# Schedule - Parallel Session 7 - Models of Uncertainty

IDL First Floor Syndicate Room - 09:00 - 10:30#### Range-dependent Utility Model with Constant Risk Aversion

Michal Lewandowski; Krzysztof Kontek

#### Abstract

The range-dependent utility model for choice under risk is a modification of expected utility theory in which utility function depends on the lottery range, i.e. the interval between the lowest and the highest outcome in the lottery support. It differs from a number of approaches assuming context dependence, in which context is usually provided by a set of comparable lotteries and not a single lottery. We present four axioms of a range-dependent preference relation to obtain axiomatic representation of the range-dependent utility model. For operational purposes as well as for prediction, a special case of the model is proposed, which is called the decision utility model. By imposing an additional axiom of shift and scale invariance, great multiplicity of range-dependent utility functions (different functions for different ranges) is reduced to a single decision utility function from which every range-dependent utility function is induced. Due to this axiom, the decision utility model belongs to the class of Constant Risk Aversion of Safra, Segal (1998). After providing axiomatic representation for the decision utility model, we analyze two crucial properties of the model: continuity and monotonicity wrt FOSD. We say that the model satisfies continuity if for every sequence of random variables converging in distribution to a given random variable, the Certainty Equivalent values of this sequence converges to the Certainty Equivalent of this random variable. We demonstrate that the model is in general discontinuous, i.e. there are jumps of indifference lines at those edges of the probability simplex which correspond to the change of lottery range (probability zero either for the highest or for the lowest lottery prize). We provide necessary and sufficient conditions for monotonicity. Intuitively, the conditions require that the marginal utility should be sufficiently small close to the lower and upper bound of the decision utility function domain and higher towards the middle of the domain. As a consequence, it is “easiest” to satisfy monotonicity if the decision utility function is S-shaped and it is “hardest” to satisfy monotonicity if it is inverse-S-shaped. Finally, we test the decision utility model by confronting it with the data. It is verified that the model accommodates a number of well known EU paradoxes, without recourse to probability weighting. The model is then fitted to experimental data. For binary lotteries the model is observationally equivalent to Yaari dual theory and it fits the Tversky, Kahnemann (1992) data well. For the case of three and more outcome-lotteries analyzed experimentally by Kontek (2015), the model fits the data better than any known model of choice under risk, including CPT. Interestingly, it is the S-shape of the decision utility function which a) best fits experimental data, b) is necessary to accommodate the EU paradoxes and c) satisfies monotonicity conditions.

**Michal Lewandowski**

#### An Additive Model of Decision Making under Uncertainty

Ying He; James S. Dyer; John C. Butler; Jianmin Jia

#### Abstract

In this paper, we model ambiguity as the anticipated receipt of different objective lotteries obtained in different states. Using Savage’s acts that map the set of states to the set of simple objective probability measures as the ambiguous lotteries faced by the decision maker, we develop preference conditions to separate “ambiguity” from “risk” to obtain an additive model of decision making under uncertainty consisting of two parts, i.e., the risk part and ambiguity part. This model explicitly captures the tradeoff between the magnitude of risk and the magnitude of ambiguity when making choice over ambiguous lotteries. The risk of an ambiguous lottery is evaluated by von Neumann and Morgenstern’s expected utility model; the ambiguity is evaluated by a nested utility model similar to the smooth ambiguity model developed by Klibanoff, Marinacci, and Mukerji (2005). A measure that ranks lotteries in terms of the magnitude of ambiguity can also be obtained using this separation. Using Taylor expansion, we show that this ambiguity measure is the variance of the compound lottery based on the second order subjective probability over possible objective lotteries. By applying our model to asset pricing under uncertainty, we show that the equity premium can be decomposed into two parts: the risk premium and the ambiguity premium, where the risk premium takes the standard form in the literature and the ambiguity premium is proportional to the ambiguity measure. Further, combining this model with the standard risk-value model developed by Jia and Dyer (1996), we obtain a risk-ambiguity-value preference model. This model is consistent with the mean-variance-ambiguity approximation of the smooth ambiguity model (Maccheroni, Marinacci, and Ruffino. 2013), but we do not need to rely on an approximation argument to obtain the additive form.

