3420:, 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.
417:, 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. 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. 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.
272:. 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 is finite, even when its expected value is infinite.
289:
we can anticipate what choices they are going to take. 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, then 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
3603:, 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 specific assumptions.
421:
researchers to be "the most brilliant axiomatic theory of utility ever developed". 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 outcome of the event. Savage assumed that each act and state are sufficient to uniquely determine an outcome. However, this assumption breaks in cases where an individual does not have enough information about the event.
1518:
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
NeumannâMorgenstern (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.
3024:
248:, 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
3467:, 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
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2653:
3437:, 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. The classical counter example to the expected value theory (where everyone makes the same "correct" choice) is the
3019:{\displaystyle {\begin{aligned}\operatorname {E} &=\operatorname {E} -b\operatorname {E} \\&=\operatorname {E} -b\operatorname {E} -a(w-\operatorname {E} )}]\\&=\operatorname {E} -be^{-a\operatorname {E} }\operatorname {E} )}]\\&={\text{expected wealth}}-b\cdot e^{-a\cdot {\text{expected wealth}}}\cdot {\text{risk}}.\end{aligned}}}
3536:
contexts are significantly different. This is demonstrated in the contrast of individual preferences under the insurance and lottery context shows the degree of indeterminacy of the expected utility theory. Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von
NeumannâMorgenstern.
425:
statement will be different for each person depending on intrinsic factors such as financial necessity or judgment about the company. For that reason, no state can rule out the performance of an act. Only when the state and the act are evaluated simultaneously, it becomes possible to determine an outcome with certainty.
424:
Additionally, he believed that outcomes must have the same utility regardless of 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
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According to the empirical results there has been almost no recognition in decision theory of the distinction between the problem of justifying its theoretical claims regarding the properties of rational belief and desire. One of the main reasons is because people's basic tastes and preferences for
3157:
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
1517:
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
288:
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
1653:
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
420:
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 by some
3535:
are considered insufficient to predict preferences and the expected utility. Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von
NeumannâMorgenstern. This is because preferences and utility functions constructed under different
3399:
A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold. In the absence of uncertainty about the threshold, expected utility maximization simplifies to maximizing the probability of achieving some fixed
260:
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.
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where it was demonstrated that people do not revise their degrees on belief in line with experimented probabilities and also that probabilities cannot be applied to single cases. On the other hand, in updating probability distributions using evidence, a standard method uses
1489:
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
50:
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
916:
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 axiom..
207:
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. These include
485:*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 the personal preferences.
3400:
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.
2576:, or in the unlikely case in which the utility function has a quadratic form. However, David E. Bell proposed a measure of risk which follows naturally from a certain class of von NeumannâMorgenstern utility functions. Let utility of wealth be given by
3130:
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.
3581:
approaches to probability treat it as a degree of belief and thus they do not draw a distinction between risk and a wider concept of uncertainty: they deny the existence of
Knightian uncertainty. They would model uncertain probabilities with
3474:
Other critics argue applying expected utility to economic and policy decisions, has engendered inappropriate valuations, particularly in scenarios in which monetary units are used to scale the utility of nonmonetary outcomes, such as deaths.
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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
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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. The main problem with the
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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.
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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.
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in which case the right side of this equation has a finite number of terms; or there could be an infinite set of discrete values, in which case the right side has an infinite number of terms.
3173:
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 ArrowâPratt measure of absolute risk aversion:
293:, according to this theorem, we should be able to know the beliefs and utilities from a person just by looking at the choices they make (which is wrong). Ramsey defines a proposition as "
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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
1048:
1596:
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
403:
113:
4469:
Subjects changed their beliefs faster by conditioning on evidence (Bayes's theorem) than by using informal reasoning, according to a classic study by the psychologist Ward
Edwards:
244:(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
1381:
437:(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
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allowing different people to disagree about the degree of risk associated with any given portfolio. Individuals sharing a given risk measure (based on given value of
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illustrates that decision making based on expected value of monetary payoffs lead to absurd conclusions. When a probability distribution function has an infinite
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There are three components in the psychology field that are seen as crucial to the development of a more accurate descriptive theory of decision under risks.
3149:, 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
3483:
Psychologists have discovered systematic violations of probability calculations and behavior by humans. This have been evidenced with examples such as the
297:" when two possible outcomes have an equal value. In other words, if the probability can be defined in terms of a preference, each proposition should have
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3523:" because they can depend on the way the problem is presented, not on the actual costs, rewards, or probabilities involved. Particular theories include
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3519:
theories to account for instances where people's choices deviate from those predicted by expected utility theory. These deviations are described as "
3978:
Lindley DV (September 1973). "The foundations of statistics (second edition), by
Leonard J. Savage. Pp xv, 310. £1·75. 1972 (Dover/Constable)".
2401:
3586:, i.e. where the uncertain probabilities are modelled as distributions whose parameters are themselves drawn from a higher-level distribution (
3736:
507:
A consequence is the description of all that is relevant to the decision maker's utility (e.g. monetary rewards, psychological factors, etc)
4825:(1964). "Foresight: its Logical Laws, Its Subjective Sources (translation of the 1937 article in French". In Kyburg HE, Smokler HE (eds.).
4795:(September 1989). "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science (translation of 1931 article)".
1511:
536:
525:
495:
The specification of every aspect of the decision problem at hand or "A description of the world leaving no relevant aspect undescribed."
5945:
3761:
3032:
5846:
5680:
4961:
5179:
4180:
Owen J, Rabinovitch R (1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice".
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3179:
3551:
may occur. Thus one must make assumptions about the probabilities, but then the expected values of various decisions can be very
912:
193:
expresses the utility of each respective payoff. Graphically the curvature of the u function captures the agent's risk attitude.
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The class of constant relative risk aversion utility functions contains three categories. Bernoulli's utility function
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779:
assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.
3555:
to the assumptions. This is particularly a problem when the expectation is dominated by rare extreme events, as in a
3500:
has suggested that humans change their beliefs faster when using
Bayesian methods than when using informal judgment.
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3456:
which showed empirically, how preferences of individuals are inconsistent among the same choices, depending on the
3696:
3528:
3393:
3389:
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213:
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to uncertainty of probability of outcomes, either not depending on probabilities of outcomes and only requiring
1537:
is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the
6016:
284:
introduced the Ramsey's Representation Theorem. This representation theorem for expected utility assumed that
319:
5839:
4996:
54:
4153:
Chamberlain G (1983). "A characterization of the distributions that imply mean-variance utility functions".
551:
assumes that an individual has well defined preferences and can always decide between any two alternatives.
5371:
5172:
4624:
Lichtenstein S, Slovic P (1971). "Reversals of preference between bids and choices in gambling decisions".
4257:
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3516:
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281:
269:
200:
preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility
3539:
In practice there will be many situations where the probabilities are unknown, and one is operating under
5665:
5305:
4846:
3711:
3616:
1354:
546:
4504:
Phillips LD, Edwards W (October 2008). "Chapter 6: Conservatism in a simple probability inference task (
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is ordinal because any monotonic increasing transformation of expected utility gives the same behavior.
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5985:
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5327:
5315:
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5300:
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1957:
217:
4983:
Schoemaker PJ (1982). "The Expected Utility Model: Its Variants, Purposes, Evidence and Limitations".
1872:
1827:
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decision maker: completeness; transitivity; independence of irrelevant alternatives; and continuity.
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2560:
Often people refer to "risk" in the sense of a potentially quantifiable entity. In the context of
1602:
264:
Daniel Bernoulli drew attention to psychological and behavioral components behind the individual's
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Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
2005:
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5813:
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5705:
5655:
5528:
5406:
5165:
4661:
Grether DM, Plott CR (1979). "Economic Theory of Choice and the Preference Reversal Phenomenon".
3489:
3471:, which has produced deviations from expected utility theory to account for the empirical facts.
2573:
2561:
294:
265:
5049:
Briggs RA (2019). "Normative Theories of Rational Choice: Expected Utility". In Zalta EN (ed.).
4928:
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884:
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624:
598:
5975:
5940:
5890:
5730:
5690:
5685:
5467:
5438:
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5217:
5148:
Schoemaker PJ (1980). "Experiments on Decisions under Risk: The Expected Utility Hypothesis.".
