# 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.