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.
Working Papers —
Sample Chapters —
Errata —
Reviews —
Frequently Asked Questions —
Japanese Translation —
Talks —
History of Probability
Subsequent to the publication of this book,
the Rutgers–Royal Holloway research group led by Shafer and Vovk
has posted over 20 working papers on gametheoretic probability and finance on
this page.
The page also includes papers by the Tokyo research group led by Takeuchi and Takemura.
For information about other recent work on gametheoretic probability,
see the online prediction wiki.
 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 gametheoretic framework.)
Copyright © 2001 by John Wiley & Sons, Inc.
This material is used by permission of John Wiley & Sons, Inc.
 Chapter 9, "GameTheoretic Probability in Finance"
(without appendixes).
Copyright © 2001 by John Wiley & Sons, Inc.
This material is used by permission of John Wiley & Sons, Inc.
 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 email
Glenn Shafer.
 Clarifications:
things we will fix
if the book has a second edition.
 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 gametheoretic 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.)

Kolmogorov's measuretheoretic framework for probability includes an axiom of continuity,
which is equivalent to countable additivity once finite additivity is assumed.
Does the gametheoretic framework require a similar axiom?
No axiom of continuity was assumed in the book.
Because the axiom of continuity for measuretheoretic 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 measuretheoretic framework.
On the other hand,
as we have shown in
Working Paper 5
for the GameTheoretic Probability and Finance Project,
an alternative definition of gametheoretic upper probability
that does satisfy the axiom of continuity can play a useful simplifying role
in the gametheoretic treatment of continuoustime processes.
A fuller answer can be downloaded from here.

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.

You do not have any large deviation inequalities in your book.
Is it possible to express them in the gametheoretic 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 gametheoretic framework.

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.
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.
 First Russian edition (1936) of Andrei Kolmogorov's monograph
on the foundation of probability, originally published in German in 1933.
The translation in this edition follows the original German text closely.
(Kolmogorov made substantial changes to the text in the second Russian edition,
published in 1974.)
This edition is rare, at least outside of Russia.
We are indebted to Alexander Shen' for creating this electronic copy.
Please choose pdf (2.9 MB)
or zipped postscript (2.9 MB).
 A counterexample
to Richard von Mises's theory of collectives
(Jean Ville, translation and introduction by Glenn Shafer).
This document presents a translation from French into English of a passage
from Jean Ville's "Etude critique de la notion de collectif" (1939).
In the passage translated here,
Jean Ville gives an example of an event that is assigned probability zero
by classical probability theory
but cannot be ruled out by conditions of the type Richard von Mises used
to define his concept of a collective.
Ville and others considered the existence of such examples
a shortcoming of von Mises's theory,
and this motivated Ville to introduce martingales into probability theory.
 Per MartinLof,
Algorithmen und zufallige Folgen.
Vier Vortrage von Per MartinLof (Stockholm)
gehalten am Mathematischen Institut
der Universitat ErlangenNurnberg,
Erlangen, 1966.
This document, dated 16 April 1966,
consists of notes taken by K. Jacobs and W. Muller
from lectures by MartinLof at Erlangen on April 5, 6, 14, and 15.
 Electronic Journal
for History of Probability and Statistics.
 Pierre Crépel's 1984 report (in French)
on the history of the theory of martingales from 1920 to 1940.
 Oscar Sheynin's
translations and articles.
This page is maintained by
Vladimir Vovk.
Last modified on 16 October 2016