Desirability and the birth of incomplete preferences

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.