expected utility hypothesis

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In economics, game theory, and decision theory the expected utility hypothesis is a theory of utilityin which "betting preferences" of people with regard to uncertain outcomes (gambles) arerepresented by a function of the payouts (whether in money or other goods), the probabilities of

occurrence, risk aversion, and the different utility of the same payout to people with different assetsor personal preferences. This theory has proved useful to explain some popular choices that seem tocontradict the expected value criterion (which takes into account only the sizes of the payouts andthe probabilities of occurrence), such as occur in the contexts of gambling and insurance. DanielBernoulli initiated this theory in 1738.

The von Neumann–Morgenstern utility theorem provides necessary and sufficient "rationality"

axioms under which the expected utility hypothesis holds.[1]

In the presence of risky outcomes, a decision maker could use the expected value criterion as a rule

of choice: higher expected value investments are simply the preferred ones. For example, supposethere is a gamble in which the probability of getting a $100 payment is 1 in 80 and the alternative,and far more likely, outcome, is getting nothing. Then the expected value of this gamble is $1.25.Given the choice between this gamble and a guaranteed payment of $1, by this simple expectedvalue theory people would choose the $100-or-nothing gamble. However, under expected utilitytheory, some people would be risk averse enough to prefer the sure thing, even though it has a lower expected value, while other less risk averse people would still choose the riskier, higher-meangamble.

Nicolas Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory canalso more accurately describe more realistic scenarios (where expected values are finite) than

Expected utility hypothesis

1 Expected value and choice under risk 2 Bernoulli's formulation 3 Infinite expected value — St. Petersburg paradox 4 von Neumann–Morgenstern formulation

4.1 The von Neumann-Morgenstern axioms 4.2 Risk aversion 4.3 Examples of von Neumann-Morgenstern utility functions 4.4 Measuring risk in the expected utility context

5 Behavioral decision science: Deviations from expected utility theory 5.1 Conservatism in updating beliefs 5.2 Irrational deviations 5.3 Preference reversals over uncertain outcomes 5.4 Uncertain probabilities

6 See also 7 References 8 Further reading

Contents

Expected value and choice under risk

Bernoulli's formulation

of 8Expected utility hypothesis - Wikipedia, the free encyclopedia

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expected value alone.

In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate moneyin proportion to its quantity, and men of good sense in proportion to the usage that they may make of

it."[2]

In 1738, Nicolas' cousin Daniel Bernoulli, published the canonical 18th Century description of thissolution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the

easurement of Risk .[3]

Daniel Bernoulli proposed that a mathematical function should be used to correct the expected valuedepending on probability. This provides a way to account for risk aversion, where the risk premiumis higher for low-probability events than the difference between the payout level of a particular outcome and its expected value.

Bernoulli's paper was the first formalization of marginal utility, which has broad application ineconomics in addition to expected utility theory. He used this concept to formalize the idea that thesame amount of additional money was less useful to an already-wealthy person than it would be to a

poor person.

Main article: St. Petersburg paradox

The St. Petersburg paradox (named after the journal in which Bernoulli's paper was published) ariseswhen there is no upper bound on the potential rewards from very low probability events. Becausesome probability distribution functions have an infinite expected value, an expected-wealthmaximizing person would pay an infinite amount to take this gamble. In real life, people do not do

this.

Bernoulli proposed a solution to this paradox in his paper: the utility function used in real life meansthat the expected utility of the gamble is finite, even if its expected value is infinite. (Thus hehypothesized diminishing marginal utility of increasingly larger amounts of money.) It has also beenresolved differently by other economists by proposing that very low probability events are neglected,

by taking into account the finite resources of the participants, or by noting that one simply cannot buy that which is not sold (and that sellers would not produce a lottery whose expected loss to themwere unacceptable).

Main article: Von Neumann–Morgenstern utility theorem

There are four axioms[4] of the expected utility theory that define a rational decision maker. Theyare completeness, transitivity, independence and continuity.

Completeness assumes that an individual has well defined preferences and can decide between twoalternatives.

Axiom (Completeness): For every A and B either A < B , A > B or A = B (this means: A is worsethan B, better, or equally good) holds.

Infinite expected value St. Petersburg paradox

von Neumann Morgenstern formulation

The von Neumann-Morgenstern axioms





Transitivity assumes that, as an individual decides according to the completeness axiom, theindividual also decides consistently.

