Expected utility hypothesis: Difference between revisions - Wikipedia

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{{Short description|Concept in economics}}

The '''expected utility hypothesis''' is a foundational assumption in [[mathematical economics]] concerning human [[preference]] when [[decision theory|decision making]] under [[uncertainty]]. It postulates that a [[rational agent]]s maximizesmaximize [[utilitarianism|utility]], asmeaning formulatedthe insubjective the mathematicsdesirability of [[game theory]], based on their [[risk aversion]]actions. [[Rational choice theory]], a cornerstone of [[microeconomics]], builds upon the expected utility ofthis individualspostulate to model aggregate social behaviour.

The expected utility hypothesis states an agent chooses between risky prospects by comparing expected utility values (i.e. the weighted sum of adding the respective utility values of payoffs multiplied by their probabilities). The summarised formula for expected utility is <math>U(p)=\sum u(x_k)p_k </math> where <math>p_k</math> is the probability that outcome indexed by <math>k</math> with payoff <math>x_k</math> is realized, and function ''u'' expresses the utility of each respective payoff.<ref>{{Cite web|title=Expected Utility Theory {{!}} Encyclopedia.com|url=https://www.encyclopedia.com/social-sciences/applied-and-social-sciences-magazines/expected-utility-theory|access-date=2021-04-28|website=www.encyclopedia.com}}</ref> Graphically the curvature of the u function captures the agent's risk attitude.

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Standard utility functions represent [[ordinal utility|ordinal]] preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility [[cardinal number|cardinal]] (though still not comparable across individuals).

Although the expected utility hypothesis is standard in economic modelling, it has been found to be violated in psychological experiments. For many years, psychologists and economic theorists have been developing new theories to explain these deficiencies.<ref>{{Cite journal|date=2011-05-01|title=Mixture models of choice under risk|url=https://wwwhal.sciencedirectarchives-ouvertes.comfr/sciencehal-00631676/articlefile/abs/pii/S0304407609002644PEER_stage2_10.1016%252Fj.jeconom.2009.10.011.pdf|journal=Journal of Econometrics|language=en|volume=162|issue=1|pages=79–88|doi=10.1016/j.jeconom.2009.10.011|issn=0304-4076|last1=Conte|first1=Anna|last2=Hey|first2=John D.|last3=Moffatt|first3=Peter G.|s2cid=33410487 }}</ref> These include [[prospect theory]], [[rank-dependent expected utility]] and [[cumulative prospect theory]], and [[bounded rationality]].

== AntecedentsJustification ==

=== Bernoulli's formulation ===▼

=== Limits of the expected value theory ===

In the early days of the calculus of probability, classic utilitarians believed that the option which has the greatest utility will produce more pleasure or happiness for the agent and therefore must be chosen<ref name=":1">{{cite journal| vauthors = Oberhelman DD |date=June 2001|title=Stanford Encyclopedia of Philosophy | veditors = Zalta EN |journal=Reference Reviews|volume=15|issue=6|pages=9|doi=10.1108/rr.2001.15.6.9.311 }}</ref> The main problem with the [[Expected value|expected value theory]] is that there might not be a unique correct way to quantify utility or to identify the best trade-offs. For example, some of the trade-offs may be intangible or qualitative. Rather than [[Incentive|monetary incentives]], other desirable ends can also be included in utility such as pleasure, knowledge, friendship, etc. Originally the total utility of the consumer was the sum of independent utilities of the goods. However, the expected value theory was dropped as it was considered too static and deterministic.<ref name=":2">{{cite book |title=Expected Utility Hypotheses and the Allais Paradox|date=1979|publisher=Springer Netherlands|isbn=978-90-481-8354-8| veditors = Allais M, Hagen O | location= Dordrecht|language=en|doi=10.1007/978-94-015-7629-1 }}</ref> The classical counter example to the expected value theory (where everyone makes the same "correct" choice) is the [[St. Petersburg paradox|St. Petersburg Paradox]]. This paradox questioned if [[Marginal utility|marginal utilities]] should be ranked differently as it proved that a "correct decision" for one person is not necessarily right for another person.<ref name=":2" />▼

