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# Expected utility hypothesis

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### Expected utility hypothesis

In economics, game theory, and decision theory the expected utility hypothesis is a hypothesis concerning people's preferences with regard to choices that have uncertain outcomes (gambles). This hypothesis states that if specific axioms are satisfied, the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble. This hypothesis has proved useful to explain some popular choices that seem to contradict the expected value criterion (which takes into account only the sizes of the payouts and the probabilities of occurrence), such as occur in the contexts of gambling and insurance. Daniel Bernoulli initiated this hypothesis in 1738. Until the mid-twentieth century, the standard term for the expected utility was the moral expectation, contrasted with "mathematical expectation" for the expected value.[1]

The

• de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974-5.
• http://psychclassics.yorku.ca/Peirce/small-diffs.htm
• Pfanzagl, J (1967). "Subjective Probability Derived from the
• Pfanzagl, J. in cooperation with V. Baumann and H. Huber (1968). "Events, Utility and Subjective Probability". Theory of Measurement. Wiley. pp. 195–220.
• Scott Plous (1993) "The psychology of judgment and decision making", Chapter 7 (specifically) and 8,9,10, (to show paradoxes to the theory).
• Ramsey, Frank Plumpton; “Truth and Probability” (PDF), Chapter VII in The Foundations of Mathematics and other Logical Essays (1931).
• Schoemaker PJH (1982). "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations". Journal of Economic Literature 20: 529–563.
de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the 1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective Probability, New York: Wiley, 1964.
• Anand P. (1993). Foundations of Rational Choice Under Risk. Oxford: Oxford University Press.
• Arrow K.J. (1963). "Uncertainty and the Welfare Economics of Medical Care". American Economic Review 53: 941–73.
• de Finetti, Bruno. "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in Erkenntnis, volume 31, September 1989.
• de Finetti, Bruno. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré,

1. ^ "Moral expectation", under Jeff Miller, Earliest Known Uses of Some of the Words of Mathematics (M), accessed 2011-03-24. The term "utility" was first introduced mathematically in this connection by Jevons in 1871; previously the term "moral value" was used.
2. ^ Anand, P. (1993) Foundations of Rational Choice Under Risk, Oxford, Oxford University Press. [1]
3. ^ http://cerebro.xu.edu/math/Sources/NBernoulli/correspondence_petersburg_game.pdf
4. ^
5. ^ Neumann, John von, and Morgenstern, Oskar, Theory of Games and Economic Behavior, Princeton, NJ, Princeton University Press, 1944, second ed. 1947, third ed. 1953.
6. ^ Arrow, K.J.,1965, "The theory of risk aversion," in Aspects of the Theory of Risk Bearing, by Yrjo Jahnssonin Saatio, Helsinki. Reprinted in: Essays in the Theory of Risk Bearing, Markham Publ. Co., Chicago, 1971, 90-109.
7. ^ Pratt, J. W. (January–April 1964). "Risk aversion in the small and in the large". Econometrica 32 (1/2): 122–136.
8. ^ Borch, K. (January 1969). "A note on uncertainty and indifference curves". Review of Economic Studies 36 (1): 1–4.
9. ^ Chamberlain, G. (1983). "A characterization of the distributions that imply mean-variance utility functions". Journal of Economic Theory 29 (1): 185–201.
10. ^ Owen, J., Rabinovitch, R. (1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice". Journal of Finance 38 (3): 745–752.
11. ^ Bell, D.E. (December 1988). "One-switch utility functions and a measure of risk". Management Science 34 (12): 1416–24.
12. ^ Daniel Kahneman; Amos Tversky (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, Vol. 47, No. 2., pp. 263-292.
13. ^ Subjects changed their beliefs faster by conditioning on evidence (Bayes's theorem) than by using informal reasoning, according to a classic study by the psychologist Ward Edwards: Edwards, Ward (1968). "Conservatism in Human Information Processing". In Kleinmuntz, B. Formal Representation of Human Judgment. Wiley. Edwards, Ward (1982). "Conservatism in Human Information Processing (excerpted)". In
Phillips, L.D.; Edwards, W.; Edwards, Ward (October 2008). "Chapter 6: Conservatism in a simple probability inference task (Journal of Experimental Psychology (1966) 72: 346-354)". In Jie W. Weiss and David J. Weiss. A Science of Decision Making:The Legacy of Ward Edwards. Oxford University Press. p. 536.
14. ^ Acting Under Uncertainty: Multidisciplinary Conceptions by George M. von Furstenberg. Springer, 1990. ISBN 0-7923-9063-6, ISBN 978-0-7923-9063-3. 485 pages.
15. ^ Lichtenstein, S.; P. Slovic (1971). "Reversals of preference between bids and choices in gambling decisions". Journal of Experimental Psychology 89 (1): 46–55.
16. ^ Grether, David M.; Plott, Charles R. (1979). "Economic Theory of Choice and the Preference Reversal Phenomenon".
17. ^ Holt, Charles (1986). "Preference Reversals and the Independence Axiom".

