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# Prospect Theory

Published Online: 15 JUN 2010

DOI: 10.1002/9780470400531.eorms0687

Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.

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## Wiley Encyclopedia of Operations Research and Management Science

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#### How to Cite

Chiu, A. and Wu, G. 2010. Prospect Theory. Wiley Encyclopedia of Operations Research and Management Science. .

#### Publication History

- Published Online: 15 JUN 2010

- Abstract
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Prospect theory is a descriptive theory of how people make choices that involve risk. The theory was developed by psychologists Kahneman and Tversky in 1979 1. In 2002, Kahneman was awarded the Nobel Prize in Economics, partially due to prospect theory's enormous influence on economics and other social sciences, and partially due to Kahneman and Tversky's research on heuristics and biases 2. (The Nobel Prize cannot be awarded posthumously, and Tversky died in 1996.) Prospect theory challenged the traditional view in economics that decision makers are rational and, in particular, expected utility maximizers. However, prospect theory's reach has not been limited to economics—the theory has been influential in understanding a wide variety of real-world phenomena, in fields ranging from business, law, and medicine to political science and public policy.

Most real-world decisions must be made without full knowledge of what will transpire in the future. Consider, for example, an oil and gas company debating whether to drill for oil, an investor deciding whether to purchase Microsoft stock, or a family choosing whether to vacation in Chicago in May. These decisions would be straightforward if the decision makers could predict the future with certainty. But people are not clairvoyant. There may or may not be oil in the ground. Investing in Microsoft stock may increase your wealth, but you may also end up losing a sizable portion of your initial investment. Weather in Chicago in May could be glorious or unbearable.

The objective of prospect theory is to describe how people make decisions when there is uncertainty about the consequences of their choices. Decision theorists distinguish between *decision under risk*, situations in which the likelihood of events are known or objective (such as a spin of a roulette wheel), and *decision under uncertainty*, situations in which the decision maker must assess the probability of the uncertain events and hence the likelihood of events are subjective (such as the outcome of a sports game) 3. Although prospect theory applies to both risk and uncertainty, we will focus here on risk for simplicity.

### 1 Expected Utility Theory

Before prospect theory, the dominant theory of how people choose among risky alternatives was expected utility theory. This theory was originally proposed by Bernoulli in 1738 4 and was formalized by von Neumann and Morgenstern in 1947 5. von Neumann and Morgenstern proposed a set of axioms, or basic conditions, that are necessary and sufficient for expected utility. Expected utility theory is a standard building block of modern economic theory 6, 7. It is regarded to be rational to be an expected utility maximizer, as this theory is based on compelling axioms about how people *should* behave 8.

Expected utility theory posits that decision makers choose the prospect that maximizes their expected (or average) utility. More formally, consider a risky prospect, *P* = (*p*_{1}, *x*_{1};…;*p*_{i}, *x*_{i};…;*p*_{n}, *x*_{n}), that offers outcome *x*_{i} with probability *p*_{i}, where ∑*p*_{i} = 1.

According to expected utility, the prospect *P* is evaluated as , with decision makers choosing the prospect with the highest expected utility.

Under expected utility, risk preferences are completely captured by the shape of the utility function. Decision makers are risk averse if *U*(*x*) is concave, and risk seeking if *U*(*x*) is convex, with most classical economic theory based on the tenet that decisions makers are risk averse. One standard interpretation for risk aversion is that the usefulness of an additional dollar decreases as a person gets wealthier, a principle known as *diminishing marginal utility* 9. A utility function that exhibits diminishing marginal utility is concave, and hence the decision maker is risk averse.

### 2 Prospect Theory

Kahneman and Tversky showed that decision makers are not rational as required by expected utility theory. The theory has been successful in making this point, largely because the theory consists of three parts that are individually attractive and work together well. First, Kahneman and Tversky produced a number of intuitive and elegant empirical demonstrations that were at odds with expected utility theory. Second, they proposed a mathematical model that could explain and organize these violations. Finally, they posited some psychological principles that suggest how people evaluate the alternatives they face. We begin by reviewing the primary empirical demonstrations that challenged the standard theory that decision makers are rational. We then describe the pieces of the mathematical model used to organize these violations. Throughout, we discuss how the psychological principles provide insight into these empirical demonstrations and review these principles at the end of this section.

