Expected utility hypothesis
Encyclopedia
In economics
, game theory
, and decision theory
the expected utility hypothesis is a theory of utility
in which "betting preferences" of people with regard to uncertain outcomes (gambles) are represented 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 assets or personal preferences. This theory 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 theory 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.
The von Neumann–Morgenstern utility theorem
provides necessary and sufficient "rationality" axioms under which the expected utility hypothesis holds.
investments are simply the preferred ones. For example, suppose there 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 expected value theory people would choose the $100-or-nothing gamble. However, under expected utility theory, 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-mean gamble.
(involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone.
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 1738, Nicolas' cousin Daniel Bernoulli
, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk.
Daniel Bernoulli proposed that a mathematical function should be used to correct the expected value
depending on probability. This provides a way to account 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
.
Bernoulli's paper was the first formalization of marginal utility
, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person.
(named after the journal in which Bernoulli's paper was published) arises when there is no upper bound on the potential rewards from very low probability events. Because some probability distribution functions have an infinite expected value
, an expected-wealth maximizing 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 means that the expected utility of the gamble is finite, even if its expected value is infinite. (Thus he hypothesized diminishing marginal utility
of increasingly larger amounts of money.) It has also been resolved 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 them were unacceptable).
Completeness assumes that an individual has well defined preferences and can always decide between any two alternatives.
This means that the individual either prefers A to B, or is indifferent between A and B, or prefers B to A.
Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.
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 the third one. The independence axiom is the most controversial one.
Continuity assumes that when there are three lotteries (A, B and C) and the individual prefers A to B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.
If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function, i.e. one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference
amounts to choosing the lottery with the highest expected utility. This result is called the von Neumann—Morgenstern utility representation theorem
.
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 expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes,with the weights being the respective probabilities. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. This is a central theme of the expected utility hypothesis in which an individual chooses not 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
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. Note, however, that while in this context the utility function is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the expected utilty function is ordinal because any monotonic increasing transformation of it gives the same behavior.
and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.
Since the risk attitudes are unchanged under affine transformation
s of u, the first derivative u is not an 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 measure of absolute risk aversion:
The Arrow–Pratt measure of relative risk aversion is:
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. They are often used in economics for simplification.
A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold (Castagnoli and LiCalzi,1996; Bordley and 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 becomes expected value maximization. Intermediate cases lead to increasing risk-aversion above some fixed threshold and increasing risk-seeking below a fixed threshold.
was originally suggested by Bernoulli (see above). It has relative risk aversion constant and equal to one, and is still sometimes assumed in economic analyses. The utility function 
exhibits constant absolute risk aversion, and for this reason is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function 
gives exactly the same preferences orderings as does 
; thus it is irrelevant that the values of 
and its expected value are 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's utility function
has relative risk aversion equal to unity. The functions
for
have relative risk aversion equal to 
. And the functions
for
also have relative risk aversion equal to 
.
See also the discussion of utility functions having hyperbolic absolute risk aversion (HARA).
, 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
. However, D. E. Bell 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
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 
, allowing different people to disagree about the degree of risk associated with any given portfolio.
, 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.
The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions.
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.
has produced several generalized expected utility
theories to account for
instances where people's choice 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.
Particular theories include prospect theory
, rank-dependent expected utility
and cumulative prospect theory
and SP/A theory
(in economics, one talks of Knightian uncertainty
). Thus one must make assumptions, but then the expected 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 not depending on probabilities of outcomes and only requiring scenario analysis
(as in minimax
or minimax regret), or being less sensitive to assumptions.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, and decision theory
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
the expected utility hypothesis is a theory of utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
in which "betting preferences" of people with regard to uncertain outcomes (gambles) are represented by a function of the payouts (whether in money or other goods), the probabilities of occurrence, risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
, and the different utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
of the same payout to people with different assets or personal preferences. This theory has proved useful to explain some popular choices that seem to contradict the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
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
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
initiated this theory 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.
The von Neumann–Morgenstern utility theorem
Von Neumann–Morgenstern utility theorem
In 1947, John von Neumann and Oskar Morgenstern exhibited four relatively modest axioms of "rationality" such that any agent satisfying the axioms has a utility function...
provides necessary and sufficient "rationality" axioms under which the expected utility hypothesis holds.
Expected value and choice under risk
In the presence of risky outcomes, a decision maker could use the expected value criterion as a rule of choice: higher expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
investments are simply the preferred ones. For example, suppose there 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 expected value theory people would choose the $100-or-nothing gamble. However, under expected utility theory, 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-mean gamble.
Bernoulli's formulation
Nicolas Bernoulli described the St. Petersburg paradoxSt. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
(involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone.
In 1728, Gabriel Cramer
Gabriel Cramer
Gabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...
, 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 1738, Nicolas' cousin Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
, published the canonical 18th Century description of this solution in Specimen theoriae novae de mensura sortis or Exposition of a New Theory on the Measurement of Risk.
