Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

Alternative Approaches to Comparative nth-Degree Risk Aversion

This paper extends the three main approaches to comparative risk aversion—the risk premium approach and the probability premium approach of Pratt (1964) [Risk aversion in the small and in the large. Econometrica 32(1-2):122–136] and the comparative statics approach of Jindapon and Neilson (2007) [Higher-order generalizations of Arrow-Pratt and Ross risk aversion: A comparative statics approach. J. Econom. Theory 136(1):719–728]—to study comparative nth-degree risk aversion. These extensions can accommodate trading off an nth-degree risk increase and an mth-degree risk increase for any m, such that [Formula: see text]. It goes on to show that, in the expected utility framework, all of these general notions of comparative nth-degree risk aversion are equivalent and can be characterized by the concept of (n/m)th-degree Ross more risk aversion of Liu and Meyer (2013) [Substituting one risk increase for another: A method for measuring risk aversion. J. Econom. Theory 148(6):2706–2718]. This paper was accepted by Han Bleichrodt, decision analysis.

Comparative Risk Aversion in the Presence of Ambiguity

This paper suggests a characterization of increases in risk aversion within the smooth ambiguity model by Klibanoff, Marinacci, and Mukerji (2005). I show that an increase in risk aversion is qualitatively different from that under expected utility, due to the incomplete separation between risk and ambiguity attitude. The analysis clarifies how ambiguity perception and attitude depend on risk aversion. (JEL D81)