  We establish an equivalence between two seemingly different theories: one is
the traditional axiomatisation of incomplete preferences on horse lotteries
based on the mixture independence axiom; the other is the theory of desirable
gambles developed in the context of imprecise probability. The equivalence
allows us to revisit incomplete preferences from the viewpoint of desirability
and through the derived notion of coherent lower previsions. On this basis, we
obtain new results and insights: in particular, we show that the theory of
incomplete preferences can be developed assuming only the existence of a worst
act---no best act is needed---, and that a weakened Archimedean axiom suffices
too; this axiom allows us also to address some controversy about the regularity
assumption (that probabilities should be positive---they need not), which
enables us also to deal with uncountable possibility spaces; we show that it is
always possible to extend in a minimal way a preference relation to one with a
worst act, and yet the resulting relation is never Archimedean, except in a
trivial case; we show that the traditional notion of state independence
coincides with the notion called strong independence in imprecise
probability---this leads us to give much a weaker definition of state
independence than the traditional one; we rework and uniform the notions of
complete preferences, beliefs, values; we argue that Archimedeanity does not
capture all the problems that can be modelled with sets of expected utilities
and we provide a new notion that does precisely that. Perhaps most importantly,
we argue throughout that desirability is a powerful and natural setting to
model, and work with, incomplete preferences, even in case of non-Archimedean
problems. This leads us to suggest that desirability, rather than preference,
should be the primitive notion at the basis of decision-theoretic
axiomatisations.