Daniel Bernoulli proposed that a mathematical function should be used to correct the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
depending on probability. This provides a way to account for risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
, where the risk premium
Risk premium
A risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset, in order to induce an individual to hold the risky asset rather than the risk-free asset...
is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
.
Bernoulli's paper was the first formalization of marginal utility
Marginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person.
Infinite expected value — St. Petersburg paradox
The St. Petersburg paradoxSt. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
(named after the journal in which Bernoulli's paper was published) arises when there is no upper bound on the potential rewards from very low probability events. Because some probability distribution functions have an infinite expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, an expected-wealth maximizing 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 means that the expected utility of the gamble is finite, even if its expected value is infinite. (Thus he hypothesized diminishing marginal utility
Marginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
of increasingly larger amounts of money.) It has also been resolved 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 them were unacceptable).
The von Neumann-Morgenstern axioms
There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity.Completeness assumes that an individual has well defined preferences and can always decide between any two alternatives.
- Axiom (Completeness): For every A and B either
, 
or 
.
This means that the individual either prefers A to B, or is indifferent between A and B, or prefers B to A.
Transitivity assumes that, as an individual decides according to the completeness axiom, the individual 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 the third 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 B and B to C, then there should be a possible combination of A and C in which the individual is then indifferent between this mix and the lottery B.
- Axiom (Continuity): Let A, B and C be lotteries with
; then there exists a probability p such that B is equally good as 
.
If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function, i.e. one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference
amounts to choosing the lottery with the highest expected utility. This result is called the von Neumann—Morgenstern utility representation theoremVon Neumann–Morgenstern utility theorem
In 1947, John von Neumann and Oskar Morgenstern exhibited four relatively modest axioms of "rationality" such that any agent satisfying the axioms has 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 expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes,with the weights being the respective probabilities. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. This is a central theme of the expected utility hypothesis in which an individual chooses not 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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
to economics because it was developed shortly after the Hicks-Allen "ordinal
Ordinal utility
Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles of goods can be ordered such that one is considered by an individual to be worse than, equal to,...
revolution" of the 1930s, and it revived the idea of cardinal utility
Cardinal utility
In economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...
in economic theory. Note, however, that while in this context the utility function is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the expected utilty function is ordinal because any monotonic increasing transformation of it gives the same behavior.
Risk aversion
The expected utility theory implies that rational individuals act as though they were maximizing expected utility, and allows for the fact that many individuals are risk averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). Risk aversion implies that their utility functions are concaveConcave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...
and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.
Since the risk attitudes are unchanged under affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s of u, the first derivative u is not an 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 measure of absolute risk aversion:
The Arrow–Pratt measure of relative risk aversion is:
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. They are often used in economics for simplification.
A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold (Castagnoli and LiCalzi,1996; Bordley and 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 becomes expected value maximization. Intermediate cases lead to increasing risk-aversion above some fixed threshold and increasing risk-seeking below a fixed threshold.
Examples of von Neumann-Morgenstern utility functions
The utility function was originally suggested by Bernoulli (see above). It has relative risk aversion constant and equal to one, and is still sometimes assumed in economic analyses. The utility function exhibits constant absolute risk aversion, and for this reason is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function gives exactly the same preferences orderings as does ; thus it is irrelevant that the values of and its expected value are 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's utility function
has relative risk aversion equal to unity. The functions
for
have relative risk aversion equal to . And the functionsfor
also have relative risk aversion equal to .See also the discussion of utility functions having hyperbolic absolute risk aversion (HARA).
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 analysisModern portfolio theory
Modern portfolio theory is a theory of investment which attempts to maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return, by carefully choosing the proportions of various assets...
, variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
is used as a risk measure for portfolio return; however, this is only valid if returns are normally distributed or otherwise jointly elliptically distributed
Elliptical distribution
In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution and inherit some of its properties.-Definition:...
. However, D. E. Bell 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
for individual-specific positive parameters a and b. Then expected utility is given byThus the risk measure is
, which differs between two individuals if they have different values of the parameter , allowing different people to disagree about the degree of risk associated with any given portfolio.Behavioral decision science: Deviations from expected utility theory
Like any mathematical modelMathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
, 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.
The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions.
Conservatism in updating beliefs
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.In updating probability distributions using evidence, a standard method uses conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
, namely the rule of Bayes. Some experiments on belief revision
Belief revision
Belief revision is the process of changing beliefs to take into account a new piece of information. The logical formalization of belief revision is researched in philosophy, in databases, and in artificial intelligence for the design of rational agents....
have suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.
Irrational deviations
Behavioral financeBehavioral finance
Behavioral economics and its related area of study, behavioral finance, use social, cognitive and emotional factors in understanding the economic decisions of individuals and institutions performing economic functions, including consumers, borrowers and investors, and their effects on market...
has produced several generalized expected utility
Generalized expected utility
The expected utility model developed by John von Neumann and Oskar Morgenstern dominated decision theory from its formulation in 1944 until the late 1970s, not only as a prescriptive, but also as a descriptive model, despite powerful criticism from Maurice Allais and Daniel Ellsberg who showed...
theories to account for
instances where people's choice deviate from those predicted by expected utility theory. These deviations are described as "irrational
Rational choice theory
Rational choice theory, also known as choice theory or rational action theory, is a framework for understanding and often formally modeling social and economic behavior. It is the main theoretical paradigm in the currently-dominant school of microeconomics...