== Risk aversion ==▼

{{further|Risk aversion}}▼

The expected utility theory takes into account that individuals may be [[Risk aversion|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 [[concave function|concave]] 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 second derivative ''u<nowiki>''</nowiki>'' 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<ref name=":4">{{cite book | vauthors = Arrow KJ | date = 1965 | chapter = The theory of risk aversion | title = Aspects of the Theory of Risk Bearing Reprinted in Essays in the Theory of Risk Bearing | veditors = Saatio YJ | publisher = Markham Publ. Co. | location = Chicago, 1971 | pages = 90–109 }}</ref><ref name=":5">{{cite journal| vauthors = Pratt JW |date=January–April 1964|title=Risk aversion in the small and in the large|journal=Econometrica|volume=32|issue=1/2|pages=122–136|doi=10.2307/1913738|jstor=1913738}}</ref> measure of absolute risk aversion:▼

: <math>\mathit{ARA}(w) =-\frac{u''(w)}{u'(w)},</math>▼

where <math>w</math> is wealth.▼

The Arrow–Pratt measure of relative risk aversion is:▼

: <math>\mathit{RRA}(w) =-\frac{wu''(w)}{u'(w)}</math>▼

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.<ref>Castagnoli and LiCalzi, 1996; Bordley and LiCalzi, 2000; Bordley and Kirkwood</ref> 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.▼

== The St. Petersburg paradox ==▼

The [[St. Petersburg Paradox|St. Petersburg paradox]] presented by [[Nicolas Bernoulli]] illustrates that decision making based on expected value of monetary payoffs lead to absurd conclusions.<ref name=":0">{{cite journal|vauthors=Aase KK|date=January 2001|title=On the St. Petersburg Paradox|journal=Scandinavian Actuarial Journal|language=en|volume=2001|issue=1|pages=69–78|doi=10.1080/034612301750077356|issn=0346-1238|s2cid=14750913}}</ref> When a probability distribution function has an infinite [[expected value]], a person who only cares about expected values of a gamble would pay an arbitrarily large finite amount to take this gamble. However, this experiment demonstrated that there is no upper bound on the potential rewards from very low probability events. In the hypothetical setup, a person flips a coin repeatedly. The participant's prize is determined by the number of times the coin lands on heads consecutively. For every time the coin comes up heads (1/2 probability), the participant's prize is doubled. The game ends when the participant flips the coin and it comes out tails. A player who only cares about expected value of the payoff should be willing to pay any finite amount of money to play because this entry cost will always be less than the expected, infinite, value of the game. However, in reality, people do not do this. "Only a few of the participants were willing to pay a maximum of $25 to enter the game because many of them were risk averse and unwilling to bet on a very small possibility at a very high price.<ref>{{cite encyclopedia | title = The St. Petersburg Paradox | url = https://stanford.library.sydney.edu.au/archives/sum2012/entries/paradox-stpetersburg/ | encyclopedia = Stanford Encyclopedia of Philosophy | date = 16 June 2008 | last1 = Martin | first1 = Robert }}</ref>▼

▲==Bernoulli's formulation==

[[Nicolaus I Bernoulli|Nicolaus Bernoulli]] described the [[St. Petersburg paradox]] (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. 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. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone. He 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. Bernoulli further proposed that it was not the goal of the gambler to maximize his expected gain but to instead maximize the logarithm of his gain.{{cn|date=August 2023}}

Daniel Bernoulli drew attention to psychological and behavioral components behind the individual's [[decision-making process]] and proposed that the utility of wealth has a [[diminishing marginal utility]]. For example, as someone gets wealthier, an extra dollar or an additional good is perceived as less valuable. In other words, desirability related with a financial gain depends not only on the gain itself but also on the wealth of the person.  Bernoulli suggested that people maximize "moral expectation" rather than expected monetary value. Bernoulli made a clear distinction between expected value and expected utility. Instead of using the weighted outcomes, he used the weighted utility multiplied by probabilities. He proved that the utility function used in real life means is finite, even when its expected value is infinite.<ref name=":2" />

