An explicit representation for disappointment aversion and other betweenness preferences

Abstract

One of the most well known models of non‐expected utility is Gul's (1991) model of disappointment aversion. This model, however, is defined implicitly, as the solution to a functional equation; its explicit utility representation is unknown, which may limit its applicability. We show that an explicit representation can be easily constructed, using solely the components of the implicit representation. We also provide a more general result: an explicit representation for preferences in the betweenness class that also satisfy negative certainty independence (Dillenberger 2010) or its counterpart. We show how our approach gives a simple way to identify the parameters of the representation behaviorally and to study the consequences of disappointment aversion in a variety of applications.

Monotone Additive Statistics

“…These preferences have the standard EDU representation, but perhaps with a negative discount rate, as we explain in Section 3.3. 2 Applied to choice under risk, our representation also bears resemblance to cautious expected utility theory (Cerreia-Vioglio, Dillenberger, and Ortoleva (2015)), in which a gamble is evaluated according to the minimum certainty equivalent across a family of utility functions. The two representations are conceptually related, as both involve uncertainty over a utility function.…”

“…The two representations are conceptually related, as both involve uncertainty over a utility function. Our axioms are, however, different in that we study preferences that are invariant to adding an independent gamble, while Cerreia-Vioglio, Dillenberger, and Ortoleva (2015) considered the effect of mixing with another gamble.…”

“…At the same time, a potential difficulty for this class of preferences is that their defining parameter, the measure μ over the coefficient of risk aversion, is infinite‐dimensional. To narrow down the parameter space, we focus on those preferences that also satisfy betweenness , a well‐known weakening of the independence axiom that has been extensively studied in the literature (see Dekel (1986), Gul (1991), Cerreia‐Vioglio, Dillenberger, and Ortoleva (2020)). We show that a preference represented by a monotone additive statistic Φ satisfies betweenness if and only if it is of the form The parameter controls the relative weights of the risk‐averse and risk‐seeking components, with increased β making the decision maker more risk‐averse.…”