  Game-theoretic upper expectations are joint (global) probability models that
mathematically describe the behaviour of uncertain processes in terms of
supermartingales; capital processes corresponding to available betting
strategies. Compared to (the more common) measure-theoretic expectation
functionals, they are not bounded to restrictive assumptions such as
measurability or precision, yet succeed in preserving, or even generalising
many of their fundamental properties. We focus on a discrete-time setting where
local state spaces are finite and, in this specific context, build on the
existing work of Shafer and Vovk; the main developers of the framework of
game-theoretic upper expectations. In a first part, we study Shafer and Vovk's
characterisation of a local upper expectation and show how it is related to
Walley's behavioural notion of coherence. The second part consists in a study
of game-theoretic upper expectations on a more global level, where several
alternative definitions, as well as a broad range of properties are derived,
e.g. the law of iterated upper expectations, compatibility with local models,
coherence properties,... Our main contribution, however, concerns the
continuity behaviour of these operators. We prove continuity with respect to
non-increasing sequences of so-called lower cuts and continuity with respect to
non-increasing sequences of finitary functions. We moreover show that the
game-theoretic upper expectation is uniquely determined by its values on the
domain of bounded below limits of finitary functions, and additionally show
that, for any such limit, the limiting sequence can be constructed in such a
way that the game-theoretic upper expectation is continuous with respect to
this particular sequence.