Axiom (Transitivity): For every A, B and C with and we must have .

Independence also pertains to well-defined preferences and assumes that two gambles mixed with a

third one maintain the same preference order as when the two are presented independently of thethird one. The independence axiom is the most controversial one.

Axiom (Independence): Let A, B, and C be three lotteries with , and let ; then

.

Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to Band B to C, then there should be a possible combination of A and C in which the individual is thenindifferent between this mix and the lottery B.

Axiom (Continuity): Let A, B and C be lotteries with A > B > C ; then there exists a probability p

such that B is equally good as pA + (1 − p)C .

If all these axioms are satisfied, then the individual is said to be rational and the preferences can berepresented by a utility function. In other words: if an individual always chooses his/her most

preferred alternative available, then the individual will choose one gamble over another if and only if there is a utility function such that the expected utility of one exceeds that of the other. The expectedutility of any gamble may be expressed as a linear combination of the utilities of the outcomes,withthe weights being the respective probabilities. Utility functions are also normally continuousfunctions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utilityfunctions. This is a central theme of the expected utility hypothesis in which an individual choosesnot the highest expected value, but rather the highest expected utility. The expected utility

maximizing individual makes decisions rationally based on the axioms of the theory.

The von Neumann–Morgenstern formulation is important in the application of set theory toeconomics because it was developed shortly after the Hicks-Allen "ordinal revolution" of the 1930s,

and it revived the idea of cardinal utility in economic theory.[citation needed ] Note, however, thatwhile in this context the utility function is cardinal, in that implied behavior would be altered by anon-linear monotonic transformation of utility, the expected utilty function is ordinal because anymonotonic increasing transformation of it gives the same behavior.

The expected utility theory implies that rational individuals act as though they were maximizingexpected utility, and allows for the fact that many individuals are risk averse,[citation needed ] meaningthat the individual would refuse a fair gamble (a fair gamble has an expected value of zero). Risk aversion implies that their utility functions are concave and show diminishing marginal wealthutility. The risk attitude is directly related to the curvature of the utility function: risk neutralindividuals have linear utility functions, while risk seeking individuals have convex utility functionsand risk averse individuals have concave utility functions. The degree of risk aversion can bemeasured by the curvature of the utility function.

Since the risk attitudes are unchanged under affine transformations of u, the first derivative u' is notan adequate measure of the risk aversion of a utility function. Instead, it needs to be normalized. This

leads to the definition of the Arrow–Pratt[5][6] measure of absolute risk aversion:

Risk aversion





The Arrow–Pratt measure of relative risk aversion is:

Special classes of utility functions are the CRRA (constant relative risk aversion) functions, whereRRA(x) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) isconstant. They are often used in economics for simplification.

A decision that maximizes expected utility also maximizes the probability of the decision'sconsequences being preferable to some uncertain threshold (Castagnoli and LiCalzi,1996; Bordleyand LiCalzi,2000;Bordley and Kirkwood, ). In the absence of uncertainty about the threshold,expected utility maximization simplifies to maximizing the probability of achieving some fixed

target. If the uncertainty is uniformly distributed, then expected utility maximization becomesexpected value maximization. Intermediate cases lead to increasing risk-aversion above some fixedthreshold and increasing risk-seeking below a fixed threshold.

Further information: Risk aversion

The utility function u(w) = log(w) was originally suggested by Bernoulli (see above). It hasrelative risk aversion constant and equal to one, and is still sometimes assumed in economic

analyses. The utility function u(w) = − e − aw exhibits constant absolute risk aversion, and for this

reason is often avoided, although it has the advantage of offering substantial mathematicaltractability when asset returns are normally distributed. Note that, as per the affine transformation

property alluded to above, the utility function K − e − aw gives exactly the same preferences

orderings as does − e − aw; thus it is irrelevant that the values of − e − aw and its expected valueare always negative: what matters for preference ordering is which of two gambles gives the higher expected utility, not the numerical values of those expected utilities.

The class of constant relative risk aversion utility functions contains three categories. Bernoulli'sutility function

u(w) = log(w)

has relative risk aversion equal to unity. The functions

u(w) = wα

for have relative risk aversion equal to 1 − α. And the functions

u(w) = − wα

for α < 0 also have relative risk aversion equal to 1 − α.

