# Expected utility hypothesis

expected utilityexpected utility theoryvon Neumann-Morgenstern utility functionexpected-utilitychoice under uncertaintydecision making under riskexpectedExpected Utility Theory (EUT)expected value of a von Neumann-Morgenstern utility functionexpected'' utility
In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)⁠—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes.wikipedia
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### Decision theory

decision sciencestatistical decision theorydecision sciences
In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)⁠—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes.
These rules may, for instance, have a procedural framework (e.g. Amos Tversky's elimination by aspects model) or an axiomatic framework (e.g. stochastic transitivity axioms), reconciling the Von Neumann-Morgenstern axioms with behavioral violations of the expected utility hypothesis, or they may explicitly give a functional form for time-inconsistent utility functions (e.g. Laibson's quasi-hyperbolic discounting).

### Von Neumann–Morgenstern utility theorem

von Neumann–Morgenstern utility functionvon Neumann and MorgensternVon Neumann–Morgenstern utility
The von Neumann–Morgenstern utility theorem provides necessary and sufficient conditions under which the expected utility hypothesis holds.
The theorem is the basis for expected utility theory.

The introduction of St. Petersburg Paradox by Daniel Bernoulli in 1738 is considered the beginnings of the hypothesis.
The classical resolution of the paradox involved the explicit introduction of a utility function, an expected utility hypothesis, and the presumption of diminishing marginal utility of money.

### Gabriel Cramer

CramerCramer, Gabriel
In 1728, Gabriel Cramer, in a letter to Nicolas Bernoulli, wrote, "the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it."
In 1728 he proposed a solution to the St. Petersburg Paradox that came very close to the concept of expected utility theory given ten years later by Daniel Bernoulli.

### Marginal utility

marginal benefitdiminishing marginal utilitymarginal revolution
Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory.
In contrast, the concept of diminishing marginal utility is meaningful in the context of cardinal utility, which in modern economics is used in analyzing intertemporal choice, choice under uncertainty, and social welfare.

### Cardinal utility

cardinalcardinal (numeric) utility functionsCardinal measures
The von Neumann–Morgenstern formulation is important in the application of set theory to economics 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.
The idea of cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where it is still applied.

### Rationality

rationalrational thoughtrational thinking
From relatively early on, it was accepted that some of these conditions would be violated by real decision-makers in practice but that the conditions could be interpreted nonetheless as 'axioms' of rational choice.
Within artificial intelligence, a rational agent is typically one that maximizes its expected utility, given its current knowledge.

### Risk aversion

risk averserisk-averserisk tolerance
According to expected value theory, people should choose the $100-or-nothing gamble; however, as stressed by expected utility theory, some people are risk averse enough to prefer the sure thing, despite its lower expected value. Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where RRA(w) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) is constant. Daniel Bernoulli proposed that a nonlinear function of utility of an outcome should be used instead of the expected value of an outcome, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value. In expected utility theory, an agent has a utility function u(c) where c represents the value that he might receive in money or goods (in the above example c could be$0 or $40 or$100).

### Prospect theory

Daniel Kahneman and Amos Tversky in 1979 presented their prospect theory which showed empirically, among other things, how preferences of individuals are inconsistent among the same choices, depending on how those choices are presented. Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory and SP/A theory.
It challenges the expected utility theory, developed by John von Neumann and Oskar Morgenstern in 1944, and earned Daniel Kahneman the Nobel Memorial Prize in Economics in 2002.

### Generalized expected utility

Expected Utility Theorygeneralizationsnot an expected utility maximizer
Behavioral finance has produced several generalized expected utility theories to account for
Generalized expected utility is a decision-making metric based on any of a variety of theories that attempt to resolve some discrepancies between expected utility theory and empirical observations, concerning choice under risky (probabilistic) circumstances.

### Bayesian probability

Bayesiansubjective probabilityBayesianism
Savage's framework has since been used in neo-Bayesian statistics (see Bayesian probability) and the field of applied statistics.
Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.

### Subjective theory of value

Economic subjectivismsubjectivesubjective value theory
In economics, game theory, and decision theory, the expected utility hypothesis—concerning people's preferences with regard to choices that have uncertain outcomes (gambles)⁠—states that the subjective value associated with an individual's gamble is the statistical expectation of that individual's valuations of the outcomes of that gamble, where these valuations may differ from the dollar value of those outcomes.

### Rational choice theory

rational choicechoice theoryrational
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.
In recent years, the most prevalent version of rational choice theory, expected utility theory, has been challenged by the experimental results of behavioral economics.

### Isoelastic utility

constant relative risk aversionIsoelasticpower utility function
Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where RRA(w) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) is constant.
When the context involves risk, the utility function is viewed as a von Neumann-Morgenstern utility function, and the parameter \eta is the degree of relative risk aversion.

### Concave function

concaveconcavityconcave down
Risk aversion implies that their utility functions are concave and show diminishing marginal wealth utility.

Daniel Bernoulli proposed that a nonlinear function of utility of an outcome should be used instead of the expected value of an outcome, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value.
Let an individual's increasing, concave von Neumann-Morgenstern utility function be u, let r f be the return on the risk-free asset, and let r be the random return on the risky asset.

The Allais paradox is a choice problem designed by to show an inconsistency of actual observed choices with the predictions of expected utility theory.

### Independence of irrelevant alternatives

independence axiomIIAindependence of irrelevant alternatives criterion
Independence of irrelevant alternatives pertains to well-defined preferences as well.
In the expected utility theory of von Neumann and Morgenstern, four axioms together imply that individuals act in situations of risk as if they maximize the expected value of a utility function.

### Lottery (probability)

lotterylotteries
In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature.

### Behavioral economics

behavioral financebehavioural economicseconomic psychology
Behavioral finance has produced several generalized expected utility theories to account for This has led to the field of behavioral finance, which has produced deviations from expected utility theory to account for the empirical facts.
Expected utility and discounted utility models began to gain acceptance, generating testable hypotheses about decision-making given uncertainty and intertemporal consumption, respectively.

### Cumulative prospect theory

Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory and SP/A theory.
CPT incorporates these observations in a modification of Expected Utility Theory by replacing final wealth with payoffs relative to the reference point, replacing the utility function with a value function that depends on relative payoff, and replacing cumulative probabilities with weighted cumulative probabilities.

### Indifference price

indifference pricing
In particular, the indifference price is the price at which an agent would have the same expected utility level by exercising a financial transaction as by not doing so (with optimal trading otherwise).

### Priority heuristic

The priority heuristic is a simple, lexicographic decision strategy that correctly predicts classic violations of expected utility theory such as the Allais paradox, the four-fold pattern, the certainty effect, the possibility effect, or intransitivities.

### Risk

risksdangerrisk-taking