=== Ramsey-theoretic approach to subjective probability ===

In 1926, [[Frank Ramsey (mathematician)|Frank Ramsey]] introduced the Ramsey's Representation Theorem. This representation theorem for expected utility assumed that [[preference]]s are defined over a set of bets where each option has a different yield. Ramsey believed that we always choose decisions to receive the best expected outcome according to our personal preferences. This implies that if we are able to understand the priorities and personal preferences of an individual we can anticipate what choices they are going to take.<ref>{{cite journal| vauthors = Bradley R |date=2004|title=Ramsey's Representation Theorem|url=http://personal.lse.ac.uk/bradleyr/pdf/Ramsey.dialectica.pdf|journal=Dialectica|volume=58|issue=4|pages = 483–498 |doi=10.1111/j.1746-8361.2004.tb00320.x}}</ref> In this model, he defined numerical utilities for each option to exploit the richness of the space of prices. The outcome of each preference is exclusive fromof each other. For example, if you study, then you can't not see your friends, however you will get a good grade in your course. In this scenario, if we analyze what are his personal preferences and beliefs weand will be able to predict which heoption a person might choose. (e.g. if someone prioritizes their social life more thanover academic results, they will go out with their friends). Assuming that the decisions of a person are [[Rationalism|rational]], according to this theorem , we should be able to know the beliefs and utilities from a person just by looking at the choices someonethey takesmake (which is wrong). Ramsey defines a proposition as "[[neutrality (philosophy)|ethically neutral]]" when two possible outcomeoutcomes hashave an equal value. In other words, if the probability can be defined in terms of a preference, each proposition should have ½{{sfrac|1|2}} in order to be indifferent between both options.<ref>{{cite web | vauthors = Elliott E | title = Ramsey and the Ethically Neutral Proposition | url = http://www.edwardjrelliott.com/uploads/7/4/4/7/74475147/[natrep]_ramsey_and_the_ethically_neutral_proposition.pdf | work = Australian National University }}</ref>

Ramsey shows that

: <math> P(E) = (1-U(m))(U(b)-U(w)) </math><ref>{{cite journal| vauthors = Briggs RA |date=2014-08-08|title=Normative Theories of Rational Choice: Expected Utility|url=https://plato.stanford.edu/archives/fall2019/entries/rationality-normative-utility/}}</ref>

=== Savage's subjective expected utility representation ===

In the 1950s, [[Leonard Jimmie Savage]], an American statistician, derived a framework for comprehending expected utility. At that point, it was considered the first and most thorough foundation to understanding the concept. Savage's framework involved proving that expected utility could be used to make an optimal choice among several acts through seven axioms.<ref name = "Savage_1951">{{cite journal| vauthors = Savage LJ |date= March 1951 |title=The Theory of Statistical Decision |journal=Journal of the American Statistical Association|volume=46|issue=253|pages=55–67|doi=10.1080/01621459.1951.10500768|issn=0162-1459}}</ref> In his book, The Foundations of Statistics, Savage integrated a normative account of decision making under risk (when probabilities are known) and under uncertainty (when probabilities are not objectively known).  Savage concluded that people have neutral attitudes towards uncertainty and that observation is enough to predict the probabilities of uncertain events.  <ref>{{cite journal| vauthors = Lindley DV |date= September 1973 |title=The foundations of statistics (second edition), by Leonard J. Savage. Pp xv, 310. £1·75. 1972 (Dover/Constable) |journal=The Mathematical Gazette|volume=57|issue=401|pages=220–221|doi=10.1017/s0025557200132589|s2cid= 164842618 |issn=0025-5572}}</ref> A crucial methodological aspect of Savage's framework is its focus on observable choices. Cognitive processes and other psychological aspects of decision making matter only to the extent that they have directly measurable implications on choice.

The theory of subjective expected utility combines two concepts: first, a personal utility function, and second, a personal probability distribution (usually based on Bayesian probability theory).  This theoretical model has been known for its clear and elegant structure and its considered forby some researchers oneto ofbe "the most brilliant axiomatic theory of utility ever developed".<ref>{{Citation|title=1. Foundations of probability theory|date=2009-01-21|doi = 10.1515/9783110213195.1 |work=Interpretations of Probability|pages=1–36 |place=Berlin, New York|publisher=Walter de Gruyter|isbn=978-3-11-021319-5 }}</ref> Instead of assuming the probability of an event, Savage defines it in terms of preferences over acts.  Savage used the states (something thata isperson not in yourdoesn't control) to calculate the probability of an event. On the other hand, he used utility and intrinsic preferences to predict the outcome of the event. Savage assumed that each act and state are enoughsufficient to uniquely determine an outcome. However, this assumption breaks in the cases where thean individual doesn'tdoes not have enough information about the event.