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241:
40:
24:
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amounts to choosing the lottery with the highest expected utility. This result is called the
414:
3396:) functions, where ARA(w) is constant. They are often used in economics for simplification.
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6011:
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5874:
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5386:
5322:
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is used as a risk measure for portfolio return; however, this is only valid if returns are
2318:
2251:
2097:
1450:
1073:
237:
165:
118:
4474:
Edwards W (1968). "Conservatism in Human Information Processing". In Kleinmuntz, B (ed.).
2500:
8:
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4972:
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Aspects of the Theory of Risk Bearing Reprinted in Essays in the Theory of Risk Bearing
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3092:, which differs between two individuals if they have different values of the parameter
2378:
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145:
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5771:
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5720:
5582:
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5462:
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4936:
Pfanzagl J, Baumann V, Huber H (1968). "Events, Utility and Subjective Probability".
4905:
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4854:
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3828:
3784:
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3413:
5041:
4948:(1993). "Chapter 7 (specifically) and 8, 9, 10, (to show paradoxes to the theory)".
4446:
4344:
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losses cannot be represented with utility as they change under different scenarios.
5935:
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3952:
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3820:
3776:
3681:
3583:
3150:
1530:
245:
4377:
Oberhelman DD (June 2001). Zalta EN (ed.). "Stanford Encyclopedia of Philosophy".
3780:
2395:
can take on any of a continuous range of values, the expected utility is given by
2238:{\displaystyle \operatorname {E} =p_{1}\cdot u(x_{1})+p_{2}\cdot u(x_{2})+\cdots }
5895:
5855:
5803:
5275:
5212:
4957:
4484:
4483:
Edwards W (1982). "Conservatism in Human Information Processing (excerpted)". In
3691:
3656:
3560:
3524:
3497:
3493:
3453:
3445:
1526:
209:
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197:
28:
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5778:
5518:
5477:
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5188:
5033:
5000:
4884:
4390:
4328:
4215:
Bell DE (December 1988). "One-switch utility functions and a measure of risk".
3636:
3572:
3430:
3417:
249:
44:
36:
4728:
4271:
Pratt JW (JanuaryâApril 1964). "Risk aversion in the small and in the large".
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5600:
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4305:
Castagnoli and LiCalzi, 1996; Bordley and LiCalzi, 2000; Bordley and Kirkwood
4085:
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3964:
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3701:
3671:
3154:
3146:
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1281:
in which the individual is then indifferent between this mix and the lottery
253:
4228:
4030:
3119:) may choose different portfolios because they may have different values of
5670:
5502:
5497:
5231:
5140:
5122:
4610:
4492:
3449:
1521:
The von NeumannâMorgenstern formulation is important in the application of
257:
4779:
Arrow KJ (1963). "Uncertainty and the Welfare Economics of Medical Care".
2487:{\displaystyle \operatorname {E} =\int _{-\infty }^{\infty }u(x)f(x)\,dx,}
2342:
is its probability. There could be either a finite set of possible values
2248:
where the left side is the subjective valuation of the gamble as a whole,
5965:
5695:
5607:
5487:
5339:
5253:
4945:
4488:
4118:
Borch K (January 1969). "A note on uncertainty and indifference curves".
3540:
2125:, the formula for expected utility, which is assumed to be maximized, is
290:
32:
1544:
39:
maximize utility, meaning the subjective desirability of their actions.
5455:
5207:
4705:
4676:
4432:
4292:
4201:
4139:
4076:
4059:
3587:
3145:
The expected utility theory takes into account that individuals may be
1522:
285:
4646:
2066:
of utility functions having hyperbolic absolute risk aversion (HARA).
408:
275:
5766:
5533:
5450:
5445:
5258:
4637:
3434:
1525:
to economics because it was developed shortly after the HicksâAllen "
4811:(1937). "La Prévision: ses lois logiques, ses sources subjectives".
4721:
Experiments on Decisions under Risk: The Expected Utility Hypothesis
4424:
4284:
4242:
Arrow KJ (1965). "The theory of risk aversion". In Saatio YJ (ed.).
4193:
4131:
515:
An act is a finite-valued function that maps states to consequences.
5558:
5553:
5428:
5071:
2565:
313:
in order to be indifferent between both options. Ramsey shows that
5824:
5105:
4690:
Holt C (1986). "Preference Reversals and the Independence Axiom".
4575:"Rationality, the Bayesian standpoint, and the Monty-Hall problem"
4025:, Berlin, New York: Walter de Gruyter, pp. 1â36, 2009-01-21,
3622:
Better understanding of the psychologically relevant outcome space
5915:
5910:
5612:
5511:
5423:
5157:
3568:
3085:{\displaystyle \operatorname {E} (e^{-a(w-\operatorname {E} w)})}
2555:
5798:
5587:
4538:
Vind K (February 2000). "von Neumann Morgenstern preferences".
3593:
4923:
Essays in Mathematical Economics In Honor of Oskar Morgenstern
3943:
Savage LJ (March 1951). "The Theory of Statistical Decision".
47:, builds this postulate to model aggregate social behaviour.
5472:
5354:
5334:
4904:
Pfanzagl J (1967). "Subjective Probability Derived from the
4107:(Third ed.). Princeton, NJ: Princeton University Press.
3378:{\displaystyle {\mathit {RRA}}(w)=-{\frac {wu''(w)}{u'(w)}}}
3259:{\displaystyle {\mathit {ARA}}(w)=-{\frac {u''(w)}{u'(w)}},}
5925:
5629:
3760:
Conte, Anna; Hey, John D.; Moffatt, Peter G. (2011-05-01).
921:
Axiom (Independence of irrelevant alternatives): For every
519:
4315:
Aase KK (January 2001). "On the St. Petersburg Paradox".
3906:"Normative Theories of Rational Choice: Expected Utility"
3310:
3307:
3191:
3188:
4969:
The Foundations of Mathematics and other Logical Essays
4510:
A Science of Decision Making:The Legacy of Ward Edwards
1533:
in economic theory. However, while in this context the
4098:
3292:
The ArrowâPratt measure of relative risk aversion is:
1821:
has relative risk aversion equal to 1. The functions
1693:
gives exactly the same preferences orderings as does
1512:
von NeumannâMorgenstern utility representation theorem
4406:"Prospect Theory: An Analysis of Decision under Risk"
3392:) functions, where RRA(w) is constant, and the CARA (
3301:
3275:
3182:
3098:
3035:
2656:
2585:
2536:
2503:
2404:
2381:
2348:
2321:
2285:
2254:
2134:
2100:
2080:
2034:
2008:
1960:
1913:
1875:
1830:
1777:
1735:
1699:
1660:
1605:
1555:
1545:
Examples of von NeumannâMorgenstern utility functions
1529:
revolution" of the 1930s, and it revived the idea of
1496:
1453:
1409:
1389:
1357:
1337:
1311:
1287:
1267:
1247:
1227:
1207:
1187:
1167:
1147:
1121:
1076:
1056:
979:
953:
927:
887:
861:
835:
815:
789:
757:
737:
717:
697:
677:
657:
627:
601:
581:
561:
530:
322:
168:
148:
121:
57:
4995:
4935:
4508:(1966) 72: 346-354)". In Weiss JW, Weiss DJ (eds.).
2121:
affects a person's utility takes on one of a set of
4246:. Chicago, 1971: Markham Publ. Co. pp. 90â109.
3625:
A psychologically richer theory of the determinants
3478:
3388:
Special classes of utility functions are the CRRA (
409:
Savage's subjective expected utility representation
276:
Ramsey-theoretic approach to subjective probability
5089:"The time resolution of the St Petersburg paradox"
4920:
4757:
4623:
4100:
3817:Expected Utility Hypotheses and the Allais Paradox
3377:
3281:
3258:
3153:and show diminishing marginal wealth utility. The
3107:
3084:
3018:
2628:
2545:
2518:
2486:
2387:
2364:
2334:
2307:
2267:
2237:
2113:
2086:
2052:
2020:
1991:
1943:
1899:
1858:
1810:
1757:
1721:
1685:
1642:
1588:
1502:
1477:
1439:
1395:
1375:
1343:
1323:
1293:
1273:
1253:
1233:
1213:
1193:
1173:
1153:
1133:
1100:
1062:
1042:
965:
939:
899:
873:
847:
821:
801:
763:
743:
723:
703:
683:
663:
639:
613:
587:
567:
428:
397:
204:(though still not comparable across individuals).