## References

Bayesian approaches to probability treat it as a degree of belief and thus they do not draw a distinction between risk and a wider concept of uncertainty: they deny the existence of Knightian uncertainty. They would model uncertain probabilities with hierarchical models, i.e. where the uncertain probabilities are modelled as distributions whose parameters are themselves drawn from a higher-level distribution (hyperpriors).

Alternative decision techniques are robust to uncertainty of probability of outcomes, either not depending on probabilities of outcomes and only requiring scenario analysis (as in minimax or minimax regret), or being less sensitive to assumptions.

If one is using the frequentist notion of probability, where probabilities are considered to be facts, then applying expected value and expected utility to decision-making requires knowing the probability of various outcomes. However, in practice there will be many situations where the probabilities are unknown, one is operating under uncertainty. In economics, one talks of Knightian uncertainty or Ambiguity. Thus one must make assumptions about the probabilities, but then the expected value of various decisions can be very sensitive to the assumptions. This is particularly a problem when the expectation is dominated by rare extreme events, as in a long-tailed distribution.

### Uncertain probabilities

Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regard to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "$bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "$ bets."[15] Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979)[16] and theoretical (e.g., Holt, 1986)[17] standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under specific assumptions.

### Preference reversals over uncertain outcomes

Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory and SP/A theory.[14]

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choices deviate from those predicted by expected utility theory. These deviations are described as "irrational" because they can depend on the way the problem is presented, not on the actual costs,rewards, or probabilities involved.

### Irrational deviations

In updating probability distributions using evidence, a standard method uses conditional probability, namely the rule of Bayes. An experiment on belief revision has suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.[13]

It is well established that humans find logic hard, mathematics harder, and probability even more challenging. Psychologists have discovered systematic violations of probability calculations and behavior by humans. Consider, for example, the Monty Hall problem.

### Conservatism in updating beliefs

The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions. This has led to the field of behavioral finance, which has produced deviations from expected utility theory to account for the empirical facts.

Like any mathematical model, expected utility theory is an abstraction and simplification of reality. The mathematical correctness of expected utility theory and the salience of its primitive concepts do not guarantee that expected utility theory is a reliable guide to human behavior or optimal practice.

## Contents

• Expected value and choice under risk 1
• Bernoulli's formulation 2
• Infinite expected value — St. Petersburg paradox 3
• Von Neumann–Morgenstern formulation 4
• The von Neumann-Morgenstern axioms 4.1
• Risk aversion 4.2
• Examples of von Neumann-Morgenstern utility functions 4.3
• Measuring risk in the expected utility context 4.4
• Criticism 5
• Conservatism in updating beliefs 5.1
• Irrational deviations 5.2
• Preference reversals over uncertain outcomes 5.3
• Uncertain probabilities 5.4
• References 7

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