Additionally, he believed that outcomes must have the same utility regardless of the state. For that reason, it is essential to correctly identify which statement is considered an outcome. For example, if someone says "I got the job" this affirmation is not considered an outcome, since the utility of the statement will be different onfor each person depending on intrinsic factors such as financial necessity or judgmentsjudgment about the company. For that reason, no state can rule out the performance of anyan act, only. Only when the state and the act are evaluated simultaneously, youit will bebecomes ablepossible to determine an outcome with certainty.<ref name = "Li_2017">{{cite journal| vauthors = Li Z, Loomes G, Pogrebna G |date=2017-05-01|title=Attitudes to Uncertainty in a Strategic Setting |journal=The Economic Journal|language=en|volume=127|issue=601|pages=809–826|doi=10.1111/ecoj.12486|issn=0013-0133|doi-access=free}}</ref>

==== Savage's representation theorem ====

The [[Leonard Jimmie Savage|Savage representation theorem]] (Savage, 1954) A preference < satisfies P1–P7 if and only if there is a finitely additive probability measure P and a function u : C → R such that for every pair of acts ''f'' and ''g''.<ref name="Li_2017" />

''f'' < ''g'' ⇐⇒ Z Ω ''u''(''f''(''ω'')) ''dP'' ≥ Z Ω ''u''(''g''(''ω'')) ''dP'' <ref name="Li_2017" />

<nowiki>*</nowiki>If and only if all the axioms are satisfied whenone can useduse the information to reduce the uncertainty about the events that are out of yourtheir control. Additionally the theorem ranks the outcome according to a utility function that reflects the personal preferences.

'''Key ingredients:'''

The key ingredients in Savage's theory are:

* ''States:'' The specification of every aspect of the decision problem at hand or  "A description of the world leaving no relevant aspect undescribed."<ref name = "Savage_1951" />

* ''Events:'' A set of states identified by someone

* ''Consequences:'' A consequence is the description of all that is relevant to the decision maker's utility (e.g. monetary rewards, psychological factors, etc)

* '''''Acts:''''' An act is a finite-valued function that maps states to consequences.

=== Von Neumann–Morgenstern utility theorem ===

{{Main|Von Neumann–Morgenstern utility theorem}}

==== The von Neumann–Morgenstern axioms ====

There are [[Von Neumann–Morgenstern utility theorem|four axioms]] of the expected utility theory that define a ''rational'' decision maker: completeness; transitivity; independence of irrelevant alternatives; and continuity.<ref>{{cite book | vauthors = von Neumann J, Morgenstern O |title=Theory of Games and Economic Behavior |url=https://archive.org/details/theoryofgameseco00vonn |url-access=registration |location=Princeton, NJ |publisher=Princeton University Press |orig-year=1944 |edition=Third |year=1953 }}</ref>

[[Completeness (order theory)|''Completeness'']] assumes that an individual has well defined preferences and can always decide between any two alternatives.

* Axiom (Completeness): For every <math>A</math> and <math>B</math> either <math>A \succeq B</math> or <math>A \preceq B</math> or both.

This means that the individual prefers <math>A</math> to <math>B</math>, <math>B</math> to <math>A</math>, or is indifferent between <math>A</math> and <math>B</math>.

[[Transitive relation|''Transitivity'']] assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.

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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 utility|ordinal]] revolution" of the 1930s, and it revived the idea of [[cardinal utility]] in economic theory.{{Citation needed|date=August 2008}} However, 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 utility function'' is ordinal because any monotonic increasing transformation of expected utility gives the same behavior.

==== Examples of von Neumann–Morgenstern utility functions ====

The utility function <math>u(w)=\log(w)</math> 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

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For general utility functions, however, expected utility analysis does not permit the expression of preferences to be separated into two parameters with one representing the expected value of the variable in question and the other representing its risk.