#### 2.1 Empirical Demonstrations

##### 2.1.1 Fourfold Pattern of Risk Attitudes

Standard economic theory assumes that individuals are *risk averse*. Consider a lottery that offers a 50% chance to win $1000 and a 50% chance to win $0. The expected value of this lottery is $500 (0.5 × $1000 + 0.5 × $0). A risk-averse individual prefers winning the expected value of this lottery, $500, for certain, to the lottery itself. Indeed, surveys show that the vast majority of people are risk averse for this choice.

But individuals are not universally risk averse. They dislike risk in some situations, while liking risk in others. For example, lotteries and casinos are billion-dollar industries in the United States. Many individuals purchase lottery tickets, even though the expected value of a state lottery ticket is often as little as half the price of the lottery ticket 10.

Kahneman and Tversky demonstrated that the purchase of lottery tickets is not an isolated example of risk-seeking behavior. They documented a pattern that has become known as the *fourfold pattern of risk attitudes* 11. Individuals are risk averse for most gains, but risk seeking for most losses. As previously mentioned, most people would choose $500 for sure over a lottery that offered them an equal chance at $1000 or $0. On the other hand, the same person typically prefers an even chance at losing $0 or losing $1000 to losing $500 for sure. This pattern, however, switches when probabilities are small. In this case, decision makers are risk seeking for gains, and risk averse for losses. Consider a choice between winning $1 for sure and a prospect that offered a 1% chance at winning $100. Gambling in this case seems pretty reasonable. After all, $1 is not much money, and the lottery offers some chance at winning a sizable amount of cash. The purchase of a lottery ticket is indeed an example of risk seeking for small probability gains. However, preferences reverse when we change “winning” to “losing” in the above example. Now, the typical preference is for losing $1 for sure over taking a prospect in which there is a 1% chance of losing $100. The attractiveness of insurance provides an example of risk aversion for small probability losses. In sum, the fourfold pattern of risk preferences is risk aversion for medium to large probability gains, risk seeking for small probability gains, risk seeking for medium to large probability losses, and risk aversion for small probability losses. This pattern is often called the *reflection effect*, because preferences flip or “reflect” when outcomes are changed from gains to losses.

##### 2.1.2 Framing

Expected utility theory assumes that preferences between prospects do not depend on the manner in which they are described. For example, $20 should be viewed the same whether you are given a single $20 bill, two $10 bills, or 20 singles. This compelling principle is known as the *invariance assumption*. However, prospect theory demonstrates that the same options can be framed in different ways to produce dramatically different preferences. In other words, our choices do not always obey the invariance assumption.

To illustrate, consider the following two-choice situations:

In situation 1, you are first given $1000. You must now choose between two options. If you choose

*A*, you will receive an additional $500 for sure. If you choose*B*, there is an equal chance that you will receive either an additional $1000 or nothing.Now consider situation 2; this time you are first given $2000. Again, there are two options. Perhaps you would like

*C*, a sure loss of $500? Or, maybe you would prefer*D*, a 50% chance at losing $1000 and a 50% chance at losing $0?

When confronted with situation 1, most people prefer *A*, the sure $500, to the gamble *B*. On the other hand, when posed with a choice between *C* and *D*, most choose *D*, the uncertain loss, over the sure loss *C*. While both choices seem reasonable in isolation, situations 1 and 2 are identical in terms of final consequences, reducing to a choice between $1500 for sure (*A* and *C*) and a lottery that offers an even chance at $1000 or $2000 (*B* and *D*).

The standard preferences described above violate the invariance assumption invoked by expected utility. However, they are consistent with the risk attitudes revealed by the fourfold pattern of risk attitudes. When the problem is framed as a gain (situation 1), people are usually risk averse. But when the problem is framed as a loss (situation 2), people tend to be risk seeking.