181:
154:
134:
107:
5020:Aase KK (2001). "On the St. Petersburg Paradox".
4497:Judgment under uncertainty: Heuristics and biases
4060:"Attitudes to Uncertainty in a Strategic Setting"
1241:, then there should be a possible combination of
5998:
4057:
4863:Selected Economic Writings of Oskar Morgenstern
4503:
4403:
4179:
3945:Journal of the American Statistical Association
3759:
2069:
4950:The psychology of judgment and decision making
4831:. Vol. 7. New York: Wiley. pp. 1â68.
3884:"Ramsey and the Ethically Neutral Proposition"
2556:Measuring risk in the expected utility context
268:and proposed that the utility of wealth has a
5840:
5173:
4879:
4865:. New York University Press. pp. 65â70.
3938:
3936:
3460:of the choices, i.e. how they are presented.
3403:
3161:Since the risk attitudes are unchanged under
1115:assumes that when there are three lotteries (
539:of the expected utility theory that define a
4660:
3737:"Expected Utility Theory | Encyclopedia.com"
3594:Preference reversals over uncertain outcomes
2639:for individual-specific positive parameters
488:The key ingredients in Savage's theory are:
4893:Memoirs of the National Academy of Sciences
4853:
4152:
3814:
3575:), or being less sensitive to assumptions.
1729:; thus it is irrelevant that the values of
1043:{\displaystyle tA+(1-t)C\succeq tB+(1-t)C,}
142:is the probability that outcome indexed by
5946:Heuristics in judgment and decision-making
5847:
5833:
5180:
5166:
5147:
5058:Hacking I (1980). "Strange Expectations".
4982:
4837:
4821:
4807:
4791:
4718:
4376:
4053:
4051:
4049:
3933:
232:
5130:
5104:
5009:Decision-Making: An Experimental Approach
4760:Foundations of Rational Choice Under Risk
4645:
4600:
4590:
4075:
4058:Li Z, Loomes G, Pogrebna G (2017-05-01).
2474:
4903:
4572:
4512:. Oxford University Press. p. 536.
4021:"1. Foundations of probability theory",
3507:
398:{\displaystyle P(E)=(1-U(m))(U(b)-U(w))}
5057:
5051:The Stanford Encyclopedia of Philosophy
4927:. Princeton University Press. pp.
4482:
4476:Formal Representation of Human Judgment
4473:
4046:
3977:
3847:
913:Independence of irrelevant alternatives
651:This means that the individual prefers
526:Von NeumannâMorgenstern utility theorem
520:Von NeumannâMorgenstern utility theorem
108:{\displaystyle U(p)=\sum u(x_{k})p_{k}}
5999:
5086:
5048:
4956:
4357:
4099:von Neumann J, Morgenstern O (1953) .
3942:
3903:
3559:. Alternative decision techniques are
5828:
5161:
4944:
4778:
4755:
4533:
4531:
4270:
4241:
4117:
4103:Theory of Games and Economic Behavior
3762:"Mixture models of choice under risk"
2647:. Then expected utility is given by
2028:have relative risk aversion equal to
1907:have relative risk aversion equal to
501:A set of states identified by someone
196:Standard utility functions represent
5019:
4962:"Chapter VII: Truth and Probability"
4813:Annales de l'Institut Henri Poincaré
4689:
4537:
4447:"Expected utility | decision theory"
4314:
4214:
3810:
3808:
3806:
5854:
5150:Experiments on Decisions under Risk
4889:"On Small Differences in Sensation"
4764:. Oxford: Oxford University Press.
4364:Stanford Encyclopedia of Philosophy
3881:
3819:. Dordrecht: Springer Netherlands.
1376:{\displaystyle A\succeq B\succeq C}
13:
5736:Microfoundations of macroeconomics
5187:
4748:
4712:
4626:Journal of Experimental Psychology
4528:
4506:Journal of Experimental Psychology
4370:
4308:
4264:
4235:
3869:10.1111/j.1746-8361.2004.tb00320.x
3606:
3304:
3185:
3065:
3036:
2933:
2904:
2887:
2855:
2822:
2792:
2772:
2751:
2713:
2692:
2661:
2445:
2440:
2405:
2135:
1944:{\displaystyle 1-\alpha \in (0,1)}
531:The von NeumannâMorgenstern axioms
14:
6038:
4828:Studies in Subjective Probability
4540:Journal of Mathematical Economics
3850:"Ramsey's Representation Theorem"
3803:
1992:{\displaystyle u(w)=-w^{\alpha }}
5809:
5808:
5797:
3815:Allais M, Hagen O, eds. (1979).
3479:Conservatism in updating beliefs
3134:
2574:jointly elliptically distributed
1900:{\displaystyle \alpha \in (0,1)}
1859:{\displaystyle u(w)=w^{\alpha }}
783:Axiom (Transitivity): For every
555:Axiom (Completeness): For every
227:
23:is a foundational assumption in
4683:
4654:
4617:
4566:
4463:
4439:
4397:
4358:Martin, Robert (16 June 2008).
4351:
4299:
4208:
4173:
4146:
4111:
4092:
3697:Rank-dependent expected utility
3529:rank-dependent expected utility
3394:constant absolute risk aversion
3390:constant relative risk aversion
2629:{\displaystyle u(w)=w-be^{-aw}}
2053:{\displaystyle 1-\alpha >1.}
429:Savage's representation theorem
214:rank-dependent expected utility
5022:Scandinavian Actuarial Journal
4985:Journal of Economic Literature
4404:Kahneman D, Tversky A (1979).
4317:Scandinavian Actuarial Journal
4023:Interpretations of Probability
4014:
3971:
3957:10.1080/01621459.1951.10500768
3897:
3891:Australian National University
3875:
3841:
3753:
3729:
3369:
3363:
3350:
3344:
3321:
3315:
3247:
3241:
3228:
3222:
3202:
3196:
3079:
3074:
3056:
3042:
2953:
2948:
2945:
2939:
2924:
2910:
2899:
2893:
2867:
2861:
2842:
2837:
2834:
2828:
2813:
2804:
2798:
2778:
2763:
2757:
2738:
2719:
2704:
2698:
2682:
2679:
2673:
2667:
2595:
2589:
2513:
2507:
2471:
2465:
2459:
2453:
2426:
2423:
2417:
2411:
2302:
2289:
2226:
2213:
2191:
2178:
2156:
2153:
2147:
2141:
1970:
1964:
1938:
1926:
1894:
1882:
1840:
1834:
1805:
1799:
1787:
1781:
1615:
1609:
1583:
1577:
1565:
1559:
1472:
1460:
1431:
1419:
1095:
1083:
1031:
1019:
1001:
989:
392:
389:
383:
374:
368:
362:
359:
356:
350:
338:
332:
326:
252:of an outcome, accounting for
92:
79:
67:
61:
1:
4861:". In Andrew Schotter (ed.).
4857:(1976). "Some Reflections on
4552:10.1016/s0304-4068(99)00004-x
4499:. Cambridge University Press.
3781:10.1016/j.jeconom.2009.10.011
3723:
1643:{\displaystyle u(w)=-e^{-aw}}
1161:) and the individual prefers
435:Savage representation theorem
4360:"The St. Petersburg Paradox"
4167:10.1016/0022-0531(83)90129-1
3662:Generalized expected utility
3517:generalized expected utility
3423:
2528:probability density function
2070:Formula for expected utility
2021:{\displaystyle \alpha <0}
1811:{\displaystyle u(w)=\log(w)}
1589:{\displaystyle u(w)=\log(w)}
1050:must hold for every lottery
731:, or is indifferent between
270:diminishing marginal utility
7:
5870:Expected utility hypothesis
5681:Civil engineering economics
5666:Statistical decision theory
5306:Income elasticity of demand
3712:Subjective expected utility
3629:
21:expected utility hypothesis
10:
6043:
5986:Evidential decision theory
5316:Price elasticity of supply
5311:Price elasticity of demand
5301:Cross elasticity of demand
5034:10.1080/034612301750077356
4940:. Wiley. pp. 195â220.
