game-theoretic LLN

A law of large numbers stated in terms of game-theoretic probability. Consider the following setup:

• Bob has some initial capital, $C_0=1$.
• For $t=1,2,\dots$
  • Bob makes a bet $M_t\in \mathbb{R}$
  • The world reveals a value $x_t\in [0,1]$.
• Bob updates his capital as $C_t=C_{t-1}+M_t(1/2-x_t)$.

The game stops if Bob goes broke (i.e.,$C_t<0$). The game-theoretic LLN states that Bob has a strategy which ensures that either

$$\underset{t\to \infty }{\lim }\frac{1}{t}\sum _{i\le t}{x}_i=1/2,\text{\hspace{1em}(1)}$$

or he becomes infinitely wealthy, ${\mathrm{liminf}}_{t\to \infty }{C}_t=\infty$. There’s nothing special about the constant 1/2 here. It could be replaced with any value between 0 and 1 and we’d simply change how Bob updates his capital.

A Bayesian interpretation would be that Bob has a belief that the average value of the observations $x_t$ is 1/2 and is betting accordingly. Or you don’t have to get Bayesian at all, and you can instead say that Bob is able force the world to act in a certain way if the world wants to prohibit him from getting infinitely wealthy.

Notice that there no probabilistic assumptions in the statement of the game-theoretic LLN. There are no distributions, no probability measures, no sigma algebras, nothing. It makes a deterministic statement about every sequence of numbers $(x_t)$, namely that there exists a betting strategy such that either (1) will happen, or Bob will become infinitely wealthy.

Remarkably, the game-theoretic LLN recovers the measure-theoretic SLLN (under the assumption of boundedness, which were assuming here). I have a blog post about this here.