##### 2.1.3 Aversion to Losses

Expected utility theory assumes that choices only reflect final outcomes. For example, if one were the beneficiary of a $100 check, but also received a $100 speeding ticket, these two events would offset one another in monetary terms. However, Kahneman and Tversky demonstrated that individuals tend to dislike losses much more than they like gains. Thus, the two events do not offset one another in psychological terms. The $100 check makes you happy, to be sure. But, your pleasurable feelings for the check pale in comparison with the pain you feel when you are reminded about your $100 speeding ticket. This dislike of losses is known as *loss aversion*. Put simply, losses loom larger than gains.

To illustrate loss aversion, consider a gamble in which you may lose $1000 or gain $1100 with equal chance. Although the expected value of this gamble is positive, most people find this prospect unappealing. The potential loss of $1000 cannot be offset by the potential gain of $1100. In order to make this gamble attractive, the gain has to be increased considerably, to about $2000. In other words, roughly speaking, the pain of a loss is about twice as much as the pleasure from a comparably sized gain 12.

##### 2.1.4 Common-Consequence Effect

Suppose that you are faced with the following choice. If you choose *A*, you receive $200 with 10% chance. If you choose *B*, you receive $100 with 20% chance. Next, imagine that we improve both options by adding to each a 10% chance of receiving $1000. Now if you choose *A*, you receive $200 with 10% chance, $1000 with 10% chance, and $0 with 80% chance. If you choose *B*, you receive $100 with 20% chance, $1000 with 10% chance, and $0 with 70% chance. It seems reasonable that your choices in these two situations be identical, that is, unaffected by the additional chance at receiving $1000. Since both of the second options have a 10% chance of receiving $1000, this common consequence is irrelevant for this choice. Expected utility theory assumes this principle—adding a common consequence to two prospects should not change which alternative the decision maker prefers. This principle is known as the *independence axiom* 13, 14.

Kahneman and Tversky documented a violation of this principle. Consider the following two choices:

Situation 1

option

*A*offers a 33% chance at $2500 and a 67% chance at $0;option

*B*offers a 34% chance at $2400 and a 66% chance at $0.

Situation 2

option

*C*offers a 33% chance at $2500, a 66% chance at $2400, and a 1% chance at $0;option

*D*offers $2400 for sure.

Most people prefer *A* to *B*, arguing that 33% and 34% seem about the same, but $2500 is clearly better than $2400. On the other hand, people like *D* over *C*, reasoning that it is not worth taking a chance with *C* since *D* has a sure chance of winning a good amount of money. To see that this pattern of choices violates the independence axiom and hence is inconsistent with expected utility, note that situation 2 is created from situation 1 by adding a common consequence to both *A* and *B*: a 66% chance at $2400.

This problem illustrates the strong preference for certainty over uncertainty. Indeed, this example has been termed the *certainty effect*. Adding a 66% chance at $2400 to *B*, a 34% chance at $2400, is particularly attractive, because it changes the prospect from possible to certain. On the other hand, for option *A*, the additional 66% chance at $2400 increases the chance of winning from 33% to 99%, or from probable to more probable.

This demonstration is also called the *common-consequence effect*, because adding a common consequence to two options changes preferences, contrary to expected utility theory 15. Indeed, a variant of this problem posited by the French economist Maurice Allais was one of the earliest challenges to the descriptive validity of expected utility 16. Allais' challenge, which has become known as the *Allais Paradox*, consisted of two pairs of gambles. The first involved a choice between either (*A*) $1 million for sure or (*B*) a 0.10 chance at $5 million, a 0.89 chance at $1 million, and 0.01 chance at $0, whereas the second was a choice between (*C*) a 0.11 chance at $1 million (and a 0.89 chance at $0) and (*D*) a 0.10 chance at $5 million (and a 0.90 chance at $0). Most people prefer *A* and *D*, a pattern that contradicts expected utility. Under expected utility theory, a choice of *A* over *B* implies that 0.11*u*(1) > 0.10*u*(5) + 0.01*u*(0), whereas a preference for *D* over *C* implies the opposite inequality, 0.10*u*(5) + 0.01*u*(0) > 0.11*u*(1).