4391:10.1108/rr.2001.15.6.9.311
4329:10.1080/034612301750077356
4155:Journal of Economic Theory
4120:Review of Economic Studies
3717:Two-moment decision models
3533:cumulative prospect theory
3404:The St. Petersburg paradox
3138:
966:{\displaystyle A\succeq B}
900:{\displaystyle A\succeq C}
874:{\displaystyle B\succeq C}
848:{\displaystyle A\succeq B}
640:{\displaystyle A\preceq B}
614:{\displaystyle A\succeq B}
523:
218:cumulative prospect theory
189:is realized, and function
5921:Principle of indifference
5883:
5862:
5792:
5759:
5638:
5195:
5013:Stanford University Press
4729:10.1007/978-94-017-5040-0
4573:Baratgin J (2015-08-11).
3992:10.1017/s0025557200132589
3825:10.1007/978-94-015-7629-1
3158:of the utility function.
3029:Thus the risk measure is
1686:{\displaystyle K-e^{-aw}}
1539:expected utility function
1440:{\displaystyle pA+(1-p)C}
5372:Incomeâconsumption curve
4781:American Economic Review
4693:American Economic Review
4664:American Economic Review
4592:10.3389/fpsyg.2015.01168
4256:: CS1 maint: location (
3980:The Mathematical Gazette
3904:Briggs RA (2014-08-08).
3557:long-tailed distribution
3452:in 1979 presented their
3169:, the second derivative
2308:{\displaystyle u(x_{i})}
1758:{\displaystyle -e^{-aw}}
1722:{\displaystyle -e^{-aw}}
1503:{\displaystyle \succeq }
1403:is equally preferred to
1305:Axiom (Continuity): Let
5981:Emotional choice theory
5706:Industrial organization
4579:Frontiers in Psychology
4451:Encyclopedia Britannica
4229:10.1287/mnsc.34.12.1416
4031:10.1515/9783110213195.1
3769:Journal of Econometrics
3490:conditional probability
266:decision-making process
233:Bernoulli's formulation
5976:Causal decision theory
5941:St. Petersburg paradox
5891:Decision-matrix method
5123:10.1098/rsta.2011.0065
4719:Schoemaker PJ (1980).
3921:Cite journal requires
3439:St. Petersburg Paradox
3410:St. Petersburg paradox
3379:
3283:
3260:
3163:affine transformations
3109:
3086:
3020:
2630:
2562:mean-variance analysis
2547:
2520:
2488:
2389:
2366:
2365:{\displaystyle x_{i},}
2336:
2315:is its valuation, and
2309:
2269:
2239:
2115:
2088:
2054:
2022:
1993:
1945:
1901:
1860:
1812:
1759:
1723:
1687:
1644:
1590:
1504:
1479:
1441:
1397:
1377:
1345:
1325:
1295:
1275:
1255:
1235:
1215:
1195:
1175:
1155:
1135:
1102:
1064:
1044:
967:
941:
901:
875:
849:
823:
803:
765:
745:
725:
705:
685:
665:
641:
615:
589:
569:
399:
242:St. Petersburg paradox
183:
156:
136:
109:
41:Rational choice theory
25:mathematical economics
6017:Motivational theories
5956:Bayesian epistemology
5676:Engineering economics
5271:Costâbenefit analysis
5060:Philosophy of Science
4938:Theory of Measurement
4843:Theory of Probability
3677:Lottery (probability)
3601:certainty equivalents
3545:Knightian uncertainty
3515:has produced several
3508:Irrational deviations
3431:expected value theory
3380:
3284:
3261:
3125:Entropic risk measure
3110:
3087:
3021:
2631:
2548:
2521:
2489:
2390:
2367:
2337:
2335:{\displaystyle p_{i}}
2310:
2279:th possible outcome,
2270:
2268:{\displaystyle x_{i}}
2240:
2116:
2114:{\displaystyle x_{i}}
2089:
2055:
2023:
1994:
1951:. And the functions
1946:
1902:
1861:
1813:
1760:
1724:
1688:
1645:
1591:
1549:The utility function
1505:
1480:
1478:{\displaystyle p\in }
1442:
1398:
1378:
1346:
1326:
1296:
1276:
1256:
1236:
1216:
1196:
1176:
1156:
1136:
1103:
1101:{\displaystyle t\in }
1065:
1045:
968:
942:
902:
876:
850:
824:
804:
766:
746:
726:
706:
686:
666:
642:
616:
590:
570:
415:Leonard Jimmie Savage
400:
184:
182:{\displaystyle x_{k}}
157:
137:
135:{\displaystyle p_{k}}
110:
35:. It postulates that
5971:Social choice theory
5875:Intertemporal choice
5493:Price discrimination
5387:Intertemporal choice
4064:The Economic Journal
3741:www.encyclopedia.com
3652:Behavioral economics
3647:Bayesian probability
3299:
3273:
3180:
3096:
3033:
2654:
2583:
2570:normally distributed
2534:
2519:{\displaystyle f(x)}
2501:
2402:
2379:
2346:
2319:
2283:
2252:
2132:
2098:
2078:
2032:
2006:
1958:
1911:
1873:
1828:
1775:
1733:
1697:
1658:
1603:
1553:
1494:
1451:
1407:
1387:
1355:
1335:
1309:
1285:
1265:
1245:
1225:
1205:
1185:
1165:
1145:
1119:
1074:
1054:
977:
951:
925:
885:
859:
833:
813:
787:
755:
735:
715:
695:
675:
655:
625:
599:
579:
559:
320:
166:
146:
119:
55:
16:Concept in economics
5906:Strategic dominance
5804:Business portal
5741:Operations research
5568:Substitution effect
5115:2011RSPTA.369.4913P
5099:(1956): 4913â4931.
3619:effect (psychology)
3584:hierarchical models
3496:. An experiment on
3435:monetary incentives
2449:
1324:{\displaystyle A,B}
1134:{\displaystyle A,B}
940:{\displaystyle A,B}
802:{\displaystyle A,B}
222:bounded rationality
43:, a cornerstone of
5951:Probability theory
5382:Indifference curve
5350:Goods and services
5291:Economies of scope
5286:Economies of scale
5087:Peters O (2011) .
4849:. New York: Wiley.
4217:Management Science
4182:Journal of Finance
4077:10.1111/ecoj.12486
3848:Bradley R (2004).
