Foundations

These are some of the known facts about game-theoretic upper probability ⚠ $P$:

• It is an outer measure [obvious].
• It is a Choquet capacity, at least in the case of a finite outcome space and one-step ahead forecasts [Vovk 2009].
• In general, it is not strongly additive, i.e., it is not guaranteed to satisfy ⚠ $P(A\cup B) + P(A\cap B) \le P(A) + P(B)$. (Therefore, the situation is similar to that in the theory of imprecise probabilities: cf. Walley (2000), page 128.) This is a simple example in the prequential framework (sequential probability forecasting of binary outcomes) with horizon 2 (i.e., the forecaster issues 2 forecasts ⚠ $p_1,p_2$ for 2 consecutive outcomes ⚠ $y_1,y_2$): ⚠ $A = \{(0,0,1/2,0),(1/2,0,0,0)\}$ and ⚠ $B = \{(0,0,1/2,0),(1/2,1,0,0)\}$ (the elements of ⚠ $A$ and ⚠ $B$ are represented in the form ⚠ $(p_1,y_1,p_2,y_2)$). In this case we have ⚠ $P(A\cup B)=1$ and ⚠ $P(A\cap B) = P(A) = P(B) = 1/2$. (For the standard prequential framework with infinite horizon just add ⚠ $00\ldots$ at the end of each element of ⚠ $A$ and ⚠ $B$.)

Bibliography

• Vladimir Vovk. Prequential probability: game-theoretic = measure-theoretic. The Game-Theoretic Probability and Finance Project,Working Paper 27, January 2009.
• Peter Walley. Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24 (2000) 125 - 148.