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:
The upper probability of any subset of the sample space can be defined as in game-theoretic probability:
This is the game-theoretic definition.
An alternative, "measure-theoretic", definition is as follows. A forecasting system is a function (intuitively, for each sequence of Reality's moves recommneds a move for Forecaster; in other words, it is a strategy for Forecaster that ignores Sceptic's moves). For each forecasting system , let be the probability distribution on corresponding to the following process: is chosen according to (, where is the empty sequence), then is chosen according to the Bernoulli distribution with parameter , then is chosen according to (), then is chosen according to the Bernoulli distribution with parameter , etc. Set
We are mostly interested in the case of a Borel , in which case the notation is unambiguous. In general, can be understood to be the outer measure of .
The open question (posed by Shafer and Shen, among others) is: Suppose is Borel set. Is it true that the two definitions coincide: ? 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 ), 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).