Abstract
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. 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. This general insight, which we explain to raise several interesting questions for further research, is our main contribution.
Highlights
The traditional risk attitude concepts of economics are defined with reference to the more primitive notion of an increase in risk
We have shown that under expected utility, weak and strong comparative risk aversion are characterized by the same condition when the compared utility functions have convex range, which was already known, and when they have finite range, which had not been hitherto established
We have shown that, under expected utility still, weak comparative risk aversion is characterized by the same condition in the finite and the convex case, introducing a new kind of comparisons to the literature
Summary
The traditional risk attitude concepts of economics are defined with reference to the more primitive notion of an increase in risk. 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. The take-home message from our conclusions on these two distinct issues is that like absolute risk attitudes, comparative risk attitudes help better understand the fundamental divide between expected and nonexpected utility, more generally, the structural properties of the main models of decision-making under risk We consider this general conceptual insight, which we explain to raise several interesting questions for further research, to be the main contribution of our investigation of risk attitudes over finite domains.
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