#### 2.2 The Prospect Theory Model

Prospect theory offered a rich set of empirical challenges to expected utility theory, including the fourfold pattern of risk attitudes, framing effects, aversion to losses, and the common-consequence effect. These demonstrations are robust, attractive, and intuitive. But do these demonstrations inform us about how individuals make decisions that involve risk in general? Kahneman and Tversky proposed a model for organizing the empirical demonstrations described above and making more accurate predictions about how individuals would behave in other choice situations.

Like expected utility theory, prospect theory assumes that individuals evaluate each alternative and then choose the alternative with the highest subjective valuation. However, prospect theory differs from expected utility theory in two substantive ways. Suppose a decision maker might choose an alternative that offers a *p* chance at outcome *x*. Expected utility theory assumes that the utility of this alternative is *pu*(*x*). Prospect theory, like expected utility, assumes that the decision maker evaluates the *x* that he/she might get and the *p* chance at winning, and then combines these two evaluations together. However, the valuation of outcomes is governed by the *value function*, *v*(*x*), and the valuation of the probabilities is governed by the *probability weighting function*, π(*p*), and thus the value of this alternative is given by π(*p*)*v*(*x*). [The valuation of more complicated alternatives with more than one nonzero outcome follows a revision of prospect theory, *cumulative prospect theory* 11.] These two pieces are described below.

##### 2.2.1 Value Function

The prospect theory value function, *v*(*x*), captures an individual's subjective perception of a particular outcome (Fig. 1). Prospect theory assumes that outcomes are coded as gains and losses, or outcomes above and below a reference point, and the value function captures how much better one gain is than another gain, how much worse one loss is than another loss, or alternatively how a particular loss compares with a gain of the same magnitude.

Three psychological principles constrain the shape of the value function. The first principle, *reference dependence*, suggests that outcomes are viewed relative to a reference point and hence coded as gains or losses. A $10,000 bonus is seen differently if an employee is expecting a $5000 bonus or a $15,000 bonus. The bonus is seen as a $5000 gain in the first case, but a $5000 loss in the second case.

The second principle, *diminishing sensitivity*, borrows from research on the psychophysics of perception. Research has shown that an individual is highly likely to discriminate between a 2 and 3 kg weight, but not very likely to notice the difference between 22 and 23 kg weights 17. Analogous findings appear in tasks ranging from the perception of line lengths to the perception of temperature. Prospect theory thus suggests that changes in value have a greater impact near the reference point than away from the reference point. Thus, there is a big difference between a $100 gain and a $200 gain, but a much smaller difference between gains of $1100 and $1200. Similarly, a loss of $100 seems quite distinct from a loss of $200, but losses of $1100 and $1200 seem pretty similar.

The principle of diminishing sensitivity away from the reference point is consistent with the reflection effect described above. Most people would choose $500 for sure over an even chance at winning $1000 or winning $0. Diminishing sensitivity suggests that the first $500 is more pleasurable than the second $500. When this choice is changed to losses, however, preferences switch. The majority of individuals prefers an even chance at losing $1000 or losing $0 to losing $500 for sure. This preference is also consistent with diminishing sensitivity, because the first $500 loss is much more painful than the second $500 loss. Thus, the value function in Fig. 1 is concave for gains and convex for losses.

The third psychological principle underlying the value function is *loss aversion*. Recall that losses loom larger than gains. A loss of $100 seems much more painful than a gain of $100 seems pleasurable. Most people dislike a prospect that gives an equal chance at winning $1000 or losing $1000. These two examples are captured by a value function that is steeper for losses than gains ( − *v*( − *x*) > *v*(*x*)), as is depicted in the value function in Fig. 1.

