Equivalence Of Game-theoretic And Measure-theoretic Probability

This is an open problem in game-theoretic probability. This problem is claimed to be solved in Vovk (2009), but this claim has not been independently verified.

Consider the following basic prediction protocol:

$\quad$ $\mathcal{K}_0:=1$.
$\quad$ FOR $n=1,2,...$:
$\quad$ $\quad$ Forecaster announces $p_n\in[0,1]$
$\quad$ $\quad$ Sceptic announces $M_n\in\mathbb{R}$
$\quad$ $\quad$ Reality announces $x_n\in\{0,1\}$
$\quad$ $\quad$ $\mathcal{K}_n:=\mathcal{K}_{n-1}+M_n(x_n-p_n)$

The upper probability $\overline{\mathbb{P}}(E)$ of any subset $E\subseteq\Pi$ of the sample space $\Pi:=([0,1]\times\{0,1\})^{\infty}$ can be defined as in game-theoretic probability:

$$
  \overline{\mathbb{P}}(E)
  :=
  1\slash{}\sup\{C>0:\liminf_{n\to\infty}\mathcal{K}_n\ge C\}.
$$

This is the game-theoretic definition.

An alternative, "measure-theoretic", definition is as follows. A forecasting system is a function $\phi:\{0,1\}^*\to[0,1]$ (intuitively, for each sequence of Reality's moves $\phi$ recommneds a move for Forecaster; in other words, it is a strategy for Forecaster that ignores Sceptic's moves). For each forecasting system $\phi$, let $\mathbb{P}^{\phi}$ be the probability distribution on $\Pi$ corresponding to the following process: $p_1$ is chosen according to $\phi$ ($p_1:=\phi(\Box)$, where $\Box$ is the empty sequence), then $y_1$ is chosen according to the Bernoulli distribution with parameter $p_1$, then $p_2$ is chosen according to $\phi$ ($p_2:=\phi(y_1)$), then $y_2$ is chosen according to the Bernoulli distribution with parameter $p_2$, etc. Set

$$
  \overline{\mathbb{P}}'(E)
  :=
  \sup_{\phi}\mathbb{P}^{\phi}(E).
$$

We are mostly interested in the case of a Borel $E$, in which case the notation $\mathbb{P}^{\phi}(E)$ is unambiguous. In general, $\mathbb{P}^{\phi}(E)$ can be understood to be the outer measure of $E$.

The open question (posed by Shafer and Shen, among others) is: Suppose $E$ is Borel set. Is it true that the two definitions coincide: $\overline{\mathbb{P}}(E)=\overline{\mathbb{P}}'(E)$? To avoid measurability issues, assume that Forecaster's moves are restricted to the rational numbers in [0,1].

When Forecaster plays according to a fixed forecasting system (for example, always plays $p_n=1\slash{}2$), the equivalence of game-theoretic and measure-theoretic probability is well known: it is a version of Ville's theorem (see Shafer and Vovk, 2001, Section 8.5).

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