See also the discussion of utility functions having hyperbolic absolute risk aversion (HARA).

Examples of von Neumann-Morgenstern utility functions

Measuring risk in the expected utility context





Often people refer to "risk" in the sense of a potentially quantifiable entity. In the context of mean-variance analysis, variance is used as a risk measure for portfolio return; however, this is only valid

if returns are normally distributed or otherwise jointly elliptically distributed.[7][8][9] However, D. E.

Bell[10] proposed a measure of risk which follows naturally from a certain class of von Neumann-

Morgenstern utility functions. Let utility of wealth be given by u(w) = w − be − aw for individual-

specific positive parameters a and b. Then expected utility is given by

Thus the risk measure is , which differs between two individuals if they have

different values of the parameter a, allowing different people to disagree about the degree of risk associated with any given portfolio.

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 donot guarantee that expected utility theory is a reliable guide to human behavior or optimal practice.Similarly, the mathematical correctness and practical salience of classical mechanics does not imply

that engineers may build bridges and dams without errors; indeed, civil engineering projects result instructural failures in dams and bridges in roughly five percent of cases[citation needed ].

The mathematical clarity of expected utility theory has helped scientists design experiments to testits adequacy, and to distinguish systematic departures from its predictions.

It is well established that humans find logic hard, mathematics harder, and probability even more

challenging[citation needed ]. Therefore, it is unsurprising that psychologists have discoveredsystematic violations of probability calculations and behavior by humans.

In updating probability distributions using evidence, a standard method uses conditional probability,namely the rule of Bayes. Some experiments on belief revision have suggested that humans change

their beliefs faster when using Bayesian methods than when using informal judgment.[11]

Behavioral finance has produced several generalized expected utility theories to account for instances where people's choice deviate from those predicted by expected utility theory. Thesedeviations are described as "irrational" because they can depend on the way the problem is

presented, not on the actual costs,rewards, or probabilities involved.

Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect

theory and SP/A theory[12]

Behavioral decision science: Deviations from expected utilitytheory

Conservatism in updating beliefs

Irrational deviations





Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimesexhibit 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 ahigh 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, theyfrequently prefer the "p bets" over "$ bets."[13] Many studies have examined this "preference

reversal," from both an experimental (e.g., Plott & Grether, 1979)[14] and theoretical (e.g., Holt,

1986)[15] standpoint, indicating that this behavior can be brought into accordance with neoclassicaleconomic theory under some specific assumptions.

Applying expected value and expected utility to decision-making requires knowing the probability of various outcomes. However, this is unknown in practice: one is operating under uncertainty (ineconomics, one talks of Knightian uncertainty). Thus one must make assumptions, but then theexpected value of various decisions is 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.

Alternative decision techniques are robust to uncertainty of probability of outcomes, either notdepending on probabilities of outcomes and only requiring scenario analysis (as in minimax or minimax regret), or being less sensitive to assumptions.

Allais paradox

Bayesian probability Behavioral economics Decision theory Subjective expected utility generalized expected utility Rank-dependent expected utility Prospect theory Risk in psychology Risk aversion ambiguity aversion Marginal utility

Two-moment decision models

1. ^ http://cepa.newschool.edu/het/essays/uncert/vnmaxioms.htm2. ^ [1]3. ^ Bernoulli, Daniel; Originally published in 1738; translated by Dr. Lousie Sommer. (January 1954).

"Exposition of a New Theory on the Measurement of Risk". Econometrica (The Econometric Society)22 (1): 22–36. doi:10.2307/1909829. http://www.math.fau.edu/richman/Ideas/daniel.htm. Retrieved2006-05-30.

4. ^ Neumann, John von, and Morgenstern, Oskar, Theory of Games and Economic Behavior , Princeton, NJ, Princeton University Press, 1944, second ed. 1947, third ed. 1953.

5. ^ Arrow, K.J.,1965, "The theory of risk aversion," in Aspects of the Theory of Risk Bearing , by YrjoJahnssonin Saatio, Helsinki. Reprinted in: Essays in the Theory of Risk Bearing , Markham Publ. Co.,Chicago, 1971, 90-109.

