Abstract
We study decision under uncertainty in an Anscombe–Aumann framework. Two binary relations characterize a decision-maker: one (in general) incomplete relation, reflecting her objective rationality, and a second complete relation, reflecting her subjective rationality. We require the latter to be an extension of the former. Our key axiom is a dominance condition. Our main theorem provides a representation of the two relations. The objectively rational relation has a Bewley-style multiple prior representation. Using this set of priors, we fully characterize the subjectively rational relation in terms of the most optimistic and most pessimistic expected utilities.
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Notes
Strictly speaking, the version of continuity which we have listed is stronger than the one that appears in GMMS, while the independence axiom is weaker. However, as GMMS note in their final technical remarks on p 769, Shapley and Baucells (1998) have shown that, in the presence of Preorder, this stronger continuity axiom and this weaker independence axiom imply the independence axiom that GMMS use. Conversely, the representation of \(\succsim ^*\) that they obtain in their theorems implies the stronger continuity axiom.
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Grant, S., Rich, P. & Stecher, J. Objective and subjective rationality and decisions with the best and worst case in mind. Theory Decis 90, 309–320 (2021). https://doi.org/10.1007/s11238-020-09777-x
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DOI: https://doi.org/10.1007/s11238-020-09777-x