Game Theoretic Probability

Master of Logic project (other coordinated projects)
30 May – 24 June 2016
Instructor: Wouter M. Koolen  (
6 EC
Project presentation slides

Probabilities arising from Backward Induction


In 1654 Pascal and Fermat corresponded on the fair division of the stakes in gambling games that got interrupted mid-way. Fermat reasoned measure-theoretically, counting the proportion of wins in imagined continued play. Pascal argued game-theoretically, establishing fair prices for all intermediate states. Pascal and Fermat arrived at the same answer, yet their methods make different assumptions and generalise in different ways.

Today, game-theoretic probability is an alternative foundation for probability theory. It goes beyond Kolmogorov's axiomatic (measure-theoretic) framework by considering an underlying strategic dimension. We will start with various classical laws of probability and re-prove them in the following form. In an infinite game where we are allowed to bet on the outcomes sequentially, we have a strategy that accumulates infinite wealth unless the sequence of outcomes satisfies the probability law. This will be the sense in which we can force the law, no matter how the outcomes arise.

We will study the (intricate) strategies underlying such theorems. Throughout the course we will see how the generality of the framework allows us to reduce assumptions, clarify interpretation, and open up applications to other areas of science. In particular we will look at one application in machine learning in the form of defensive forecasting.


The first two weeks are devoted to six lectures with associated homework. The last two weeks will be individual projects with one progress meeting midway.

Preliminary Schedule

Lectures are from 10:00 to 12:00, CWI, rooms L016, L017 or L120.

30 May
Lecture 1 (chapters 1 and 2)
Historical perspective.
Kolmogorov axioms.
Binomial pricing.
Chapter 1
Set 1
1 June
Lecture 2 (chapters 3 and 4)
Bounded strong law of large numbers.
Unbounded strong law of large numbers.
Chapter 3
Set 2
3 June
Lecture 3 (chapter 5)
Strong laws: law of the iterated logarithm
Definitions Set 3
6 June
Lecture 4 (chapter 6 and 7)
Weak laws: law of large numbers
Weak laws: central limit theorem
Definitions Set 4
8 June
Lecture 5 (chapter 8)
Game-theoretic and measure-theoretic probability
Definitions Set 5
10 June
Lecture 6 (not in book)
Application: On-line prediction and defensive forecasting
17 JuneProgress meeting

Reports are due on June 24.


An introductory probability course is indispensable. Familiarity with two-player zero-sum games will be an advantage. A good level of mathematical maturity is required, the strategies that will be considered are not simple.


Students will be assessed on the basis of six homeworks and a final report. The written report should provide either some (small) original result or a survey of some of the literature with a new perspective.

Each homework counts for 10% and the final report counts for 40% of the grade.



GTP The course will cover the first half of the the book Probability and Finance: It's Only a Game! by Glenn Shafer and Vladimir Vovk. I will lecture from the book but do my best to keep the course self-contained. In particular I will post lecture notes.

Website dedicated to the book here. You can read chapters 1, 3 and 9 online.

On-line prediction Wiki

The on-line prediction wiki has a section about Game-Theoretic Probability here.

Project ideas

To get inspiration and ideas for your project topic, you can browse the list of working papers on the book website. A broader selection of related topics can be found in the archives of the Workshop on GTP and related topics below.

It is perfectly fine to do a project on one of the related topics, including imprecise probabilities, prequential statistics, on-line prediction, algorithmic randomness and conformal prediction.