Probability and Finance: It's Only a Game!
by Glenn Shafer and Vladimir Vovk. New York: Wiley, 2001

• A new mathematics for probability: game theory instead of measure theory.
• A new philosophy of probability, synthesizing the objective and the subjective.
• A new mathematics for finance theory: game theory instead of stochastic models.

Sample Chapters

• Table of contents.
• Short abstracts of all chapters (in Microsoft Word).
• Chapter 1, "Probability and Finance as a Game". Copyright © 2001 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
• Chapter 3, "The Bounded Strong Law of Large Numbers". (This is a mathematical introduction to the game-theoretic framework.) Copyright © 2001 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
• Chapter 9, "Game-Theoretic Probability in Finance" (without appendixes). Copyright © 2001 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.

Errata

• Errata1: errors that were corrected in the second printing.
• Errata2: errors in the second printing known to the authors. Note especially the error concerning Borel on page 37. If you see other errors, please e-mail Glenn Shafer.
• Clarifications: things we will fix if the book has a second edition.

Praise from the Reviewers

• Reading this book will generously reward every person interested in randomness and finance, and more generally in understanding some of the "representations of reality" that are assumed in everyday life! Antonio Gualtierotti in Mathematical Reviews

  This thorough and glowing review appeared in Issue 2002k. To access it online through a subscribing library, use the code 2002k:60008 or click here.

• Invoking my prerogative as a reviewer, I declare that, in my opinion, the authors' approach is entirely original. It seems to me that their predecessors, for all their splendor, had nothing like it.... I would evaluate the book as a classic work of the highest rank, which explains entirely new ideas in a way that it is easy and pleasant to read page by page, from beginning to end. Valery Nikolaevich Tutubalin, Theory of Probability and Its Applications, Vol.47, No.3, 2002

  Download the full text in Russian. Download the full text in English. Download our response. Download Professor Tutubalin's response, in Russian, to our response (Microsoft Word).

• The first part of the book presents an interesting new mathematical and philosophical framework for probability: game theory instead of measure theory. It is based on a sequential game between an idealized scientist and "the world". The second half, on finance, illustrates the potential of the new framework. It suggests greater use of the market mechanisms and less use of stochastic models in the pricing of financial derivatives. In particular, it shows how purely game-theoretic probability can replace models in the efficient market hypothesis. Ulrich Horst in Zentralblatt MATH 0985.91024 (Vol.985, No.10, 2002)

• ...a creative, entertaining and imaginative book... David J. Hand in Short Book Reviews, August 2002

• It is a pleasure to salute an impressively original approach, for which the authors deserve our thanks. N. H. Bingham, commenting on an early draft.

  Download Professor Bingham's full comments, with our response.

• There are many photographs of celebrities (including one of the authors), which makes the book even more interesting. Freddy Delbaen in Journal of the American Statistical Association (Vol.97, No.459, 2002)

  Unfortunately, there are many errors in Professor Delbaen's review. Download our response.

• Shafer... and Vovk... explain how probability can be based on game theory rather than measure theory, and how doing so allows it to be used in finance without a lot of distracting an confusing assumptions about randomness. Reference & Research Book News, November 2001

• The first half of this truly original book introduces a novel approach to probability, founded on game theory rather than measure theory. In an admirably clear, scholarly and engaging manner, it traces its historical antecedents, expounds its advantages, develops its technicalities, and addresses its philosophical implications. The second half goes on to do the same for financial modelling. This is a book that should utterly change the way we think about its two topics. A. Philip Dawid, University College London, UK. (The quote is from the Wiley web site. Philip Dawid is now at the University of Cambridge.)

Frequently Asked Questions

1. Kolmogorov's measure-theoretic framework for probability includes an axiom of continuity, which is equivalent to countable additivity once finite additivity is assumed. Does the game-theoretic framework require a similar axiom?

   No axiom of continuity was assumed in the book. Because the axiom of continuity for measure-theoretic probabilities is not well motivated, our being able to prove the classical limit theorems of probability without it should count as an advantage of our framework over the measure-theoretic framework. On the other hand, as we have shown in Working Paper 5 for the Game-Theoretic Probability and Finance Project, an alternative definition of game-theoretic upper probability that does satisfy the axiom of continuity can play a useful simplifying role in the game-theoretic treatment of continuous-time processes. A fuller answer can be downloaded from here.

2. Do you use the principle of no arbitrage in your approach to finance? The principle of no arbitrage involves probabilities, so if you use it, then you are using probabilities; if you do not use it, on the other hand, you must be foregoing one of the most important insights of modern finance theory.

   The short answer is that we use a form of the principle of no arbitrage that avoids appealing to the doubtful assumption that markets are stochastic. This note is a more extended answer.

3. You do not have any large deviation inequalities in your book. Is it possible to express them in the game-theoretic framework?

   A large deviation inequality can be extracted from the proof of the validity part of the law of the iterated logarithm in Section 5.2. This note makes the obvious observation that Hoeffding's original proof of his inequality remains valid in the game-theoretic framework. See also this arXiv report.

4. In "Probability and Finance: It's Only a Game!", you used games in which Forecaster offers Skeptic a cone of gambles. In Working Paper 29, you instead allow Skeptic to buy any function of Reality's move as a payoff and price this function using what you call a superexpectation functional. What is the relation between these two formulations?

   They are equivalent, provided that conditions on the cone are properly matched with conditions on the superexpectation functional. The attached note explains how this can be done.

Japanese Translation

An abridged Japanese translation appeared in 2006 (translated by Masayuki Kumon and edited by Kei Takeuchi).

For the English version of the authors' preface to the Japanese edition, click here.

Order from Iwanami Shoten (Tokyo) or amazon.co.jp.