Uncertainty and preference is often modeled using linear previsions and linear
orders. Some more expressive models use sets of probabilities, lower previsions,
or partial orders (see, e.g., the work of Seidenfeld et al. and Walley). In the
discussion of these more expressive models, or even to justify them, alternative
representations in terms of sets of so-called acceptable, favorable, or
desirable gambles appear (cf. the work of Williams, Seidenfeld et al., and
Walley). Such ‘sets of gambles’-based models are attractive because of their
geometric nature.
We generalize these ‘sets of gambles’-based models by considering a pair of
sets, one with accepted gambles and one with rejected gambles. We develop a
framework based on a small number of axioms—No Confusion, Deductive Closure, No
Limbo, and Indifference to Status Quo—and provide an interesting
characterization of the resulting models. Furthermore, we define a pair of
equivalent gamble relations that generalize the partial orders mentioned
earlier; the corresponding characterization result is also given.