" 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
Prospect theory
Prospect theory is a theory that describes decisions between alternatives that involve risk i.e. where the probabilities of outcomes are known. The model is descriptive: it tries to model real-life choices, rather than optimal decisions.-Model:...
, rank-dependent expected utility
Rank-dependent expected utility
The rank-dependent expected utility model is a generalized expected utility model of choice under uncertainty, designed to explain the behaviour observed in the Allais paradox, as well as for the observation that many people both purchase lottery tickets and insure against losses .A...
and cumulative prospect theory
Cumulative prospect theory
Cumulative prospect theory is a model for descriptive decisions under risk which was introduced by Amos Tversky and Daniel Kahneman in 1992 . It is a further development and variant of prospect theory...
and SP/A theory
Preference reversals over uncertain outcomes
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." Many studies have examined this "preference reversal," from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under some specific assumptions.Uncertain probabilities
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 uncertaintyUncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
(in economics, one talks of Knightian uncertainty
Knightian uncertainty
In economics, Knightian uncertainty is risk that is immeasurable, not possible to calculate.Knightian uncertainty is named after University of Chicago economist Frank Knight , who distinguished risk and uncertainty in his work Risk, Uncertainty, and Profit:- Common-cause and special-cause :The...
). Thus one must make assumptions, but then the expected value of various decisions is very sensitive
Sensitivity analysis
Sensitivity analysis is the study of how the variation in the output of a statistical model can be attributed to different variations in the inputs of the model. Put another way, it is a technique for systematically changing variables in a model to determine the effects of such changes.In any...
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 not depending on probabilities of outcomes and only requiring scenario analysis
Scenario analysis
Scenario analysis is a process of analyzing possible future events by considering alternative possible outcomes . Thus, the scenario analysis, which is a main method of projections, does not try to show one exact picture of the future. Instead, it presents consciously several alternative future...
(as in minimax
Minimax
Minimax is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a worst case scenario. Alternatively, it can be thought of as maximizing the minimum gain...
or minimax regret), or being less sensitive to assumptions.
See also
- Allais paradoxAllais paradoxThe Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...
- Ambiguity aversionAmbiguity aversionIn decision theory and economics, ambiguity aversion describes an attitude of preference for known risks over unknown risks. People would rather choose an option with fewer unknown elements than with many unknown elements. It is demonstrated in the Ellsberg paradox In decision theory and...
- Bayesian probabilityBayesian probabilityBayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
- Behavioral economics
- Decision theoryDecision theoryDecision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
- Generalized expected utilityGeneralized expected utilityThe expected utility model developed by John von Neumann and Oskar Morgenstern dominated decision theory from its formulation in 1944 until the late 1970s, not only as a prescriptive, but also as a descriptive model, despite powerful criticism from Maurice Allais and Daniel Ellsberg who showed...
- Indifference priceIndifference priceIn finance, indifference pricing is a method of pricing financial securities with regards to a utility function. Also known as the reservation price or private valuation. Particularly the indifference price is the price that an agent would have the same expected utility level between exercising a...
- Lottery (probability)Lottery (probability)In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probability that a certain outcome arises from a given state of nature...
- Marginal utilityMarginal utilityIn economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
- Prospect theoryProspect theoryProspect theory is a theory that describes decisions between alternatives that involve risk i.e. where the probabilities of outcomes are known. The model is descriptive: it tries to model real-life choices, rather than optimal decisions.-Model:...
- Rank-dependent expected utilityRank-dependent expected utilityThe rank-dependent expected utility model is a generalized expected utility model of choice under uncertainty, designed to explain the behaviour observed in the Allais paradox, as well as for the observation that many people both purchase lottery tickets and insure against losses .A...
- Risk aversionRisk aversionRisk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....
- Risk in psychology
- Subjective expected utilitySubjective expected utilitySubjective expected utility is a method in decision theory in the presence of risk, promoted by L. J. Savage in 1954 following previous work by Ramsey and von Neumann...
- Two-moment decision modelsTwo-moment decision modelsIn decision theory, economics, and finance, a two-moment decision model is a model that describes or prescribes the process of making decisions in a context in which the decision-maker is faced with random variables whose realizations cannot be known in advance, and in which choices are made based...
Further reading
- Scott PlousScott PlousScott Plous, Ph.D., is a Professor of Psychology at the Department of Psychology of Wesleyan University. He is also a Faculty Associate of the Tufts University Center for Animals and Public Policy and founder of the Social Psychology Network....
(1993) "The psychology of judgment and decision making", Chapter 7 (specifically) and 8,9,10, (to show paradoxes to the theory).