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Representing Suppositional Decision Theories with Sets of Desirable Gambles

The sets of desirable gambles framework has been well-studied as a tool for representing decision-making with imprecise probabilistic beliefs -- under the assumption of act-state independence. The question of this paper is: can we use sets of desirable gambles to represent decisions where the states do depend (e.g., causally or probabilistically) on the acts? In particular, I investigate two possible routes for representing suppositional decision theories with sets of desirable gambles, concluding that while one route works only for the subclass of SDTs representable by general imaging, the other route can represent any SDT whatsoever. After giving a fairly flat-footed representation, I investigate whether it's equivalent to a construction directly from the local (suppositional) desirability judgments; it isn't, but this latter construction represents a different aggregation rule applied to the same "credal committee". Finally, I extend the representation to model uncertainty about the supposition rule itself, in addition to imprecise credences.