3707:Risk in psychology
3687:Priority heuristic
3667:Indifference price
3642:Ambiguity aversion
3513:Behavioral finance
3485:Monty Hall problem
3469:behavioral finance
3465:mathematical model
3375:
3279:
3256:
3108:{\displaystyle a,}
3105:
3082:
3016:
3014:
2626:
2546:{\displaystyle x.}
2543:
2516:
2484:
2432:
2385:
2362:
2332:
2305:
2265:
2235:
2111:
2084:
2050:
2018:
1989:
1941:
1897:
1856:
1808:
1755:
1719:
1683:
1640:
1586:
1500:
1475:
1437:
1393:
1373:
1351:be lotteries with
1341:
1321:
1291:
1271:
1251:
1231:
1211:
1191:
1171:
1151:
1131:
1098:
1060:
1040:
963:
937:
897:
871:
845:
819:
799:
761:
741:
721:
701:
681:
661:
637:
611:
585:
565:
395:
238:Nicolaus Bernoulli
179:
152:
132:
105:
6022:Optimal decisions
5994:
5993:
5822:
5821:
5784:Political economy
5583:Supply and demand
5463:Pareto efficiency
4872:978-0-8147-7771-8
4771:978-0-19-823303-9
4738:978-94-017-5042-4
4519:978-0-19-532298-9
4379:Reference Reviews
4040:978-3-11-021319-5
3834:978-90-481-8354-8
3565:scenario analysis
3414:Nicolas Bernoulli
3373:
3282:{\displaystyle w}
3251:
3007:
2997:
2969:
2388:{\displaystyle x}
2087:{\displaystyle x}
1396:{\displaystyle B}
1344:{\displaystyle C}
1294:{\displaystyle B}
1274:{\displaystyle C}
1254:{\displaystyle A}
1234:{\displaystyle C}
1214:{\displaystyle B}
1194:{\displaystyle B}
1174:{\displaystyle A}
1154:{\displaystyle C}
1063:{\displaystyle C}
973:, the preference
822:{\displaystyle C}
764:{\displaystyle B}
744:{\displaystyle A}
724:{\displaystyle A}
704:{\displaystyle B}
684:{\displaystyle B}
664:{\displaystyle A}
588:{\displaystyle B}
568:{\displaystyle A}
295:ethically neutral
155:{\displaystyle k}
6034:
6027:Expected utility
5936:Ellsberg paradox
5901:Expected utility
5849:
5842:
5835:
5826:
5825:
5812:
5811:
5802:
5801:
5544:Returns to scale
5402:Market structure
5182:
5175:
5168:
5159:
5158:
5153:
5144:
5134:
5108:
5083:
5054:
5045:
5016:
4992:
4979:
4977:
4971:. Archived from
4966:
4953:
4941:
4932:
4926:
4900:
4876:
4850:
4845:. Translated by
4832:
4816:
4804:
4788:
4775:
4763:
4756:Anand P (1993).
4743:
4742:
4716:
4710:
4709:
4687:
4681:
4680:
4658:
4652:
4651:
4649:
4638:10.1037/h0031207
4621:
4615:
4614:
4604:
4594:
4570:
4564:
4563:
4535:
4526:
4523:
4500:
4479:
4467:
4461:
4460:
4458:
4457:
4443:
4437:
4436:
4410:
4401:
4395:
4394:
4374:
4368:
4367:
4355:
4349:
4348:
4312:
4306:
4303:
4297:
4296:
4279:(1/2): 122â136.
4268:
4262:
4261:
4255:
4247:
4239:
4233:
4232:
4212:
4206:
4205:
4177:
4171:
4170:
4150:
4144:
4143:
4115:
4109:
4108:
4106:
4096:
4090:
4089:
4079:
4070:(601): 809â826.
4055:
4044:
4043:
4018:
4012:
4011:
3986:(401): 220â221.
3975:
3969:
3968:
3940:
3931:
3930:
3924:
3919:
3917:
3909:
3901:
3895:
3894:
3888:
3879:
3873:
3872:
3854:
3845:
3839:
3838:
3812:
3801:
3800:
3766:
3757:
3751:
3750:
3748:
3747:
3733:
3682:Marginal utility
3617:decision framing
3543:. In economics,
3384:
3382:
3381:
3376:
3374:
3372:
3362:
3353:
3343:
3331:
3314:
3313:
3288:
3286:
3285:
3280:
3265:
3263:
3262:
3257:
3252:
3250:
3240:
3231:
3221:
3212:
3195:
3194:
3114:
3112:
3111:
3106:
3091:
3089:
3088:
3083:
3078:
3077:
3025:
3023:
3022:
3017:
3015:
3008:
3005:
3000:
2999:
2998:
2995:
2970:
2967:
2959:
2952:
2951:
2903:
2902:
2848:
2841:
2840:
2744:
2737:
2736:
2635:
2633:
2632:
2627:
2625:
2624:
2552:
2550:
2549:
2544:
2525:
2523:
2522:
2517:
2493:
2491:
2490:
2485:
2448:
2443:
2394:
2392:
2391:
2386:
2371:
2369:
2368:
2363:
2358:
2357:
2341:
2339:
2338:
2333:
2331:
2330:
2314:
2312:
2311:
2306:
2301:
2300:
2274:
2272:
2271:
2266:
2264:
2263:
2244:
2242:
2241:
2236:
2225:
2224:
2206:
2205:
2190:
2189:
2171:
2170:
2120:
2118:
2117:
2112:
2110:
2109:
2093:
2091:
2090:
2085:
2074:When the entity
2059:
2057:
2056:
2051:
2027:
2025:
2024:
2019:
1998:
1996:
1995:
1990:
1988:
1987:
1950:
1948:
1947:
1942:
1906:
1904:
1903:
1898:
1865:
1863:
1862:
1857:
1855:
1854:
1817:
1815:
1814:
1809:
1764:
1762:
1761:
1756:
1754:
1753:
1728:
1726:
1725:
1720:
1718:
1717:
1692:
1690:
1689:
1684:
1682:
1681:
1649:
1647:
1646:
1641:
1639:
1638:
1595:
1593:
1592:
1587:
1535:utility function
1531:cardinal utility
1509:
1507:
1506:
1501:
1484:
1482:
1481:
1476:
1446:
1444:
1443:
1438:
1402:
1400:
1399:
1394:
1382:
1380:
1379:
1374:
1350:
1348:
1347:
1342:
1330:
1328:
1327:
1322:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1200:
1198:
1197:
1192:
1180:
1178:
1177:
1172:
1160:
1158:
1157:
1152:
1140:
1138:
1137:
1132:
1107:
1105:
1104:
1099:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
972:
970:
969:
964:
946:
944:
943:
938:
906:
904:
903:
898:
880:
878:
877:
872:
854:
852:
851:
846:
828:
826:
825:
820:
808:
806:
805:
800:
770:
768:
767:
762:
750:
748:
747:
742:
730:
728:
727:
722:
710:
708:
707:
702:
690:
688:
687:
682:
670:
668:
667:
662:
646:
644:
643:
638:
620:
618:
617:
612:
594:
592:
591:
586:
574:
572:
571:
566:
404:
402:
401:
396:
312:
310:
309:
306:
303:
246:marginal utility
188:
186:
185:
180:
178:
177:
161:
159:
158:
153:
141:
139:
138:
133:
131:
130:
114:
112:
111:
106:
104:
103:
91:
90:
6042:
6041:
6037:
6036:
6035:
6033:
6032:
6031:
6007:Belief revision
5997:
5996:
5995:
5990:
5896:Decision matrix
5879:
5858:
5856:Decision theory
5853:
5823:
5818:
5796:
5788:
5755:
5634:
5276:Deadweight loss
5213:Consumer choice
5191:
5186:
5156:
4975:
4964:
4873:
4772:
4751:
4749:Further reading
4746:
4739:
4717:
4713:
4688:
4684:
4659:
4655:
4622:
4618:
4571:
4567:
4536:
4529:
4520:
4485:Daniel Kahneman
4470:
4468:
4464:
4455:
4453:
4445:
4444:
4440:
4425:10.2307/1914185
4408:
4402:
4398:
4375:
4371:
4356:
4352:
4313:
4309:
4304:
4300:
4285:10.2307/1913738
4269:
4265:
4249:
4248:
4240:
4236:
4223:(12): 1416â24.
