  To model discrete-time finite-state uncertain processes, we argue for the use
of a global belief model in the form of an upper expectation that is the most
conservative one under a set of basic axioms. Our motivation for these axioms,
which describe how local and global belief models should be related, is based
on two possible interpretations for an upper expectation: a behavioural one
similar to Walley's, and an interpretation in terms of upper envelopes of
linear expectations. We show that the most conservative upper expectation
satisfying our axioms, that is, our model of choice, coincides with a
particular version of the game-theoretic upper expectation introduced by Shafer
and Vovk. This has two important implications: it guarantees that there is a
unique most conservative global belief model satisfying our axioms; and it
shows that Shafer and Vovk's model can be given an axiomatic characterisation
and thereby provides an alternative motivation for adopting this model, even
outside their game-theoretic framework. Finally, we relate our model to the
upper expectation resulting from a traditional measure-theoretic approach. We
show that this measure-theoretic upper expectation also satisfies the proposed
axioms, which implies that it is dominated by our model or, equivalently, the
game-theoretic model. Moreover, if all local models are precise, all three
models coincide.
