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Expected utility hypothesis
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== Justification == === 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 the 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 maximize the logarithm of his gain instead.{{citation needed|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, an extra dollar or an additional good is perceived as less valuable as someone gets wealthier. In other words, desirability related to a financial gain depends on the gain itself and the person's wealth. 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 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 Ramsey's Representation Theorem. This representation theorem for expected utility assumes that [[preference]]s are defined over a set of bets where each option has a different yield. Ramsey believed that we should always make decisions to receive the best-expected outcome according to our personal preferences. This implies that if we can understand an individual's priorities and preferences, we can anticipate their choices.<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 = 483â498 |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 of each other. For example, if you study, you can not see your friends. However, you will get a good grade in your course. In this scenario, we analyze personal preferences and beliefs and will be able to predict which option a person might choose (e.g., if someone prioritizes their social life over 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 of a person just by looking at the choices they make (which is wrong). Ramsey defines a proposition as "[[neutrality (philosophy)|ethically neutral]]" when two possible outcomes have an equal value. In other words, if the probability can be defined as a preference, each proposition should have {{sfrac|1|2}} 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. 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=55â67|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. ÂŁ1·75. 1972 (Dover/Constable) |journal=The Mathematical Gazette|volume=57|issue=401|pages=220â221|doi=10.1017/s0025557200132589|s2cid= 164842618 |issn=0025-5572}}</ref> A crucial methodological aspect of Savage's framework is its focus on observable choicesâcognitive processes and other psychological aspects of decision-making matter only to the extent that they directly impact 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 is considered by some researchers to be "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=1â36 |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 a person doesn't control) to calculate the probability of an event. On the other hand, he used utility and intrinsic preferences to predict the event's outcome. Savage assumed that each act and state were sufficient to determine an outcome uniquely. However, this assumption breaks in cases where an individual does not have enough information about the event. Additionally, he believed that outcomes must have the same utility regardless of state. Therefore, it is essential to identify which statement is an outcome correctly. 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 for each person depending on intrinsic factors such as financial necessity or judgment about the company. Therefore, no state can rule out the performance of an act. Only when the state and the act are evaluated simultaneously is it possible 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=809â826|doi=10.1111/ecoj.12486|issn=0013-0133|doi-access=free}}</ref> ==== Savage's representation theorem ==== [[Savage's subjective expected utility model|Savage's representation theorem]] (Savage, 1954): A preference < satisfies P1âP7 if and only if there is a finitely additive probability measure P and a function u : C â R such that for every pair of acts ''f'' and ''g''.<ref name="Li_2017" /> ''f'' < ''g'' ââ Z Ω ''u''(''f''(''Ï'')) ''dP'' â„ Z Ω ''u''(''g''(''Ï'')) ''dP'' <ref name="Li_2017" /> <nowiki>*</nowiki>If and only if all the axioms are satisfied, one can use the information to reduce the uncertainty about the events that are out of their control. Additionally, the theorem ranks the outcome according to a utility function that reflects personal preferences. 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 describes everything 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 NeumannâMorgenstern utility theorem === {{Main|Von NeumannâMorgenstern utility theorem}} ==== The von NeumannâMorgenstern axioms ==== There are [[Von NeumannâMorgenstern 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. * Axiom (Transitivity): For every <math>A, B</math> and <math>C</math> with <math>A \succeq B</math> and <math> B \succeq C</math> we must have <math> A \succeq C</math>. ''[[Independence of irrelevant alternatives]]'' pertains to well-defined preferences as well. It assumes that two gambles mixed with an irrelevant third one will maintain the same order of preference as when the two are presented independently of the third one. The independence axiom is the most controversial.{{Citation needed|date=September 2016}}. * Axiom (Independence of irrelevant alternatives): For every <math>A, B</math> such that <math>A \succeq B</math>, the preference <math>tA+(1-t)C \succeq t B+(1-t)C,</math> must hold for every lottery <math>C</math> and real <math>t \in [0, 1]</math>. ''Continuity'' assumes that when there are three lotteries (<math>A, B</math> and <math>C</math>) and the individual prefers <math>A</math> to <math>B</math> and <math>B</math> to <math>C</math>. There should be a possible combination of <math>A</math> and <math>C</math> in which the individual is then indifferent between this mix and the lottery <math>B</math>. * Axiom (Continuity): Let <math>A, B</math> and <math>C</math> be lotteries with <math>A \succeq B \succeq C</math>. Then <math>B</math> is equally preferred to <math>pA+(1-p)C</math> for some <math>p\in [0,1]</math>. If all these axioms are satisfied, then the individual is rational. A utility function can represent the preferences, i.e., one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference <math>\succeq</math> amounts to choosing the lottery with the highest expected utility. This result is the [[Von NeumannâMorgenstern utility theorem|von NeumannâMorgenstern utility representation theorem]]. In other words, if an individual's behavior always satisfies the above axioms, then there is a utility function such that the individual will choose one gamble over another if and only if 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 called von NeumannâMorgenstern (vNM). 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 theory's axioms. The von NeumannâMorgenstern formulation is important in the application of [[set theory]] to economics because it was developed shortly after the HicksâAllen "[[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 nonlinear 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 NeumannâMorgenstern 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 :<math> u(w)= -e^{-aw}</math> It 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 <math>K-e^{-aw}</math> gives the same preferences orderings as does <math>-e^{-aw}</math>; thus it is irrelevant that the values of <math>-e^{-aw}</math> 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 :<math> u(w) = \log(w)</math> Has relative risk aversion equal to 1. The functions :<math> u(w) = w^{\alpha}</math> for <math>\alpha \in (0,1)</math> have relative risk aversion equal to <math>1-\alpha\in (0,1)</math>. And the functions :<math> u(w) = -w^{\alpha}</math> for <math>\alpha < 0</math> have relative risk aversion equal to <math>1-\alpha >1.</math> See also [[Risk aversion#Absolute risk aversion|the discussion]] of utility functions having hyperbolic absolute risk aversion (HARA).
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