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

Sample ChaptersErrataReviewsFrequently Asked QuestionsJapanese Translation

Sample Chapters


Praise from the Reviewers

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

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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.

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This page is maintained by Vladimir Vovk.   Last modified on 4 February 2018