Econometrica Vol. 90 No. 2 2022
Model and Predictive Uncertainty: A Foundation for Smooth Ambiguity Preferences
Abstract
Smooth ambiguity preferences (Klibanoff, Marinacci, and Mukerji (2005)) describe a decision maker who evaluates each act f according to the twofold expectation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>V</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>f</a:mi> <a:mo stretchy="false">)</a:mo> <a:mo>=</a:mo> <a:msub> <a:mrow> <a:mo>∫</a:mo> </a:mrow> <a:mrow> <a:mi mathvariant="script">P</a:mi> </a:mrow> </a:msub> <a:mi>ϕ</a:mi> <a:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">(</a:mo> <a:msub> <a:mrow> <a:mo>∫</a:mo> </a:mrow> <a:mrow> <a:mi mathvariant="normal">Ω</a:mi> </a:mrow> </a:msub> <a:mi>u</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>f</a:mi> <a:mo stretchy="false">)</a:mo> <a:mspace width="0.2em"/> <a:mi mathvariant="normal">d</a:mi> <a:mi>p</a:mi> <a:mo stretchy="true" maxsize="5.2ex" minsize="5.2ex">)</a:mo> <a:mspace width="0.2em"/> <a:mi mathvariant="normal">d</a:mi> <a:mi>μ</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>p</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> defined by a utility function u , an ambiguity index ϕ , and a belief μ over a set <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi mathvariant="script">P</u:mi> </u:math> of probabilities. We provide an axiomatic foundation for the representation, taking as a primitive a preference over Anscombe–Aumann acts. We study a special case where <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mi mathvariant="script">P</x:mi> </x:math> is a subjective statistical model that is point identified, that is, the decision maker believes that the true law <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>p</ab:mi> <ab:mo>∈</ab:mo> <ab:mi mathvariant="script">P</ab:mi> </ab:math> can be recovered empirically. Our main axiom is a joint weakening of Savage's sure‐thing principle and Anscombe–Aumann's mixture independence. In addition, we show that the parameters of the representation can be uniquely recovered from preferences, thereby making operational the separation between ambiguity attitude and perception, a hallmark feature of the smooth ambiguity representation.
- DOI
- 10.3982/ecta18009
- Volume
- 90
- Issue
- 2
- Pages
- 551-584
- Language
- en
- Sources
- bibtex:phds-export.bib openalex crossref