##### 2.2.2 Probability Weighting Function

Prospect theory assumes that individuals do not weigh outcomes by their probability, as in expected utility theory, but by some distortion of probabilities. This distortion of probability is captured by prospect theory's probability weighting function, π(*p*) (Fig. 2). Interestingly, the psychology governing the distortion of probability involves two of the three principles used to understand the value function, reference dependence and diminishing sensitivity away from a reference point. For probability, there are two obvious reference points, certainty and impossibility, or 100% chance and 0% chance. The distortion of probability shown in the probability weighting function captures the diminishing sensitivity away from these two reference points. People are most sensitive to changes in probability when they are near 0% or 100% than when the change applies to intermediate probabilities. Consider adding a chance to win a large prize, such as a new car. Which of the following appears to be the least significant increase: increasing your chance of winning from 0% to 1%, from 33% to 34%, or from 99% to 100%? Most people regard the change from 0% to 1% as significant, because it changes the chance of winning from impossible to possible 18. The change from 99% to 100%, too, seems noteworthy, because it shifts the odds from likely to certain. In contrast, the change from 33% to 34% seems negligible. Put simply, there is a categorical difference between impossible and possible or between likely and certain, but the difference between possible and slightly more possible is a matter of degree. The probability weighting function in Fig. 2 exhibits diminishing sensitivity: π(0.01) − π(0) > π(0.34) − π(0.33) and π(1) − π(0.99) > π(0.34) − π(0.33). The function is thus concave for small probabilities and convex for medium and large probabilities.

Diminishing sensitivity implies that low probabilities are typically given more weight than they would receive using expected utility. This overweighting is consistent with risk seeking for low probability gains (such as lottery tickets) and risk aversion for low probability losses (such as insurance). Medium to high probabilities are typically given less weight than they would receive using expected value. Such underweighting is consistent with risk aversion for medium to high probability gains, and risk seeking for medium to high probability losses. The probability weighting function also helps us understand the common-consequence effect described above. In the first choice, the probabilities of winning a prize are 33% and 34%, whereas the probabilities of winning with the second choice are 99% and 100%. The probability weighting function suggests that the first difference is trivial (thus decision makers look at outcomes to make this choice), whereas the second difference is consequential (and thus sufficient to drive the decision).

#### 2.3 Psychological Principles

Prospect theory invokes three psychological principles: reference dependence, diminishing sensitivity, and loss aversion. The first two principles produce the characteristic form of the value function and the probability weighting function shown in Figs 1 and 2. For the value function, outcomes are viewed relative to a reference point, with outcomes above the reference point coded as gains and outcomes below the reference point coded as losses. The valuation of both gains and losses exhibits diminishing sensitivity away from the reference point. For the probability weighting function, probabilities are viewed relative to the reference points of impossibility (0) and certainty 1. Changes away from these two reference points also exhibit diminishing sensitivity.

### 3 Applications

Prospect theory has been useful for understanding real-world phenomena in a variety of domains. A few applications are discussed below to provide a flavor of the broad reach the theory has had. For a description of other applications, see Ref. 19.

#### 3.1 Mental Accounting

Although prospect theory was developed to explain how people make choices involving risk, the principles of prospect theory have been used to understand how people choose among certain alternatives as well. One influential application of prospect theory is known as *mental accounting* 20, 21. Mental accounting is a theory of how individuals and households make consumer and financial decisions. Most of the implications of mental accounting follow from the shape of prospect theory's value function.

Mental accounting suggests that people prefer to segregate gains and aggregate losses. Consider, for example, two pleasurable outcomes, a $100 check from your aunt and $50 won in a school raffle. Most people find it more enjoyable to view these outcomes separately, $100 *plus* $50, than together, $150. The attractiveness of segregating gains follows from the shape of the value function and the psychological principle of diminishing sensitivity away from the reference point of zero. Next consider two painful outcomes, a $100 speeding ticket and a $150 tax liability. It seems clear that aggregating the losses is a good idea psychologically. In other words, a loss of $250 seems less painful than two separate losses of $100 and $150. Mental accounting suggests that aggregating losses is more attractive than segregating losses. This simple principle helps us understand a number of real-world phenomena, including the attractiveness of flat-rate pricing plans. Most people find it less painful to pay $70 for a monthly health club membership than $10 each time they work out 22.