**Ying He**

#### Quasi-homotheticity Preferences over Lotteries

Thomas Epper

#### Abstract

We introduce a general characterization of Allais paradoxes. Our characterization motivates a generalized homotheticity condition which precludes particular Allais paradoxes while permitting others. Preferences over lotteries are quasi-homothetic (qHT(L)) if a scaling of two lotteries relative to a homothetic center does not revert preferences. The class of preferences obeying the qHT(L) axiom entails existing proposals to weaken independence as special cases. For example, if the homothetic center is a lottery inferior to the two lotteries under comparison, it precludes the classic common ratio effect. Similarly, if the homothetic center lies at infinity, it precludes the classic common consequence effect. A homothetic center being equal of the original two lotteries leads to the betweenness axiom. If the homothetic set consists of all lotteries in the n-simplex, we obtain mixture independence. If qHT(L) holds for comonotonic sets, it is equivalent to comonotonic independence. qHT(L) also yields to a number of interesting and novel results. First, if preference are homothetic with respect to two distinct homothetic centers not lying on the same indifference curve, they obey mixture independence. Second, a preference relation on a specific subdomain of the n-simplex can never imply both common ration and common consequence violations. Third, we propose a class of quasi-homothetic non-expected utility theories. Adopting an extended version of Bergson’s theorem, i.e. additive separability together with qHT(L) and the standard axioms, is equivalent to probability distorted non-expected utility model with a particular weighting function. Quadratic preferences (i.e. preferences obeying the mixture symmetry axiom) are a subclass of this representation. Restricting preferences to comonotonic sets of lotteries, further constrains this class of preferences to rank-dependent utility representations. Fourth, there is a direct link between violations of qHT(L) and dynamic inconsistency. Fifth, qHT(L) preferences can be recovered from choices using triangulation. We derive some direct tests of qHT and the large domain of theories it encompasses. qHT(L) helps to understand the drivers of various behavioral patterns documented in the literature. For example, we find that preferences for late resolution of uncertainty and preferences for one-shot resolution of uncertainty are driven by certain violations of qHT(L), and do not depend on other characteristics of the representation. Also, qHT(L) is crucial for the aggregation of preferences. Representative agent results are systematically biased as compared to population means when the qHT(L) axiom fails.

**Thomas Epper**

#### Local Prior Expected Utility: a Basis for Utility Representations under Uncertainty

Christian Nauerz; Andreas Perea

#### Abstract

One way of avoiding the Ellsberg paradox is to relax the Independence Axiom of Anscombe et al. (1963) (AA). This approach yielded two of the most well-known models of ambiguity: Schmeidler’s (1989) Choquet Expected Utility (CEU), and Gilboa and Schmeidler’s (1989) Maximin Expected Utility (MEU). Under both approaches the decision maker acts as if he maximizes utility with respect to a set of priors. In this paper we identify necessary assumptions on the preference relation to obtain a representation under which the decision maker maximizes utility with respect to a set of priors, called Local Prior Expected Utility (LEU). Moreover, we show that the prior used to evaluate a certain act is equal to the gradient of some appropriately defined utility mapping. We argue that the equality is not a mere technicality but coherent with the qualitative interpretation of a probabilistic belief. Based on this result we provide a unified approach to MEU, CEU, and AA and characterize the respective sets of priors. A preference relation on acts that satisfies the standard axioms Weak Order, Monotonicity, Continuity, Risk Independence, and Non-Degeneracy can be represented by a functional D that maps vectors of statewise utilities to “overall” utilities. We show that Monotonicity of D guarantees the existence of its Gâteaux derivative almost everywhere relying on a deep mathematical theorem by Chabrillac et al. (1987). Using only the Gâteaux derivative of D as an analytical tool, we prove characterization results corresponding to LEU, CEU, MEU, and AA. Within our approach we clearly identify the structure of the set of priors needed in these environments. As a basis for our approach, we identify an axiom called Independence of Certainty Equivalents that is weaker than Gilboa and Schmeidler’s (1989) C-Independence and weaker than Schmeidler’s (1989) Comonotonic Independence, but together with Weak Order, Monotonicity, Continuity, Risk Independence, Non-Degeneracy, and Uncertainty Aversion induces the same restrictions on the preference relation on acts as the Gilboa-Schmeidler axioms. We then show that without assuming Uncertainty Aversion we can represent a decision maker’s preferences by taking expectations of an affine utility function u with respect to a (possibly) different prior for every act, which we call LEU. The prior used is equal to the gradient of D at the vector of utilities induced by the act if the Gâteaux derivative exists. Moreover, we show that relaxing the Independence Axiom further by requiring invariance with respect to translations but not invariance to rescaling still results in a prior representation, where the gradient is take at a possibly different location. In case the Gâteaux derivative does not exist at a particular vector, we can approximate it by the Gâteaux derivatives of nearby acts.

**Christian Nauerz**