4213:
4209:
4194:10.2307/2328079
4178:
4174:
4151:
4147:
4132:10.2307/2296336
4116:
4112:
4097:
4093:
4056:
4047:
4041:
4020:
4019:
4015:
3976:
3972:
3941:
3934:
3922:
3920:
3911:
3910:
3902:
3898:
3886:
3880:
3876:
3852:
3846:
3842:
3835:
3813:
3804:
3764:
3758:
3754:
3745:
3743:
3735:
3734:
3730:
3726:
3721:
3692:Prospect theory
3657:Decision theory
3632:
3609:
3607:Recommendations
3596:
3525:prospect theory
3510:
3498:belief revision
3481:
3454:prospect theory
3446:Daniel Kahneman
3426:
3406:
3355:
3354:
3336:
3332:
3330:
3303:
3302:
3300:
3297:
3296:
3274:
3271:
3270:
3233:
3232:
3214:
3213:
3211:
3184:
3183:
3181:
3178:
3177:
3143:
3137:
3097:
3094:
3093:
3049:
3045:
3034:
3031:
3030:
3013:
3012:
3004:
2996:expected wealth
2994:
2984:
2980:
2968:expected wealth
2966:
2957:
2956:
2917:
2913:
2880:
2876:
2846:
2845:
2785:
2781:
2742:
2741:
2726:
2722:
2685:
2657:
2655:
2652:
2651:
2614:
2610:
2584:
2581:
2580:
2558:
2535:
2532:
2531:
2502:
2499:
2498:
2444:
2436:
2403:
2400:
2399:
2380:
2377:
2376:
2353:
2349:
2347:
2344:
2343:
2326:
2322:
2320:
2317:
2316:
2296:
2292:
2284:
2281:
2280:
2259:
2255:
2253:
2250:
2249:
2220:
2216:
2201:
2197:
2185:
2181:
2166:
2162:
2133:
2130:
2129:
2123:discrete values
2105:
2101:
2099:
2096:
2095:
2079:
2076:
2075:
2072:
2033:
2030:
2029:
2007:
2004:
2003:
1983:
1979:
1959:
1956:
1955:
1912:
1909:
1908:
1874:
1871:
1870:
1850:
1846:
1829:
1826:
1825:
1776:
1773:
1772:
1743:
1739:
1734:
1731:
1730:
1707:
1703:
1698:
1695:
1694:
1671:
1667:
1659:
1656:
1655:
1628:
1624:
1604:
1601:
1600:
1554:
1551:
1550:
1547:
1495:
1492:
1491:
1452:
1449:
1448:
1408:
1405:
1404:
1388:
1385:
1384:
1356:
1353:
1352:
1336:
1333:
1332:
1310:
1307:
1306:
1286:
1283:
1282:
1266:
1263:
1262:
1246:
1243:
1242:
1226:
1223:
1222:
1206:
1203:
1202:
1186:
1183:
1182:
1166:
1163:
1162:
1146:
1143:
1142:
1120:
1117:
1116:
1075:
1072:
1071:
1055:
1052:
1051:
978:
975:
974:
952:
949:
948:
926:
923:
922:
886:
883:
882:
860:
857:
856:
834:
831:
830:
814:
811:
810:
788:
785:
784:
756:
753:
752:
736:
733:
732:
716:
713:
712:
696:
693:
692:
676:
673:
672:
656:
653:
652:
626:
623:
622:
600:
597:
596:
580:
577:
576:
560:
557:
556:
533:
528:
522:
431:
411:
321:
318:
317:
307:
304:
301:
300:
298:
278:
235:
230:
210:prospect theory
173:
169:
167:
164:
163:
147:
144:
143:
126:
122:
120:
117:
116:
99:
95:
86:
82:
56:
53:
52:
37:rational agents
29:decision making
17:
12:
11:
5:
6040:
6030:
6029:
6024:
6019:
6014:
6009:
5992:
5991:
5989:
5988:
5983:
5978:
5973:
5968:
5963:
5958:
5953:
5948:
5943:
5938:
5933:
5931:Allais paradox
5928:
5923:
5918:
5913:
5908:
5903:
5898:
5893:
5887:
5885:
5881:
5880:
5878:
5877:
5872:
5866:
5864:
5860:
5859:
5852:
5851:
5844:
5837:
5829:
5820:
5819:
5817:
5816:
5806:
5793:
5790:
5789:
5787:
5786:
5781:
5779:Macroeconomics
5776:
5775:
5774:
5763:
5761:
5757:
5756:
5754:
5753:
5748:
5743:
5738:
5733:
5728:
5723:
5718:
5713:
5708:
5703:
5698:
5693:
5688:
5683:
5678:
5673:
5668:
5663:
5658:
5653:
5648:
5642:
5640:
5636:
5635:
5633:
5632:
5627:
5626:
5625:
5620:
5610:
5605:
5604:
5603:
5594:
5580:
5575:
5570:
5565:
5556:
5551:
5546:
5541:
5536:
5531:
5526:
5521:
5516:
5515:
5514:
5509:
5500:
5495:
5490:
5485:
5480:
5478:Price controls
5470:
5465:
5460:
5459:
5458:
5453:
5448:
5443:
5442:
5441:
5436:
5426:
5421:
5420:
5419:
5414:
5399:
5397:Market failure
5394:
5389:
5384:
5379:
5374:
5369:
5364:
5363:
5362:
5357:
5347:
5342:
5337:
5332:
5331:
5330:
5320:
5319:
5318:
5313:
5308:
5303:
5293:
5288:
5283:
5278:
5273:
5268:
5267:
5266:
5261:
5256:
5251:
5250:
5249:
5239:
5234:
5224:
5215:
5210:
5205:
5199:
5197:
5193:
5192:
5189:Microeconomics
5185:
5184:
5177:
5170:
5162:
5155:
5154:
5145:
5084:
5072:10.1086/288956
5066:(4): 562â567.
5055:
5046:
5017:
4993:
4980:
4978:on 2006-10-14.
4954:
4942:
4933:
4913:Utility Theory
4901:
4877:
4871:
4851:
4834:
4833:
4818:
4817:
4805:
4789:
4776:
4770:
4752:
4750:
4747:
4745:
4744:
4737:
4711:
4700:(3): 508â515.
4682:
4671:(4): 623â638.
4653:
4616:
4565:
4546:(1): 109â122.
4527:
4525:
4524:
4518:
4501:
4480:
4462:
4438:
4419:(2): 263â292.
4396:
4369:
4350:
4307:
4298:
4263:
4234:
4207:
4188:(3): 745â752.
4172:
4161:(1): 185â201.
4145:
4110:
4091:
4045:
4039:
4013:
3970:
3951:(253): 55â67.
3932:
3923:|journal=
3896:
3874:
3863:(4): 483â498.
3840:
3833:
3802:
3752:
3727:
3725:
3722:
3720:
3719:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3637:Allais paradox
3633:
3631:
3628:
3627:
3626:
3623:
3620:
3608:
3605:
3595:
3592:
3573:minimax regret
3509:
3506:
3480:
3477:
3425:
3422:
3418:expected value
3405:
3402:
3386:
3385:
3371:
3368:
3365:
3361:
3358:
3352:
3349:
3346:
3342:
3339:
3335:
3329:
3326:
3323:
3320:
3317:
3312:
3309:
3306:
3278:
3267:
3266:
3255:
3249:
3246:
3243:
3239:
3236:
3230:
3227:
3224:
3220:
3217:
3210:
3207:
3204:
3201:
3198:
3193:
3190:
3187:
3139:Main article:
3136:
3133:
3104:
3101:
3081:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3048:
3044:
3041:
3038:
3027:
3026:
3011:
3003:
2993:
2990:
2987:
2983:
2979:
2976:
2973:
2965:
2962:
2960:
2958:
2955:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2916:
2912:
2909:
2906:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2879:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2849:
2847:
2844:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2784:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2745:
2743:
2740:
2735:
2732:
2729:
2725:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2686:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2659:
2637:
2636:
2623:
2620:
2617:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2557:
2554:
2542:
2539:
2515:
2512:
2509:
2506:
2495:
2494:
2483:
2480:
2477:
2473:
2470:
2467:
2464:
2461:
2458:
2455:
2452:
2447:
2442:
2439:
2435:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2407:
2384:
2361:
2356:
2352:
2329:
2325:
2304:
2299:
2295:
2291:
2288:
2262:
2258:
2246:
2245:
2234:
2231:
2228:
2223:
2219:
2215:
2212:
2209:
2204:
2200:
2196:
2193:
2188:
