Asymmetry of Risk and Value of Information

Abstract

The von Neumann and Morgenstern theory postulates that rational choice under uncertainty is equivalent to maximization of expected utility (EU). This view is mathematically appealing and natural because of the affine structure of the space of probability measures. Behavioural economists and psychologists, on the other hand, have demonstrated that humans consistently violate the EU postulate by switching from risk-averse to risk-taking behaviour. This paradox has led to the development of descriptive theories of decisions, such as the celebrated prospect theory, which uses an $S$-shaped value function with concave and convex branches explaining the observed asymmetry. Although successful in modelling human behaviour, these theories appear to contradict the natural set of axioms behind the EU postulate. Here we show that the observed asymmetry in behaviour can be explained if, apart from utilities of the outcomes, rational agents also value information communicated by random events. We review the main ideas of the classical value of information theory and its generalizations. Then we prove that the value of information is an $S$-shaped function, and that its asymmetry does not depend on how the concept of information is defined, but follows only from linearity of the expected utility. Thus, unlike many descriptive and `non-expected' utility theories that abandon the linearity (i.e. the `independence' axiom), we formulate a rigorous argument that the von Neumann and Morgenstern rational agents should be both risk-averse and risk-taking if they are not indifferent to information.Comment: 16 pages, 2 figure

Derivation and Application of the Subjective–Objective Probability Relationship from Entropy: The Entropy Decision Risk Model (EDRM)

“…Yang and Qui in Normalized Expected Utility-Entropy Measure of Risk, apply an additive entropy term in an attempt to model Prospect Theory and introduce the concept of redundancy, but do not subsequently make application [37]. Roman Belavkin, in Asymmetry of Risk and Value of Information, discusses many topics and even suggests application of entropy to Prospect Theory [38] and, in an earlier work, The Use of Entropy for Analysis and Control of Cognitive Models, Belavkin suggests the use of redundancy in estimating system accumulated information [39]. Even Tversky discussed information theory entropy as a measure of decision uncertainty in his paper, On the Optimal Number of Alternatives at a Choice Point, but this was not explored further [40].…”