  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.
