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.
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Notes
- 1.
This property is sometimes called completeness, but this term often has other meanings in order theory (e.g. complete partial order) or topology (e.g. complete metric space).
- 2.
We use the notion of a closed functional, because the topology in Y is not defined.
- 3.
An affine functional h and a linear functional \(u(y) = h(y) - h(0)\) have isomorphic level sets.
- 4.
Note that \(\underline{u}_{S}(\lambda )\neq -\overline{u}_{S}(\lambda )\) in general, and one of the branches may be empty.
References
Allais, M.: Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’École americaine. Econometrica 21, 503–546 (1953)
Belavkin, R.V.: Optimal measures and Markov transition kernels. J. Global Optim. 55, 387–416 (2013)
Bernoulli, D.: Commentarii acad. Econometrica 22, 23–36 (1954)
Bourbaki, N.: Eléments de mathématiques. Intégration. Hermann, Paris (1963)
Debreu, G.: Representation of a preference relation by a numerical function. In: Thrall, R.M., Coombs, C.H., Davis, R.L. (eds.) Decision Process. Wiley, New York (1954)
Ellsberg, D.: Risk, ambiguity, and the Savage axioms. Q. J. Econom. 75(4), 643–669 (1961)
Grishanin, B.A., Stratonovich, R.L.: Value of information and sufficient statistics during an observation of a stochastic process (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 6, 4–14 (1966)
Huck, S., Müller, W.: Allais for all: revisiting the paradox in a large representative sample. J. Risk Uncertainty 44(3), 261–293 (2012)
Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263–292 (1979)
Khinchin, A.I.: Mathematical Foundations of Information Theory. Dover, New York (1957)
Kreps, D.M.: Notes on the Theory of Choice. Westview Press, Colorado (1988)
Kullback, S.: Information Theory and Statistics. Wiley, New York (1959)
List, J.A., Haigh, M.S.: A simple test of expected utility theory using professional traders. PNAS 102(3), 945–948 (2005)
Loomes, G., Sugden, R.: Regret theory: an alternative theory of rational choice under uncertainty. Econom. J. 92(368), 805–824 (1982)
Machina, M.J.: “Expected utility” Analysis without the independence axiom. Econometrica 50(2), 277–323 (1982)
Machina, M.J.: States of the world and the state of decision theory, chap. 2. In: Meyer, D. (ed.) The Economics of Risk. W. E. Upjohn Institute for Employment Research, Kalamazoo (2003)
Machina, M.J.: Nonexpected utility theory. In: Teugels, J.L., Sundt, B. (eds.) Encyclopedia Of Actuarial Science, vol. 2, pp. 1173–1179. Wiley, Chichester (2004)
von Neumann, J., Morgenstern, O.: Theory of games and economic behavior, first edn. Princeton University Press, Princeton (1944)
Rockafellar, R.T.: Conjugate duality and optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics, Philadelphia (1974)
Savage, L.: The Foundations of Statistics. Wiley, New York (1954)
Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)
Starmer, C.: Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk. J. Econ. Liter. pp. 332–382 (2000)
Stratonovich, R.L.: On value of information (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 5, 3–12 (1965)
Stratonovich, R.L.: Value of information during an observation of a stochastic process in systems with finite state automata (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 5, 3–13 (1966)
Stratonovich, R.L.: Extreme problems of information theory and dynamic programming (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 5, 63–77 (1967)
Stratonovich, R.L.: Information Theory (in Russian). Sovetskoe Radio, Moscow (1975)
Stratonovich, R.L., Grishanin, B.A.: Value of information when an estimated random variable is hidden (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 3, 3–15 (1966)
Stratonovich, R.L., Grishanin, B.A.: Game-theoretic problems with information constraints (in Russian). Izvestiya USSR Acad. Sci. Tech. Cybern. 1, 3–12 (1968)
Tikhomirov, V.M.: Analysis II,chap. In: Convex Analysis. Encyclopedia of Mathematical Sciences, vol. 14, pp. 1–92. Springer, New York (1990)
Tversky, A., Kahneman, D.: The framing of decisions and the psychology of choice. Science 211, 453–458 (1981)
Wald, A.: Statistical Decision Functions. Wiley, New York (1950)
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This work was supported by UK EPSRC grant EP/H031936/1.
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Belavkin, R.V. (2014). Asymmetry of Risk and Value of Information. In: Vogiatzis, C., Walteros, J., Pardalos, P. (eds) Dynamics of Information Systems. Springer Proceedings in Mathematics & Statistics, vol 105. Springer, Cham. https://doi.org/10.1007/978-3-319-10046-3_1
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