▲== Risk aversion ==

==Criticism== ▼

▲{{furthermain|Risk aversion}}

In empirical applications, a number of violations of expected utility theory have been shown to be systematic and these falsifications have deepened understanding of how people actually decide. [[Daniel Kahneman]] and [[Amos Tversky]] in 1979 presented their [[prospect theory]] which showed empirically, how preferences of individuals are inconsistent among the same choices, depending on how those choices are presented.<ref>{{cite journal | vauthors = Kahneman D, Tversky A | title = Prospect Theory: An Analysis of Decision under Risk. | journal = Econometrica | year = 1979 | volume = 47 | issue = 2 | pages = 263–292 | doi = 10.2307/1914185 | jstor = 1914185 | url = http://www.dklevine.com/archive/refs47656.pdf }}</ref> This is mainly because people are different in terms of their preferences and parameters. Additionally, personal behaviors may be different between individuals even when they are facing the same choice problem.▼

▲The expected utility theory takes into account that individuals may be [[Risk aversion|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 [[concave function|concave]] 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 second derivative ''u<nowiki>''</nowiki>'' 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<ref name=":4">{{cite book | vauthors = Arrow KJ | date = 1965 | chapter = The theory of risk aversion | title = Aspects of the Theory of Risk Bearing Reprinted in Essays in the Theory of Risk Bearing | veditors = Saatio YJ | publisher = Markham Publ. Co. | location = Chicago, 1971 | pages = 90–109 }}</ref><ref name=":5">{{cite journal| vauthors = Pratt JW |date=January–April 1964|title=Risk aversion in the small and in the large|journal=Econometrica|volume=32|issue=1/2|pages=122–136|doi=10.2307/1913738|jstor=1913738}}</ref> measure of absolute risk aversion:

▲: <math>\mathit{ARA}(w) =-\frac{u''(w)}{u'(w)},</math>

▲where <math>w</math> is wealth.

▲The Arrow–Pratt measure of relative risk aversion is:

▲: <math>\mathit{RRA}(w) =-\frac{wu''(w)}{u'(w)}</math>

▲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.<ref>Castagnoli and LiCalzi, 1996; Bordley and LiCalzi, 2000; Bordley and Kirkwood</ref> 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.

▲== The St. Petersburg paradox ==

▲The [[St. Petersburg Paradox|St. Petersburg paradox]] presented by [[Nicolas Bernoulli]] illustrates that decision making based on expected value of monetary payoffs lead to absurd conclusions.<ref name=":0">{{cite journal|vauthors=Aase KK|date=January 2001|title=On the St. Petersburg Paradox|journal=Scandinavian Actuarial Journal|language=en|volume=2001|issue=1|pages=69–78|doi=10.1080/034612301750077356|issn=0346-1238|s2cid=14750913}}</ref> When a probability distribution function has an infinite [[expected value]], a person who only cares about expected values of a gamble would pay an arbitrarily large finite amount to take this gamble. However, this experiment demonstrated that there is no upper bound on the potential rewards from very low probability events. In the hypothetical setup, a person flips a coin repeatedly. The participant's prize is determined by the number of times the coin lands on heads consecutively. For every time the coin comes up heads (1/2 probability), the participant's prize is doubled. The game ends when the participant flips the coin and it comes out tails. A player who only cares about expected value of the payoff should be willing to pay any finite amount of money to play because this entry cost will always be less than the expected, infinite, value of the game. However, in reality, people do not do this. "Only a few of the participants were willing to pay a maximum of $25 to enter the game because many of them were risk averse and unwilling to bet on a very small possibility at a very high price.<ref>{{cite encyclopedia | title = The St. Petersburg Paradox | url = https://stanford.library.sydney.edu.au/archives/sum2012/entries/paradox-stpetersburg/ | encyclopedia = Stanford Encyclopedia of Philosophy | date = 16 June 2008 | last1 = Martin | first1 = Robert }}</ref>

