Abstract This paper extends the subjective expected utility model of decision making under uncertainty to include incomplete beliefs and tastes. The main results are two axiomatizations of the multi-prior expected multi-utility representations of preference relation under uncertainty. The paper also introduces new axiomatizations of Knightian uncertainty and expected multi-utility model with complete beliefs.

Abstract Many violations of the Independence axiom of Expected Utility can be traced to subjects' attraction to risk-free prospects. The key axiom in this paper, Negative Certainty Independence JEL: D80, D81 Keywords: Preferences under risk, Allais paradox, Negative Certainty Independence, Incomplete preferences, Cautious Completion, Multi-Utility representation. * We thank Faruk Gul, Fabio Maccheroni, Massimo Marinacci, Efe Ok, Wolfgang Pesendorfer, Gil Riella, and Todd Sarver for very useful comments and suggestions. The co-editor and four anonymous referees provided valuable comments that improved the paper significantly. We greatly thank Selman Erol for his help in the early stages of the project. Cerreia-Vioglio gratefully acknowledges the financial support of ERC (advanced grant, BRSCDP-TEA). Ortoleva gratefully acknowledges the financial support of NSF grant SES-1156091.

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This is a study of the representations of subjective expected utility preferences that admits state-dependent incompleteness, and subjective expected utility preferences displaying noncomparability of acts from distinct sources. The notions familiar events and sources are defined and characterized. The relation greater familiarity on sources and increasing familiairity of a source are also defined and characterized.

We study the relation between ambiguity aversion and the Allais paradox. To this end, we introduce a novel definition of hedging which applies to objective lotteries as well as to uncertain acts, and we use it to define a novel axiom that captures a preference for hedging which generalizes the one of Schmeidler (1989). We argue how this generalized axiom captures both aversion to ambiguity, and attraction towards certainty for objective lotteries. We show that this axiom, together with other standard ones, is equivalent to two representations both of which generalize the MaxMin Expected Utility model of Gilboa and Schmeidler (1989). In both, the agent reacts to ambiguity using multiple priors, but does not use expected utility to evaluate objective lotteries. In our first representation, the agent treats objective lotteries as ‘ambiguous objects,’ and use a set of priors to evaluate them. In the second, equivalent representation, lotteries are evaluated by distorting probabilities as in the Rank Dependent Utility model, but using the worst from a set of such distortions. Finally, we show how a preference for hedging is not sufficient to guarantee an Ellsberg-like behavior if the

represent preliminary reports on work in progress and should therefore be neither quoted nor referred to in published work without the written consent of the author. Universal interactive preferences

The state-preference framework for modeling choice under uncertainty, in which objects of choice are allocations of wealth or commodities across states of the world, is a natural one for modeling “smooth” ambiguity-averse preferences. It does not require reference to objective probabilities, personalistic consequences, or counterfactual acts, and it allows for state-dependence of utility and unobservable background risk. The decision maker’s local beliefs are encoded in her risk neutral probabilities (her relative marginal rates of substitution between states) and her local risk preferences are encoded in the matrix of derivatives of the risk neutral probabilities. This matrix plays a central but generally unappreciated role in the modeling of risk attitudes in the state-preference framework. It can be computed by inverting a bordered Slutsky matrix and vice versa, it generalizes the Arrow-Pratt measure for approximating local risk premia, and its structure reveals whether the decision maker’s risk preferences are ambiguity-averse as well as risk averse. Two versions of the smoothambiguity model are analyzed — the source-dependent risk aversion model and the second-order uncertainty (KMM) model — and it is shown that in both cases the overall premium for local uncertainty can be decomposed as the sum of a risk premium and an ambiguity premium.

A paradigmatic problem in social choice theory deals with the aggregation of subjective preferences of individuals — repre-sented as rankings of alternatives — into a social ranking. We are interested in settings where individuals are uncertain about their own preferences, and represent their uncertainty as distributions over rankings. Under the classic objective of minimizing the (expected) sum of Kendall tau distances be-tween the input rankings and the output ranking, we establish that preference elicitation is surprisingly straightforward and near-optimal solutions can be obtained in polynomial time. We show, both in theory and using real data, that ignoring uncertainty altogether can lead to suboptimal outcomes. 1

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This paper axiomatizes expected multi-utility representations of incomplete preferences under risk and under uncertainty. The von Neumann-Morgenstern expected utility model with incomplete preferences is revisited using a “constructive” approach, as opposed to earlier treatments that use convex analysis.