2184:
2180:
2177:
2174:
2169:
2165:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2108:
2104:
2083:
2071:
2068:
2064:the discussion
2049:
2046:
2043:
2040:
2037:
2017:
2014:
2011:
2000:
1999:
1986:
1982:
1978:
1975:
1972:
1969:
1966:
1963:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1867:
1866:
1853:
1849:
1845:
1842:
1839:
1836:
1833:
1819:
1818:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1752:
1749:
1746:
1742:
1738:
1716:
1713:
1710:
1706:
1702:
1680:
1677:
1674:
1670:
1666:
1663:
1651:
1650:
1637:
1634:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1546:
1543:
1499:
1487:
1486:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1392:
1372:
1369:
1366:
1363:
1360:
1340:
1320:
1317:
1314:
1290:
1270:
1250:
1230:
1210:
1190:
1170:
1150:
1130:
1127:
1124:
1110:
1109:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1059:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
962:
959:
956:
936:
933:
930:
909:
908:
896:
893:
890:
870:
867:
864:
844:
841:
838:
818:
798:
795:
792:
760:
740:
720:
700:
680:
660:
649:
648:
636:
633:
630:
610:
607:
604:
584:
564:
532:
529:
524:Main article:
521:
518:
517:
516:
508:
502:
496:
430:
427:
413:In the 1950s,
410:
407:
406:
405:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
337:
334:
331:
328:
325:
277:
274:
250:expected value
240:described the
234:
231:
229:
226:
176:
172:
151:
129:
125:
102:
98:
94:
89:
85:
81:
78:
75:
72:
69:
66:
63:
60:
45:microeconomics
15:
9:
6:
4:
3:
2:
6039:
6028:
6025:
6023:
6020:
6018:
6015:
6013:
6010:
6008:
6005:
6004:
6002:
5987:
5984:
5982:
5979:
5977:
5974:
5972:
5969:
5967:
5964:
5962:
5961:Risk aversion
5959:
5957:
5954:
5952:
5949:
5947:
5944:
5942:
5939:
5937:
5934:
5932:
5929:
5927:
5924:
5922:
5919:
5917:
5914:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5888:
5886:
5882:
5876:
5873:
5871:
5868:
5867:
5865:
5861:
5857:
5850:
5845:
5843:
5838:
5836:
5831:
5830:
5827:
5815:
5807:
5805:
5800:
5795:
5794:
5791:
5785:
5782:
5780:
5777:
5773:
5770:
5769:
5768:
5765:
5764:
5762:
5758:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5711:Institutional
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5687:
5684:
5682:
5679:
5677:
5674:
5672:
5669:
5667:
5664:
5662:
5659:
5657:
5656:Computational
5654:
5652:
5649:
5647:
5644:
5643:
5641:
5637:
5631:
5628:
5624:
5621:
5619:
5616:
5615:
5614:
5611:
5609:
5606:
5602:
5601:Law of supply
5598:
5595:
5593:
5592:Law of demand
5589:
5586:
5585:
5584:
5581:
5579:
5578:Social choice
5576:
5574:
5571:
5569:
5566:
5564:
5563:Excess supply
5560:
5557:
5555:
5552:
5550:
5549:Risk aversion
5547:
5545:
5542:
5540:
5537:
5535:
5532:
5530:
5527:
5525:
5522:
5520:
5517:
5513:
5510:
5508:
5504:
5501:
5499:
5496:
5494:
5491:
5489:
5486:
5484:
5483:Price ceiling
5481:
5479:
5476:
5475:
5474:
5471:
5469:
5466:
5464:
5461:
5457:
5454:
5452:
5449:
5447:
5444:
5440:
5439:Complementary
5437:
5435:
5432:
5431:
5430:
5427:
5425:
5422:
5418:
5415:
5413:
5410:
5409:
5408:
5405:
5404:
5403:
5400:
5398:
5395:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5373:
5370:
5368:
5365:
5361:
5358:
5356:
5353:
5352:
5351:
5348:
5346:
5343:
5341:
5338:
5336:
5333:
5329:
5326:
5325:
5324:
5321:
5317:
5314:
5312:
5309:
5307:
5304:
5302:
5299:
5298:
5297:
5294:
5292:
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5265:
5262:
5260:
5257:
5255:
5252:
5248:
5245:
5244:
5243:
5240:
5238:
5235:
5233:
5230:
5229:
5228:
5225:
5223:
5222:non-convexity
5219:
5216:
5214:
5211:
5209:
5206:
5204:
5201:
5200:
5198:
5194:
5190:
5183:
5178:
5176:
5171:
5169:
5164:
5163:
5160:
5151:
5146:
5142:
5138:
5133:
5128:
5124:
5120:
5116:
5112:
5107:
5102:
5098:
5094:
5090:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5056:
5052:
5047:
5043:
5039:
5035:
5031:
5027:
5023:
5018:
5014:
5010:
5006:
5002:
4998:
4994:
4990:
4986:
4981:
4974:
4970:
4963:
4959:
4955:
4951:
4947:
4943:
4939:
4934:
4930:
4925:
4924:
4918:
4917:Martin Shubik
4914:
4911:
4907:
4902:
4898:
4894:
4890:
4886:
4882:
4878:
4874:
4868:
4864:
4860:
4856:
4855:Morgenstern O
4852:
4848:
4844:
4840:
4836:
4835:
4830:
4829:
4824:
4820:
4819:
4814:
4810:
4806:
4802:
4798:
4794:
4790:
4786:
4782:
4777:
4773:
4767:
4762:
4761:
4754:
4753:
4740:
4734:
4730:
4726:
4722:
4715:
4707:
4703:
4699:
4695:
4694:
4686:
4678:
4674:
4670:
4666:
4665:
4657:
4648:
4643:
4639:
4635:
4631:
4627:
4620:
4612:
4608:
4603:
4598:
4593:
4588:
4584:
4580:
4576:
4569:
4561:
4557:
4553:
4549:
4545:
4541:
4534:
4532:
4521:
4515:
4511:
4507:
4502:
4498:
4494:
4490:
4486:
4481:
4477:
4472:
4471:
4466:
4452:
4448:
4442:
4434:
4430:
4426:
4422:
4418:
4414:
4407:
4400:
4392:
4388:
4384:
4380:
4373:
4365:
4361:
4354:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4318:
4311:
4302:
4294:
4290:
4286:
4282:
4278:
4274:
4267:
4259:
4253:
4245:
4238:
4230:
4226:
4222:
4218:
4211:
4203:
4199:
4195:
4191:
4187:
4183:
4176:
4168:
4164:
4160:
4156:
4149:
4141:
4137:
4133:
4129:
4125:
4121:
4114:
4105:
4104:
4095:
4087:
4083:
4078:
4073:
4069:
4065:
4061:
4054:
4052:
4050:
4042:
4036:
4032:
4028:
4024:
4017:
4009:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3974:
3966:
3962:
3958:
3954:
3950:
3946:
3939:
3937:
3928:
3915:
3907:
3900:
3892:
3885:
3878:
3870:
3866:
3862:
3858:
3851:
3844:
3836:
3830:
3826:
3822:
3818:
3811:
3809:
3807:
3798:
3794:
3790:
3786:
3782:
3778:
3774:
3770:
3763:
3756:
3742:
3738:
3732:
3728:
3718:
3715:
3713:
3710:
3708:
3705:
3703:
3702:Risk aversion
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3672:Loss function
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3634:
3624:
3621:
3618:
3614:
3613:
3612:
3604:
3602:
3591:
3589:
3585:
3580:
3576:
3574:
3570:
3566:
3562:
3558:
3554:
3550:
3546:
3542:
3537:
3534:
3530:
3526:
3522:
3518:
3514:
3505:
3501:
3499:
3495:
3494:rule of Bayes
3492:, namely the
3491:
3486:
3476:
3472:
3470:
3466:
3461:
3459:
3455:
3451:
3447:
3442:
3440:
3436:
3432:
3421:
3419:
3415:
3412:presented by
3411:
3401:
3397:
3395:
3391:
3366:
3359:
