Objective and subjective rationality and decisions with the best and worst case in mind
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Objective and subjective rationality and decisions with the best and worst case in mind Simon Grant1
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Patricia Rich2
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Jack Stecher3
Accepted: 21 September 2020 Ó Springer Science+Business Media, LLC, part of Springer Nature 2020
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. Keywords Ambiguity Incomplete preferences Optimism Pessimism
1 Introduction It is well known that ambiguity affects decision-making, often in important ways. It is less well understood, however, where ambiguity preferences come from; there are many theories of ambiguity preference and modeling approaches are diverse. Although it is common to model agents as pessimistic or cautious in the face of ambiguity, this is far from the only attitude with theoretical plausibility or empirical & Simon Grant [email protected] Patricia Rich [email protected] Jack Stecher [email protected] 1
Australian National University, Canberra, Australia
2
University of Bayreuth, Bayreuth, Germany
3
University of Alberta, Edmonton, Canada
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support. Ju¨rgen Eichberger—in whose honor this special issue appears—was an early proponent of the view that some agents are instead optimistic. He developed models accommodating a wider range of ambiguity attitudes, and showed that this kind of heterogeneity can have significant economic implications (Chateauneuf et al. 2007; Eichberger et al. 2008). We are in full agreement with Eichberger on this point; in a companion paper (Grant et al. 2020), we develop a meta-utility theory in which the worst-case and best-case expected utilities of an act (with respect to the set of priors C and the utility index u) serve as sufficient statistics to describe a decision-maker’s preferences. The worst-case and best-case expected utilities, being cardinal, act like any other quantities, and can serve as inputs to an ordinal utility function. For example, the arithmetic weighted average expected utility model of Hurwicz (1951) is a meta-utility function in which the worst-case and best-case expected utilities are perfect substitutes. The minimum expected utility theory of Wald (1950), Gilboa and Schmeidler (1989) is an important special case, and Binmore’s (2009) geometric weighted average expected utility theory is likewise subsumed. We dub this family of preferences Ordinal Hurzwicz Expected Utility, as they extend Hurwicz
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