▲==Criticism==

▲In the early days of the calculus of probability, classic utilitarians believed that the option which has the greatest utility will produce more pleasure or happiness for the agent and therefore must be chosen.<ref name=":1">{{cite journal| vauthors = Oberhelman DD |date=June 2001|title=Stanford Encyclopedia of Philosophy | veditors = Zalta EN |journal=Reference Reviews|volume=15|issue=6|pages=9|doi=10.1108/rr.2001.15.6.9.311 }}</ref> The main problem with the [[Expected value|expected value theory]] is that there might not be a unique correct way to quantify utility or to identify the best trade-offs. For example, some of the trade-offs may be intangible or qualitative. Rather than [[Incentive|monetary incentives]], other desirable ends can also be included in utility such as pleasure, knowledge, friendship, etc. Originally the total utility of the consumer was the sum of independent utilities of the goods. However, the expected value theory was dropped as it was considered too static and deterministic.<ref name=":2">{{cite book |title=Expected Utility Hypotheses and the Allais Paradox|date=1979|publisher=Springer Netherlands|isbn=978-90-481-8354-8| veditors = Allais M, Hagen O | location= Dordrecht|language=en|doi=10.1007/978-94-015-7629-1 }}</ref> The classical counter example to the expected value theory (where everyone makes the same "correct" choice) is the [[St. Petersburg paradox|St. Petersburg Paradox]]. This paradox questioned if [[Marginal utility|marginal utilities]] should be ranked differently as it proved that a "correct decision" for one person is not necessarily right for another person.<ref name=":2" />

▲In empirical applications, a number of violations of expected utility theory have been shown to be systematic and these falsifications have deepened understanding of how people actually decide. [[Daniel Kahneman]] and [[Amos Tversky]] in 1979 presented their [[prospect theory]] which showed empirically, how preferences of individuals are inconsistent among the same choices, depending on howthe those[[Framing (social sciences)|framing]] of the choices, i.e. how they are presented.<ref>{{cite journal | vauthors = Kahneman D, Tversky A | title = Prospect Theory: An Analysis of Decision under Risk. | journal = Econometrica | year = 1979 | volume = 47 | issue = 2 | pages = 263–292 | doi = 10.2307/1914185 | jstor = 1914185 | url = http://www.dklevine.com/archive/refs47656.pdf }}</ref> This is mainly because people are different in terms of their preferences and parameters. Additionally, personal behaviors may be different between individuals even when they are facing the same choice problem.

Like any [[mathematical model]], expected utility theory is a 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. This has led to the field of [[behavioral finance]], which has produced deviations from expected utility theory to account for the empirical facts.

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# Better understanding of the psychologically relevant outcome space

# A psychologically richer theory of the determinants

=== Mixture models of choice under risk ===

In this model Conte (2011) found that behaviour differs between individuals and for the same individual at different times. Applying a Mixture Model fits the data significantly better than either of the two preference functionals individually.<ref>{{cite journal| vauthors = Conte A, Hey JD, Moffatt PG |date= May 2011 |title=Mixture models of choice under risk |journal=Journal of Econometrics|language=en|volume=162|issue=1|pages=79–88|doi=10.1016/j.jeconom.2009.10.011|s2cid= 33410487 |url= https://hal.archives-ouvertes.fr/hal-00631676/file/PEER_stage2_10.1016%252Fj.jeconom.2009.10.011.pdf }}</ref> Additionally it helps to estimate preferences much more accurately than the old economic models because it takes heterogeneity into account. In other words, the model assumes that different agents in the population have different functionals. The model estimate the proportion of each group to consider all forms of heterogeneity.

=== Psychological expected utility model:<ref>{{cite journal| vauthors = Caplin A, Leahy J |date=2001-02-01|title=Psychological Expected Utility Theory and Anticipatory Feelings |journal=The Quarterly Journal of Economics|language=en|volume=116|issue=1|pages=55–79|doi=10.1162/003355301556347|issn=0033-5533}}</ref> ===

Caplin (2001) expanded the standard prize space to include the influence on preferences and decisions of anticipatory emotions such as suspense and anxiety. He replaced the standard prize space with a space of "psychological states". In this research he opens up a variety of psychologically interesting phenomena to rational analysis. This model explained how time inconsistency arises naturally in the presence of anticipations and also how emotions may change the result of choices. For example, this model finds that anxiety is anticipatory and that the desire to reduce anxiety motivates many decisions. A better understanding of the psychologically relevant outcome space will facilitate theorists to develop richer theory of determinants.

== See also ==