6. ^ Pratt, J. W. (January–April 1964). "Risk aversion in the small and in the large". Econometrica 32 (1/2):

Preference reversals over uncertain outcomes

Uncertain probabilities

See also

References





122–136. doi:10.2307/1913738. http://jstor.org/stable/1913738.7. ^ Borch, K. (January 1969). "A note on uncertainty and indifference curves". Review of Economic

Studies 36 (1): 1–4. doi:10.2307/2296336. http://jstor.org/stable/2296336.8. ^ Chamberlain, G. (1983). "A characterization of the distributions that imply mean-variance utility

functions". Journal of Economic Theory 29: 185–201. doi:10.1016/0022-0531(83)90129-1.9. ^ 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. doi:10.2307/2328079.

http://jstor.org/stable/2328079.10. ^ Bell, D.E. (December 1988). "One-switch utility functions and a measure of risk". Management Science 34: 1416–24. doi:10.1287/mnsc.34.12.1416.

11. ^ Subjects changed their beliefs faster by conditioning on evidence (Bayes's theorem) than by usinginformal 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 Daniel Kahneman, Paul Slovic and Amos Tversky. Judgment under uncertainty:

Heuristics and biases. Cambridge University Press.Phillips, L.D.; Edwards, W. (October 2008). "Chapter 6: Conservatism in a simple probability inferencetask ( Journal of Experimental Psychology (1966) 72: 346-354)". In Jie W. Weiss and David J. Weiss. AScience of Decision Making:The Legacy of Ward Edwards. Oxford University Press. pp. 536.ISBN 9780195322989.

12. ^ Acting Under Uncertainty: Multidisciplinary Conceptions by George M. von Furstenberg. Springer,1990. ISBN 0-7923-9063-6, 9780792390633. 485 pages

13. ^ Lichtenstein, S.; P. Slovic (1971). "Reversals of preference between bids and choices in gamblingdecisions". Journal of Experimental Psychology 89 (1): 46–55. ISSN 0096-3445.http://psycnet.apa.org/journals/xge/89/1/46/.

14. ^ Grether, David M.; Plott, Charles R. (1979). "Economic Theory of Choice and the Preference ReversalPhenomenon". American Economic Review 69 (4): 623–638. JSTOR 1808708.

15. ^ Holt, Charles (1986). "Preference Reversals and the Independence Axiom". American Economic Review 76 (3): 508–515. JSTOR 1813367.

Charles Sanders Peirce and Joseph Jastrow (1885). "On Small Differences in Sensation". Memoirs of the National Academy of Sciences 3: 73–83.

http://psychclassics.yorku.ca/Peirce/small-diffs.htm Ramsey, Frank Plumpton; “Truth and Probability” (PDF), Chapter VII in The Foundations of

Mathematics and other Logical Essays (1931). 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,

de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of the1937 article in French) in H. E. Kyburg and H. E. Smokler (eds), Studies in Subjective

Probability, New York: Wiley, 1964.

de Finetti, Bruno. Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974-5.

Donald Davidson, Patrick Suppes and Sidney Siegel (1957). Decision-Making: An Experimental Approach. Stanford University Press.

Pfanzagl, J (1967). "Subjective Probability Derived from the Morgenstern-von NeumannUtility Theory". In Martin Shubik. Essays in Mathematical Economics In Honor of Oskar

Morgenstern. Princeton University Press. pp. 237–251. Pfanzagl, J. in cooperation with V. Baumann and H. Huber (1968). "Events, Utility and

Subjective Probability". Theory of Measurement . Wiley. pp. 195–220. Morgenstern, Oskar (1976). "Some Reflections on Utility". In Andrew Schotter. Selected

Economic Writings of Oskar Morgenstern. New York University Press. pp. 65–70.ISBN 0814777716.





Schoemaker PJH (1982). "The Expected Utility Model: Its Variants, Purposes, Evidence andLimitations". Journal of Economic Literature 20: 529–563.

Anand P. (1993). Foundations of Rational Choice Under Risk . Oxford: Oxford UniversityPress. ISBN 0198233035.

Arrow K.J. (1963). "Uncertainty and the Welfare Economics of Medical Care". American Economic Review 53: 941–73.

Scott Plous (1993) "The psychology of judgment and decision making", Chapter 7(specifically) and 8,9,10, (to show paradoxes to the theory).

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