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.

The expectation is an example of a descriptive statistic that is monotone with respect to stochastic dominance, and additive for sums of independent random variables. We provide a complete characterization of such statistics, and explore a number of applications to models of individual and group decision‐making. These include a representation of stationary monotone time preferences, extending the work of Fishburn and Rubinstein (1982) to time lotteries. This extension offers a new perspective on risk attitudes toward time, as well as on the aggregation of multiple discount factors. We also offer a novel class of non‐expected utility preferences over gambles which satisfy invariance to background risk as well as betweenness, but are versatile enough to capture mixed risk attitudes.

We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951), we show that generically an experiment is more informative than another in large samples if and only if it has higher Rényi divergences. We apply our analysis to the problem of measuring the degree of dissimilarity between distributions by means of divergences. A useful property of Rényi divergences is their additivity with respect to product distributions. Our characterization of Blackwell dominance in large samples implies that every additive divergence that satisfies the data processing inequality is an integral of Rényi divergences.