How to Gamble if You Must: Inequalities for Stochastic Processes

This is a revised and updated republication of the 1976 Dover publication. The title for that edition had shrunk to Inequalities for Stochastic Processes, but the original title has been restored here. It was five years after Jimmie Savage died when that last edition was published. In its preface Dubins wrote a very moving tribute to Savage, and that also appears in this edition. Curiously, Dubins actually included an explicit provision in his will that any future publication of the book would leave the contents intact.

Sudderth and Gilat, the editor-updaters of the current edition, have respected that request but have added a few helpful pieces as appendices that bring the book up to date. They incorporated a bibliographic supplement to fill in references that have appeared during the 38 years since the previous edition. They have also addressed the status of the many open problems scattered throughout the book. Some of these now have complete or partial solutions; the editors discuss them and point to relevant references.

The third item that the editors have added is an appendix on finite additivity. Most of us are accustomed to the conventional approach to measure theory with countable additivity. Dubins and Savage chose the more general approach with finitely additive measures. They foresaw fundamental issues for some gambling problems with countable additivity since they wanted measures defined on all subsets of the underlying space. So the editors decided to offer some guidance to help readers understand the complications that finite additivity introduces.

The object of the book is to teach probability at the graduate level using gambling, not just in a series of examples, but making it the main feature. The work that led to the book was driven by a question Savage asked Dubins: How could one play the casino game of Red-and-Black wisely, and is there an optimal strategy?

In many aspects the book is an extended response to that question and a consideration of its many ramifications. A gambler starts with a fortune \(f_0\) and, as play progresses, he moves through a sequence of fortunes \(f_1,f_2,\dots\), a subset of the space of fortunes. A gamble is a probability measure on subsets of fortunes, and a gambling house is a function that associates with each \(f\) a set of gambles from which the gambler is allowed to choose when his fortune is \(f\). A model of gambling strategy is gradually developed as a finitely-additive, discrete time stochastic process. It is remarkable how much can be built on a simple formulation like this. Indeed many aspects of gambling can be translated pretty directly into the language of stochastic processes. The authors handle that remarkably well.

Very little background in formal probability is needed to start reading. This is not a book you’re likely to read cover-to-cover, but it’s most definitely worth a look. Pascal — not a novice when it came to probability — wrote that we have no choice about gambling, that we must bet. But of course he wasn’t talking about casinos.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.