Mental accounting also suggests that different ways of framing the same activity can be viewed very differently. Suppose that you have decided to purchase a new car for $22,000 and are contemplating purchasing a stereo upgrade. A car salesperson can frame the stereo as a $300 purchase or an increase in the total price of the new car from $22,000 to $22,300. It is clear that the salesperson will be more effective at convincing you to purchase the upgrade if he/she frames the upgrade as an increase in price. Again, the principle at work is diminishing sensitivity of the value function.

Consider one final example of mental accounting. Imagine that you can walk 10 minutes to save $5. Would you do so? People give different answers if the purchase is a $25 calculator or a $125 jacket. A $5 discount seems more significant when it is applied to the inexpensive calculator rather than the expensive jacket. Of course, this behavior is not rational, since the decision to walk 10 minutes boils down to a choice of whether walking 10 minutes is worth $5.

#### 3.2 Status Quo Bias

In many situations, the decision maker must decide whether to choose a particular alternative or stick with the status quo. However, the status quo is frequently chosen much more often than it should be if the evaluation of the alternative were made simply based on the positive and negative features of that alternative. One study found a strong status quo bias for the choice of automobile insurance plans. Prior to deregulation in the early 1990s, Pennsylvania had an expensive insurance plan that allowed a driver to sue another driver in many situations. New Jersey had a cheaper plan that permitted its driver to sue only in limited situations. As a result of deregulation, citizens of Pennsylvania were permitted to trade down to the cheaper plan, and people in New Jersey were allowed to trade up to the more expensive plan. Nevertheless, 75–80% of drivers stuck with their original plan 23.

How does prospect theory help us understand why people tend to prefer the status quo? When giving up the status quo for another alternative there are, typically, trade-offs to be made. An option will have advantages and disadvantages relative to the status quo. The advantages are likely to be perceived as gains relative to the status quo, with the disadvantages perceived as losses. Because of loss aversion, losses appear more significant than gains of comparable magnitude. The principle that losses loom larger than gains explains why people have a tendency to remain at the status quo—a phenomenon known as the *status quo bias*.

A related demonstration, called the *endowment effect*, shows once again the asymmetry between losing and gaining 24. In one study, half of the participants are given a lottery ticket and half of the participants are given $2 25. In each case, the participants can keep what they were given or trade it for what the others were given. However, very few participants chose to switch. Those who were given lottery tickets largely stuck with the lottery tickets, but those who were given the cash mostly elected to keep the cash. Once again, prospect theory's principle of loss aversion can explain this phenomenon. When people are endowed with an item, it becomes the reference point from which other options are evaluated. Since losses loom larger than gains, giving up an item that one possesses is perceived as more painful than acquiring an item of comparable magnitude that one does not already possess.

#### 3.3 Other Applications

Prospect theory provides insights into numerous other real-world decisions. We sketch a few applications below to give a flavor of the power of prospect theory. First, in horse track betting, favorites are usually underbet, and longshots are typically overbet 26. This behavior can be explained by the probability weighting function. Second, consider an investor who bought two stocks at $50. One has increased in value to $75, and the other has fallen in price to $25. If the investor has to sell one stock, which would he/she sell? Financial economists have shown that investors overwhelmingly sell the winner rather than the loser 27. The *disposition effect* suggests that selling the loser is unattractive, because the investor must come to terms with his/her losses.

Finally, we consider choices of retirement plans. Employers often provide a default plan, or a plan that will be chosen if the employee does not make a choice. This plan is often a sensible plan, but it is often chosen by the vast majority of employees, even when another plan might make more sense 28. To see why prospect theory can explain this choice, we assume that the default plan is the reference point for evaluating another retirement plan. A second plan will be better on some dimensions and worse on other dimensions. Because of loss aversion, the dimensions on which the second plan is inferior loom large compared to the dimensions on which it is superior, providing a substantial advantage to the default plan.

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### Further Reading

- Prospect theory on the brain? Toward a cognitive neuroscience of decision under risk. Cogn Brain Res 2005;23(1):34–50. , , .
- Decision under risk Prospect theory for risk and ambiguity. Cambridge: Cambridge University Press; 2009. .
- Koehler DJ, Harvey N, editors. Blackwell handbook of judgment and decision making. Oxford: Blackwell; 2004. pp. 399–423. , , . In: