. The number of technical terms and notations introduced is large in spite of economy drives, but only a few are in use in any one section and many are shortlived. The Subject Index and Index of Symbols will lead promptly to a definition that has been forgotten or skipped.

To illustrate the not unusual system of internal references, "Theorem 2.12.1 denotes the first theorem of Section 2.12, that is, of Section 12 of Chapter 2. Within Section 2.12, it is called simply Theorem 1, and elsewhere within Chapter 2, it is called Theorem 12.1. Similarly, (2.12.1) is the full name of the first displayed formula in Section 2.12; the shorter names (1) and (12.1)" are used within Section 12.1 and elsewhere in Chapter 2, respectively. This system is facilitated by showing a section number at the top of almost every pair of pages.

The mark signifies the end of a proof.

LESTER E. DUBINS

LEONARD J. SAVAGE

ACKNOWLEDGMENTS

We have benefited from conversations with many friends, whose specific contributions will be mentioned in context. The help of David Blackwell, Gerard Debreu, Bruno de Finetti, Aryeh Dvoretzky, Paul R. Halmos, Lawrence Jackson, Jesse Marcum, John Myhill, Donald Ornstein, Roger A. Purves, and W. Forrest Stinespring was particularly valuable and encouraging.

It was David Blackwell who suggested that we put our work on gambling into the form of a book. His counsel and suggestions have contributed much, and his continued interest in the project has kept it fresh and rewarding for us.

Energetic criticism by the students in a seminar based on a preliminary version has made this book better. For this, we particularly thank Vida Greenberg, Martin Helling, Lawrence Jackson, Michel Jean, and Roger A. Purves. Criticism of a late version by Bruno de Finetti, Aryeh Dvoretzky, and Thomas Ferguson initiated many improvements.

Countless sentences are clearer and more graceful, thanks to Louise Forsyth’s suggestions.

This book entailed an enormous amount of loyal and skilled secretarial work, and we are grateful to all who participated in the many drafts. The final and most demanding one was prepared by Mrs. Jaynel E. Moore.

Dubins has worked on this book at the University of Chicago, the Carnegie Institute of Technology, the Institute for Advanced Study, the University of Michigan, and the University of California at Berkeley; Savage has worked on it at the University of Chicago, the University of Michigan, Yale University, the University of California at Berkeley, and at the RAND Corporation.

Our work was given generous financial support by several agencies: the Air Force Office of Scientific Research and the National Science Foundation (through a grant and through its Regular Postdoctoral Fellowship program) for Dubins’ participation; the Office of Naval Research, the Air Force Office of Scientific Research, the National Science Foundation, the John Simon Guggenheim Memorial Foundation, and the Michigan Institute of Science and Technology for Savage’s. A grant from the Ford Foundation to Savage has defrayed travel and certain other expenses, thus making it more practical for us to work together.

The passage on the quotation page is from (Bachelier 1914, page 6), with the permission of the publisher, Flammarion et Cie.

CONTENTS

1.   INTRODUCTION

1.   THE PROBLEM

2.   PREVIEW

3.   ANTECEDENTS

2.   FORMULATION OF THE ABSTRACT GAMBLER’S PROBLEM

1.   INTRODUCTION

2.   FORTUNES AND HISTORIES

3.   GAMBLES

4.   GAMBLING HOUSE

5.   STRATEGIES

6.   THE INTEGRAL GENERATED BY A STRATEGY

7.   FINITARY MAPPINGS

8.   INDUCTIVELY INTEGRABLE FUNCTIONS

9.   STOP RULES, POLICIES, AND TERMINAL FORTUNES

10.   THE UTILITY OF FORTUNES, POLICIES, AND HOUSES

11.   INCOMPLETE STOP RULES

12.   FUNCTIONS THAT MAJORIZE U

13.   DIGRESSION ABOUT COUNTABLE ADDITIVITY

14.   U IS EXCESSIVE FOR T.

15.   LIMITED PLAYING TIME

16.   CONTINUOUS GAMBLING HOUSES

3.   STRATEGIES

1.   INTRODUCTION

2.   THE UTILITY OF A STRATEGY

3.   STRATEGIC UTILITY

4.   STAGNANT STRATEGIES

5.   OPTIMAL STRATEGIES

6.   THRIFTY STRATEGIES

7.   EQUALIZING STRATEGIES

8.   OPTIMAL STRATEGIES AGAIN

9.   STATIONARY FAMILIES

4.   CASINOS WITH FIXED GOALS

1.   INTRODUCTION

2.   THE CASINO INEQUALITY

3.   EXPLORATION OF THE CASINO INEQUALITY

4.   THE THREE SPECIAL TYPES OF CASINOS

5.   HOW NICE CASINOS ARE

6.   THE CASINO-FUNCTION SEMIGROUP

7.   RICH MAN’S, POOR MAN’S, AND INCLUSIVE CASINOS

8.   THE SPECIAL CASINO INEQUALITIES DO NOT IMPLY THE GENERAL ONE.

5.   RED-AND-BLACK

1.   INTRODUCTION

2.   THE UTILITY OF BOLD STRATEGIES

3.   FOR w < 1/2, BOLD PLAY IS OPTIMAL.

4.   OTHER OPTIMAL STRATEGIES FOR w < 1/2

5.   LIMITED PLAYING TIME

6.   THE TAXED COIN

6.   PRIMITIVE CASINOS

1.   INTRODUCTION

2.   THE UTILITY OF BOLD STRATEGIES

3.   BOLD PLAY IS OPTIMAL FOR SUBFAIR PRIMITIVE CASINOS.

4.   DIGRESSION ABOUT N

5.   OTHER OPTIMAL STRATEGIES FOR W < T

6.   LIMITED PLAYING TIME

7.   PRIMITIVE CASINOS IN RELATION TO GENERAL CASINOS

8.   MORE ABOUT PRIMITIVE CASINOS FUNCTIONS

9.   UNIFORM ROULETTE

7.   SOME GENERAL PRINCIPLES

1.   INTRODUCTION

2.   COMPOSITION

3.   PERMITTED SETS OF GAMBLES

4.   FULL HOUSES

5.   PERMITTED HOUSES

8.   HOUSES ON THE REAL LINE

1.   INTRODUCTION

2.   DOMINATION

3.   MONOTONE HOUSES

4.   INCREASING FULL HOUSES

5.   FULL-HOUSE CASINOES

6.   STATIONARY FULL HOUSES

7.   EXPONENTIAL HOUSES

9.   THREE PARTICULAR KINDS OF CASINOS

1.   INTRODUCTION

2.   INCOME-TAX CASINOS

3.   THE CASINO THAT TAKES A CUT OF THE STAKE

4.   HOUSES OF INEQUITY

10.   ONE-LOTTERY STRATEGIES

1.   INTRODUCTION

2.   LIMITED PLAYING TIME

3.   CONVENTIONAL LOTTERIES

4.   GENERAL LOTTERIES

5.   PROPORTIONAL STRATEGIES

11.   FAIR CASINOS

1.   INTRODUCTION

2.   FAIR LOTTERIES CAN GENERATE SUBFAIR CASINOS

3.   WHEN IS A CASINO FAIR?

4.   FAIR, SIMPLE-LOTTERY CASINOS

5.   FAIR, ONE-GAME CASINOS

6.   OPTIMAL STRATEGIES FOR FAIR CASINOS

12.   THE SCOPE OF GAMBLER’S PROBLEMS

1.   INTRODUCTION

2.   FORTUNES

3.   OPINION AS A COMPONENT OF FORTUNE

4.   NONLEAVABLE HOUSES

5.   DISCOUNTING THE FUTURE

6.   A DIFFERENT APPROACH TO THE INDEFINITE FUTURE

7.   DYNAMIC PROGRAMMING

8.   BAYESIAN STATISTICS

9.   STOCHASTIC PROCESSES

Bibliography

Bibliographic Supplement (1975)

Bibliographic Supplement (2014)

Index of Persons

Subject Index

Index of Symbols

Index of Open Problems

Remarks on Open Problems

Appendix on Finite Additivity

Each chapter leads most naturally to the next. Some chapters, however, lead fairly naturally to later ones, as this diagram indicates.

1

INTRODUCTION

1.  THE PROBLEM.

Imagine yourself at a casino with $1, 000. For some reason, you desperately need $10, 000 by morning; anything less is worth nothing for your purpose. What ought you do? The only thing possible is to gamble away your last cent, if need be, in an attempt to reach the target sum of $10, 000. There may be a moment of moral confusion and discouragement. For who has not been taught how wrong and futile it is to gamble, especially when short of funds? Yet, gamble you must or forgo all chance of the great purpose that can be achieved only at the price of $10, 000 payable at dawn. The question is how to play, not whether.

As is well known, any policy of compounding bets that are subfair to you must decrease your expected wealth. Consequently, no matter how you play, your chance of converting $1, 000 into $10, 000 will be less than 1/10. How close to 1/10 can you make it and by what strategy? That is the sort of problem this