Asymmetry of risk and value of information

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.