3356:
3347:
3340:
3337:
3333:
3327:
3324:
3318:
3295:
3294:
3293:
3290:
3276:
3253:
3244:
3237:
3234:
3225:
3218:
3215:
3208:
3205:
3199:
3176:
3175:
3174:
3172:
3168:
3164:
3159:
3156:
3155:risk attitude
3152:
3148:
3142:
3141:Risk aversion
3135:Risk aversion
3132:
3128:
3126:
3122:
3118:
3102:
3099:
3071:
3068:
3062:
3059:
3053:
3050:
3046:
3039:
3009:
3001:
2991:
2988:
2985:
2981:
2977:
2974:
2971:
2963:
2961:
2942:
2936:
2930:
2927:
2921:
2918:
2914:
2907:
2896:
2890:
2884:
2881:
2877:
2873:
2870:
2864:
2858:
2852:
2850:
2831:
2825:
2819:
2816:
2810:
2807:
2801:
2795:
2789:
2786:
2782:
2775:
2769:
2766:
2760:
2754:
2748:
2746:
2733:
2730:
2727:
2723:
2716:
2710:
2707:
2701:
2695:
2689:
2687:
2676:
2670:
2664:
2650:
2649:
2648:
2646:
2642:
2621:
2618:
2615:
2611:
2607:
2604:
2601:
2598:
2592:
2586:
2579:
2578:
2577:
2575:
2572:or otherwise
2571:
2567:
2563:
2553:
2540:
2537:
2529:
2510:
2504:
2481:
2478:
2475:
2468:
2462:
2456:
2450:
2437:
2433:
2429:
2420:
2414:
2408:
2398:
2397:
2396:
2382:
2373:
2359:
2354:
2350:
2327:
2323:
2297:
2293:
2286:
2278:
2260:
2256:
2232:
2229:
2221:
2217:
2210:
2207:
2202:
2198:
2194:
2186:
2182:
2175:
2172:
2167:
2163:
2159:
2150:
2144:
2138:
2128:
2127:
2126:
2124:
2106:
2102:
2081:
2067:
2065:
2060:
2047:
2044:
2041:
2038:
2035:
2015:
2012:
2009:
1984:
1980:
1976:
1973:
1967:
1961:
1954:
1953:
1952:
1935:
1932:
1929:
1923:
1920:
1917:
1914:
1891:
1888:
1885:
1879:
1876:
1851:
1847:
1843:
1837:
1831:
1824:
1823:
1822:
1802:
1796:
1793:
1790:
1784:
1778:
1771:
1770:
1769:
1766:
1750:
1747:
1744:
1740:
1736:
1714:
1711:
1708:
1704:
1700:
1678:
1675:
1672:
1668:
1664:
1661:
1635:
1632:
1629:
1625:
1621:
1618:
1612:
1606:
1599:
1598:
1597:
1580:
1574:
1571:
1568:
1562:
1556:
1542:
1540:
1536:
1532:
1528:
1524:
1519:
1515:
1513:
1497:
1469:
1466:
1463:
1457:
1454:
1434:
1428:
1425:
1422:
1416:
1413:
1410:
1390:
1370:
1367:
1364:
1361:
1358:
1338:
1318:
1315:
1312:
1304:
1303:
1302:
1288:
1268:
1248:
1228:
1208:
1188:
1168:
1148:
1128:
1125:
1122:
1114:
1092:
1089:
1086:
1080:
1077:
1057:
1037:
1034:
1028:
1025:
1022:
1016:
1013:
1010:
1007:
1004:
998:
995:
992:
986:
983:
980:
960:
957:
954:
934:
931:
928:
920:
919:
918:
915:
914:
894:
891:
888:
881:we must have
868:
865:
862:
842:
839:
836:
816:
796:
793:
790:
782:
781:
780:
778:
777:
772:
758:
738:
718:
698:
678:
658:
634:
631:
628:
608:
605:
602:
582:
562:
554:
553:
552:
550:
549:
544:
542:
538:
527:
514:
513:
509:
506:
505:Consequences:
503:
500:
497:
494:
491:
490:
489:
486:
484:
480:
476:
472:
468:
464:
460:
456:
452:
448:
444:
440:
436:
426:
422:
418:
416:
386:
380:
377:
371:
365:
353:
347:
344:
341:
335:
329:
323:
316:
315:
314:
296:
292:
287:
283:
273:
271:
267:
262:
259:
255:
254:risk aversion
251:
247:
243:
239:
228:Justification
225:
223:
219:
215:
211:
205:
203:
199:
194:
192:
174:
170:
149:
127:
123:
100:
96:
87:
83:
76:
73:
70:
64:
58:
48:
46:
42:
38:
34:
30:
26:
22:
5869:
5746:Optimization
5731:Mathematical
5691:Experimental
5686:Evolutionary
5671:Econometrics
5617:
5529:Public goods
5503:Price system
5498:Price signal
5412:Monopolistic
5281:Distribution
5196:Major topics
5149:
5096:
5092:
5063:
5059:
5050:
5028:(1): 69â78.
5025:
5021:
5008:
4988:
4984:
4973:the original
4968:
4949:
4937:
4922:
4896:
4892:
4862:
4842:
4839:de Finetti B
4827:
4823:de Finetti B
4812:
4809:de Finetti B
4800:
4796:
4793:de Finetti B
4784:
4780:
4759:
4720:
4714:
4697:
4691:
4685:
4668:
4662:
4656:
4632:(1): 46â55.
4629:
4625:
4619:
4582:
4578:
4568:
4543:
4539:
4509:
4505:
4496:
4493:Amos Tversky
4475:
4465:
4454:. Retrieved
4450:
4441:
4416:
4413:Econometrica
4412:
4399:
4382:
4378:
4372:
4363:
4353:
4323:(1): 69â78.
4320:
4316:
4310:
4301:
4276:
4273:Econometrica
4272:
4266:
4243:
4237:
4220:
4216:
4210:
4185:
4181:
4175:
4158:
4154:
4148:
4123:
4119:
4113:
4102:
4094:
4067:
4063:
4022:
4016:
3983:
3979:
3973:
3948:
3944:
3914:cite journal
3899:
3890:
3877:
3860:
3856:
3843:
3816:
3775:(1): 79â88.
3772:
3768:
3755:
3744:. Retrieved
3740:
3731:
3610:
3597:
3577:
3538:
3511:
3502:
3482:
3473:
3462:
3450:Amos Tversky
3443:
3427:
3407:
3398:
3387:
3291:
3268:
3170:
3166:
3160:
3144:
3129:
3120:
3116:
3028:
2644:
2640:
2638:
2559:
2496:
2374:
2276:
2247:
2094:whose value
2073:
2061:
2001:
1868:
1820:
1767:
1652:
1548:
1538:
1534:
1520:
1516:
1488:
1112:
1111:
911:
910:
776:Transitivity
775:
773:
650:
548:Completeness
547:
545:
540:
534:
511:
510:
504:
498:
492:
487:
482:
478:
474:
470:
466:
462:
458:
454:
450:
446:
442:
438:
432:
423:
419:
412:
282:Frank Ramsey
279:
263:
258:risk premium
256:, where the
236:
206:
195:
190:
162:with payoff
49:
20:
18:
6012:Game theory
5966:Game theory
5696:Game theory
5661:Development
5608:Uncertainty
5488:Price floor
5468:Preferences
5407:Competition
5377:Information
5340:Externality
5323:Equilibrium
5264:Transaction
5242:Opportunity
5203:Aggregation
4910:von Neumann
4906:Morgenstern
4489:Paul Slovic
3882:Elliott E.
3588:hyperpriors
3541:uncertainty
3289:is wealth.
3147:risk-averse
3123:. See also
537:four axioms
286:preferences
33:uncertainty
27:concerning
6001:Categories
5726:Managerial
5646:Behavioral
5519:Production
5456:Oligopsony
5296:Elasticity
5208:Budget set
4997:Davidson D
4991:: 529â563.
4797:Erkenntnis
4647:1794/22312
4456:2021-04-28
4126:(1): 1â4.
3857:Dialectica
3746:2021-04-28
3724:References
3615:Theory of
3521:irrational
1523:set theory
1113:Continuity
947:such that
535:There are
5863:Decisions
5767:Economics
5639:Subfields
5534:Rationing
5451:Oligopoly
5446:Monopsony
5434:Bilateral
5367:Household
5218:Convexity
5106:1011.4404
5080:224830682
4958:Ramsey RP
4885:Jastrow J
4881:Peirce CS
4787:: 941â73.
4560:0304-4068
4337:0346-1238
4252:cite book
4086:0013-0133
4008:164842618
4000:0025-5572
3965:0162-1459
3789:0304-4076
3553:sensitive
3549:ambiguity
3463:Like any
3424:Criticism
3328:−
3209:−
3069:
3063:−
3051:−
3040:
3002:⋅
2992:⋅
2986:−
2978:⋅
2972:−
2937:
2931:−
2919:−
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