Glenn Shafer is University Professor at Rutgers University.

Vladimir Vovk is Professor in the Department of Computer Science at Royal Holloway, University of London.

Shafer and Vovk are the authors of Probability and Finance: It’s Only a Game, published by Wiley, and are co-authors of Algorithmic Learning in a Random World. Shafer’s other previous books include A Mathematical Theory of Evidence and The Art of Causal Conjecture, while Vovk’s work include various articles and chapters such as Conformal predictive decision making in Proceedings of Machine Learning Research.

The two joined together in a new project which, originally, set out to be a new edition of Probability and Finance, has since become a new book altogether entitled Game-Theoretic Foundations for Probability and Finance, due to be published in **June** of this year. This new book provides an in-depth coverage of past discussions, while also exploring the new research and growth that game-theority probability has had since 2001. As in the earlier book, Shafer and Vovk show that game-theoretic and measure-theoretic probability provide equivalent descriptions of coin tossing, the archetype of probability theory, while generalizing this archetype in different directions. Now we show that the two descriptions are equivalent on a larger central core, including all discrete-time stochastic processes that have only finitely many outcomes on each round, and we present an even broader array of new ideas.

To hear more about the topic, Fran McMahon sat down with one of its authors, Glenn Shafer, to learn more.

**1. After publishing the book Probability and Finance: It’s Only a Game in 2001, you and Professor Vovk are now publishing Game-Theoretic Foundations for Probability and Finance, which is said to contain fifteen years’ worth of further research on this topic. What led to this follow up?**

Our game-theoretic picture of probability is a complement, and sometimes an alternative, to the established theory of probability, which has been developed and perfected over centuries. When we published Probability and Finance, we knew we could not fully explore this new picture’s potential in a single book. So, we always expected to develop the subject further. As it turned out, it produced an entirely new book, which adds both a general abstract theory and many more applications. It deals with continuous-time finance in a completely different way, more productive and suited to a wider audience.

**2. What were your main objectives during the writing process?**

Our book is very ambitious. We want to persuade others who use mathematical probability (mathematicians, computer scientists, theoretical and applied statisticians, engineers, etc.) to use the game-theoretic picture. We also want to draw attention to the light this picture throws on the philosophical meaning of probability and even on its history.

Mathematical probability has a wider range of applications than any other area of applied mathematics. So, it’s very ambitious to say, “We’re going to do this in a different way that will provide insights the usual way doesn’t provide.” Our goal is to bring out the simplicity and power of the game-theoretic picture so clearly that it will be appreciated both by readers new to mathematical probability and by those expert in the existing theory.

**3. How is game theory related to probability theory and finance theory?**

Game theory has become ubiquitous. It is fundamental to theoretical economics and increasingly important in the other social sciences, in computer science, and even in pure mathematics. It is a remarkably broad mathematical framework, because games can have any number of players and any number of different rules for how they play, what information they have, and what and how they win. Our book uses only games of a very special kind—perfect-information games. In these games, players move in turn and see each other’s’ moves as they play. We show that the notion of probability emerges from certain simple perfect-information games.

**4. If there is one piece of information or advice that you would want your audience to take away from your book, what would that be?**

Remember that when probabilities are numbers, they are about betting.

Everyone knows that the theory of mathematical probability began with calculations in games of chance. But that was long ago. Bringing back the language of betting can help everyone understand what is really going on.

**5. Who should read the book and why?**

The book has something to say to everyone who develops and uses mathematical probability: pure mathematicians who are proving new theorems in probability, applied mathematicians who are using probability in statistics, machine learning, engineering, or business, and philosophers who are exploring the meaning of probability across the sciences.

The book is ideal for study at the graduate level. Each chapter has theorems and proofs, but most chapters also include a great deal of supplementary explanation. Each chapter has a section at the end entitled “Context”, which discusses the relation between the chapter’s ideas and similar ideas in the established theory of probability, in other fields, and in earlier centuries. Given the book’s breadth, from finance and statistics to physics, together with its broad view of abstract theory, from measure theory to imprecise probabilities, these “context” sections constitute an overview of mathematical probability that it would be difficult to put together from other sources.

**6. Why, do you think, this book may be of interest now?**

The book relates our theory to the established theory with a depth that was not possible in 2001. This explanation of the relationship between the two theories makes the new book essential to a full understanding of the established theory.

World-wide interest in game-theoretic probability itself has grown substantially since we published Probability and Finance in 2001. Workshops on the theory have been held and Britain, North America, and Japan. Many of the new results we report are due to a research group in Japan that was initiated by Kei Takeuchi and subsequently led by Akimichi Takemura.

The largest group of mathematicians now working on game-theoretic probability is in the area of mathematical finance. Many of these researchers call their work “probability-free”, because they do not add probabilities to the game of finance in the way the established theory does. The fourth part of our book, which is devoted to this topic, is the result of interaction between Vovk, the Japanese school, and European students of probability-free finance. I expect that this part of the book will be foundational for further work in probability-free finance.

The abstract theory in the second half of the book shows that game-theoretic probability is also closely related to imprecise probability, a field of research that has become increasingly active across the world during the past twenty years. Much of the research now published under the rubric of imprecise probabilities could equally well be called research in game-theoretic probability. I believe that our book will be essential reading for students of imprecise probability.

The game-theoretic picture of probability is also poised to become important in mathematical statistics and philosophy. In statistics, testing-by-betting is poised to address the current acute dissatisfaction with p-values. On the philosophical side, our betting interpretation of probability gives new depth to the much maligned and misunderstood “frequentist’’ interpretation.

**7. In addition to being a report on your research, your book is also a teaching resource. Exercises for the student are given at the end of each chapter. What led you to include exercises?**

We have so much new to say that we needed to leave some of the details for the reader to work out. I packaged some of these details as exercises in order to provide the reader with some guidance. We then found it natural to add simpler exercises to help readers with less background get started.

The result will enable researchers interested in game-theoretic probability and its applications to include it in their teaching, whether or not they use other books as primary textbooks.

**8. What is it about the area of game-theoretic probability and finance that fascinates you?**

I’ve always been most interested in the philosophical and historical foundations of the subject. What does probability mean? Is it objective or subjective? Why do people say it is based on measure theory? How do we understand what statistical evidence is telling us? I believe that the game-theoretic picture provides the best starting point for addressing these questions.

**9. What are you working on now?**

There is a lot more mathematics to do within the game-theoretic framework, and I believe that this framework will also be important in the coming decades in many fields where probability is used. Some of the likely directions are indicated by the working papers at the website www.probabilityandfinance.com, which Professor Vovk and I have maintained since we published Probability and Finance in 2001, including a paper that discusses how testing-by-betting can replace p-values.

I am also turning my attention to a number of historical projects that I set aside in order to complete the book. The first is my biography of Jean Ville (1910-1989), the French mathematician whose 1939 dissertation launched the mathematical notion of a martingale and inspired much of our thinking about game-theoretic foundations. My article on Ville’s early years (The education of Jean André Ville, Electronic Journal for the History of Probability and Statistics, 5(1)) appeared in 2009, but his life during and after World War II is also fascinating and merits recounting. I also plan a book based on the article on Kolmogorov’s foundations for probability that Professor Vovk and I published in 2006 (The sources of Kolmogorov’s Grundbegriffe, Statistical Science,21:70-98) and another book on the game-theoretic foundations for probability advanced by Blaise Pascal and Christiaan Huygens.