game-theoretic probability

Game-theoretic probability (GTP) is a foundation for probability theory, distinct from the measure-theoretic foundations laid out by Kolmogorov in the 1920s. Whereas Kolmogorov conceptualized probability as a subset of measure theory, game-theoretic probability lays its foundations in game-theory.

GTP makes statements by means of iterated games between various players, often consisting of reality, skeptic, and possibly several other players. The moves made by reality are the “observations,” and skeptic is betting on the result of the moves. A GTP statement often has the following form: Either (i) the observations obey some property, or (ii) the skeptic can make infinite money.

For concrete examples, see

• game-theoretic LLN
• game-theoretic hypothesis testing
• game-theoretic convergence of opinions
• game-theoretic concentration inequalities

You can recover many other results of classical/modern probability theory using the game-theoretic framework. Examples not included in these notes include stochastic calculus, zero-one laws, pricing formulas in finance, and others. See the probability and finance project for more examples.

Some key differences between GTP and measure-theoretic probability (MTP):

• The definition of a probability measure comes first in MTP, and then expectations are defined as integrals with respects to measures. In GTP, the definition of expectations comes first, and then probabilities are defined afterwards.
• GTP statements are usually made with no reference to distributions, randomness, measures, etc. They are therefore typically much easier to understand than MTP statements.
• GTP is inherently sequential.

game-theoretic statistics is the application of game-theoretic probability to statistical procedures, such as hypothesis testing.

Gamble spaces, prices, and probabilities

Like measure-theoretic probability, GTP introduces various notation and concepts to formalize the discussion.

Let $\Omega$ be the outcome space, i.e., the set of things that can happen. Elements $\omega \in \Omega$ are called outcomes and subsets $A\subset \Omega$ are called events.

A variable is a mapping $X:\Omega \to \overline{\mathbb{R}}$, where $\overline{\mathbb{R}}$ is the extended reals. A useful way to think about variables is as contracts. Eg, $X$ might be the contract “if this coin lands heads then the bearer obtains 1 dollar”. Here $\Omega =\{H,T\}$ and $X(H)=1$, $X(T)=0$. Then we can start asking questions such as: “How much are you willing to pay $X$ for?”

A gamble is also a mapping $Z:\Omega \to \overline{\mathbb{R}}$, but it’s useful to differentiate gambles from variables. Think of the outcome $Z(\omega )$ is how much money is obtained when the outcome is $\omega$ when using gambling strategy $Z$.

The set of all gambles under consideration is $\mathcal{Z}$ and together $(\Omega ,\mathcal{Z})$ form the Gamble space.
Using the concept of a gamble space, we can define game-theoretic expectations which in turns defines game-theoretic probabilities.

Just as there are concentration results for random variables in MTP, so too there are concentration results in GTP: game-theoretic concentration inequalities.

Arbitrage-free and positive-linearity

A gamble space is called arbitrage-free and positive linear if for all $Z\in \mathcal{Z}$ with $Z\ne 0$, $Z\ngeqq 0$. That is, there exists some $\omega$ such that $Z(\omega )<0$. This simply means that the bettor can make free money regardless of the outcome.

A gamble space is positive-linear if for all $Z_1,Z_2\in \mathcal{Z}$ and positive constants $c_1,c_2$ we have $c_1{Z}_1+c_2{Z}_2\in \mathcal{Z}$.

Almost-sure events

What does it mean to say that something happens with probability 1 in GTP?

Here’s a result. Suppose $(\Omega ,\mathcal{Z})$ is arbitrage-free and positive-linear. Let $A\subset \Omega$. If there exists some $Z\in \mathcal{Z}$ bounded from below such that $Z(\omega )=\infty$ for all $\omega \notin A$, then $\underline{\mathbb{P}}(A)=\overline{\mathbb{P}}(A)=1$. This gives a notion of a probability 1 event: If we can gamble on it and make infinite wealth if it does not occur. This is arguably the main machinery used in game-theoretic statistics.