Prospect theory: Kahneman and Tversky's experiments, the value function and the fourfold pattern

Choose one option in each pair before reading on.

1. A 33% chance of 2,500, a 66% chance of 2,400 and a 1% chance of nothing, or 2,400 for certain.
2. A 33% chance of 2,500, or a 34% chance of 2,400.

Of the 72 people who answered these two questions for Daniel Kahneman and Amos Tversky, 82% took the certain 2,400 in the first pair and 83% took the 33% chance of 2,500 in the second. A majority, 61%, gave both answers. The second pair is the first with a 66% chance of winning 2,400 removed from both options, so under expected utility theory a person who prefers the sure 2,400 in the first pair must prefer the 34% chance of 2,400 in the second. The majority did the opposite. Kahneman and Tversky called this the certainty effect, and it is the first result reported in "Prospect theory: an analysis of decision under risk", published in Econometrica in March 1979.

Prospect theory is a descriptive model of choice under risk. It holds that people value outcomes as gains and losses measured from a reference point, and that they weight outcomes by decision weights that differ from the stated probabilities. The 1992 revision, cumulative prospect theory, put numbers on both ideas.

Who developed prospect theory and what it challenged

The 1979 paper appeared in volume 47 of Econometrica, pages 263 to 291. In 2002 Kahneman shared the Nobel memorial prize in economic sciences with the experimental economist Vernon Smith, for bringing insights from psychology into economics, above all on judgment and decision-making under uncertainty. The prize announcement named prospect theory as the model he formulated with Tversky, who had died in 1996. Kahneman died on 27 March 2024.

The target of the 1979 paper was expected utility theory, which economists used both as a rule for rational choice and as a description of real behavior. The paper summarized it in three tenets:

• Expectation. The value of a gamble is the probability-weighted average of the utilities of its outcomes.
• Asset integration. A gamble is judged by the final wealth it leaves, current assets included.
• Risk aversion. Utility is a concave function of wealth, so a sure amount is preferred to a gamble with the same expected value.

The French economist Maurice Allais had already built a counterexample to the first tenet in 1953. Kahneman and Tversky produced a whole series of them with hypothetical questions: students and university faculty answered choice problems in anonymous booklets of at most a dozen problems, with the order of problems and the left and right position of the options varied between versions. Amounts were in Israeli pounds, at a time when the median net monthly income of a family was about 3,000 pounds, so many of the amounts were close to a month's income. Some problems were also given to students and faculty in Stockholm and at the University of Michigan, and the pattern of answers was essentially the same.

The 1979 choice problems and how people answered

The table lists the problems that carry the argument of the paper. An option written as "4,000 at 80%" pays 4,000 with probability 80% and nothing otherwise. The last column gives the share of respondents who chose the option named.

Problem 9 was put to 95 Stanford University students. The other problems used the samples described above. Every effect in the paper was observed in several problems with different amounts and probabilities, and the ones printed are illustrations.

The certainty effect

Problems 3 and 4 give a simpler version of the first pair. Problem 4 is Problem 3 with every chance of winning divided by four: 80% becomes 20% and certainty becomes 25%. Expected utility theory treats that as a neutral change, so the preferred option should stay the same. The majority switched from the sure 3,000 to the 20% chance of 4,000. Cutting the chance of winning from 100% to 25% had a much bigger effect on the option's appeal than cutting it from 80% to 20%.

Problems 5 and 6 repeat the result with holidays in place of money. Problems 7 and 8 show a related pattern with no certainty involved. When the chances of winning are substantial, 90% against 45%, people pick the likelier prize. When winning is merely possible, 0.2% against 0.1%, most pick the bigger prize, although the ratio of probabilities is the same.

The reflection effect

Problems 3', 4', 7' and 8' are the same gambles with gains turned into losses, and in each pair the preference reverses. Risk aversion over gains came with risk seeking over losses: 92% preferred an 80% chance of losing 4,000 to a certain loss of 3,000, although the gamble has the worse expected value, a loss of 3,200.

The reflection also rules out a popular explanation of the certainty effect, a general dislike of variance. The certain loss of 3,000 has both the better expected value and no variance at all, and almost everyone rejected it. Our article on loss aversion follows this pair into brokerage accounts, where it shows up as selling winners early and holding losers.

Probabilistic insurance

Problem 9 asked students to imagine a property insurance policy they found barely worth its premium, then offered a variant: pay half the premium, and if damage occurs, there is a 50% chance of paying the other half and being fully covered and a 50% chance of getting the premium back and bearing the whole loss. Expected utility theory with a concave utility function says this cheaper, partial cover must be preferred to the regular policy. 80% of respondents refused it. Cutting the probability of a loss from p to half of p was worth much less to them than cutting it from half of p to zero. Kahneman and Tversky pointed out that a burglar alarm, new tires or quitting smoking also reduce a risk without removing it.

The isolation effect

In Problem 10, the two-stage game gives a 20% overall chance of 4,000 against a 25% chance of 3,000, the same final prospects as Problem 4. Yet 78% of 141 respondents chose the 3,000, the answer given in Problem 3. People ignored the first stage because it was common to both options and judged only the second, where 3,000 was certain.

Problems 11 and 12 are identical in final wealth. In both, one option leaves a 50% chance of 2,000 and a 50% chance of 1,000, and the other leaves 1,500 for certain. The gift was ignored because it was common to both options, and the answers followed the reflection effect: a sure 500 when the choice was framed as a gain, the gamble when it was framed as a loss. From this pair Kahneman and Tversky drew the central claim of the theory: people attach value to changes in wealth, and final asset positions play little part in the choice.

Lottery tickets and insurance

Problems 14 and 14' explain why the same person buys both. 72% preferred a 1 in 1,000 chance of 5,000 to its expected value of 5, which is what buying a lottery ticket amounts to. 83% preferred to lose 5 for certain than face a 1 in 1,000 chance of losing 5,000, which is what paying an insurance premium amounts to. The combination is hard to explain with expected utility theory. Prospect theory explains both with one assumption: very small probabilities are overweighted.

Editing, then evaluation: the two phases of a choice

Prospect theory splits a decision into two phases. In the editing phase, a person simplifies the options before comparing them. The paper lists six operations:

• Coding sorts outcomes into gains and losses around a reference point.
• Combination merges identical outcomes: 200 at 25% twice becomes 200 at 50%.
• Segregation splits off a sure part: 300 at 80% and 200 at 20% becomes a sure 200 plus 100 at 80%.
• Cancellation drops what the options share, the source of the isolation effect.
• Simplification rounds numbers, so 101 at 49% is read as an even chance of 100, and discards outcomes that are extremely unlikely.
• Detection of dominance throws out options that are worse on every count.

The order of these steps can change the result. After rounding, 500 at 20% plus 101 at 49% appears to dominate 500 at 15% plus 99 at 51%, although no such dominance exists in the unrounded numbers.

In the evaluation phase, each edited prospect gets an overall value, written V, and the prospect with the highest V is chosen. For a gamble paying x with probability p and y with probability q, V equals π(p)v(x) plus π(q)v(y), where v is the value function and π the decision weight. It has the same bilinear shape as expected utility, with two substitutions: v is defined on gains and losses, and π replaces the probability. For a gamble whose outcomes are all gains or all losses, the sure part is valued first and only the risky part is weighted. A 25% chance of 400 with a 75% chance of 100 is worth v(100) plus π(0.25) times the difference between v(400) and v(100).

The value function: reference point, curvature and loss aversion

The 1979 paper drew the value function as an S-shaped curve with three properties. It is defined on deviations from the reference point. It is generally concave for gains and often convex for losses. It is steeper for losses than for gains, and steepest at the reference point itself.

The curvature comes from diminishing sensitivity, a principle taken from perception. Telling apart changes in room temperature of 30 and 60 is easier than telling apart changes of 130 and 160, and in the same way the difference between gains of 100 and 200 feels larger than the difference between gains of 1,100 and 1,200. The same holds for losses. Problems 13 and 13' test it directly: 82% preferred two separate chances of 4,000 and 2,000 to one chance of 6,000, and 70% preferred one chance of losing 6,000 to two separate chances of losing 4,000 and 2,000. Both answers mean v(6,000) is worth less than v(4,000) plus v(2,000), and the loss of 6,000 hurts less than the two smaller losses combined.

The steepness for losses is loss aversion. Most people find a 50-50 bet to win or lose the same amount unattractive, and more so as the amount grows. The paper also cited a review of utility functions measured for 30 business decision makers in five earlier studies: most were concave for gains and convex for losses, only three people were risk averse on both sides, and all but one had a steeper curve for losses. The size of the asymmetry, usually quoted as about 2, is covered with the recent meta-analyses in our loss aversion article.

Strictly, the value function also depends on wealth, and the paper expected it to become more linear as assets grow. In practice, the certainty equivalent of a 50% chance of 1,000 fell between 300 and 400 for most people across a wide range of asset positions, so one value function was a fair approximation.

When the reference point moves

The reference point is usually the status quo, but the paper gave cases where it is an expectation. An unexpected tax deduction from a monthly paycheck feels like a loss, and an entrepreneur who loses less than expected in a slump can experience the result as a gain. It can also lag behind reality. A person who has lost 2,000 in a venture and now chooses between a sure 1,000 and an even chance of 2,000 may still code the choice as a sure loss of 1,000 against an even chance of losing 2,000, which favors the gamble. Kahneman and Tversky cited a known regularity at the racetrack as support: betting on long shots increases over the course of the betting day, which fits the idea that a failure to adapt to losses makes people seek risk.

The probability weighting function

Decision weights are inferred from choices. They measure how much a chance counts in the decision, and they are distinct from how likely a person thinks the event is. Someone offered a fair coin toss for 1,000 knows the probability is 50%, yet the weight derived from choices, π(0.5), is likely to be below 0.5. The 1979 paper separated this overweighting from the overestimation of rare events that people also show when they have to judge probabilities themselves. In real life the two can add up.

The paper set out four properties of π:

1. Overweighting of small probabilities. From Problem 14, the weight on a 0.1% chance is larger than 0.001.
2. Subadditivity at low probabilities. From Problem 8, halving a small probability less than halves its weight.
3. Subcertainty. The weights of complementary events add up to less than 1. From Problems 1 and 2, π(0.66) plus π(0.34) is below 1.
4. Subproportionality. For a fixed ratio of probabilities, the ratio of their weights is closer to 1 when the probabilities are small, which produces the pattern in Problems 7 and 8.

The resulting curve is flat in the middle, where people are fairly insensitive to changes in probability, and jumps near 0 and 1. An example the paper credits to Zeckhauser makes the jump concrete. Forced to play Russian roulette, most people would pay far more to have the only bullet removed from the gun, taking the risk of death from 1 in 6 to zero, than to go from four bullets to three. Extremely unlikely events are either ignored or given too much weight, and the difference between near-certainty and certainty is either neglected or exaggerated.

Later experiments measured the curve in money. Subjects in a 1999 experiment gave an average certainty equivalent of 10 dollars for a 5% chance of 100 dollars, twice its expected value, and 63 dollars for a 90% chance of 100 dollars, well below its expected value of 90.

Cumulative prospect theory and the 1992 parameters

The 1979 model had two technical limits. It applied to gambles with at most two outcomes other than zero, and without the editing step it could predict the choice of dominated gambles, which the paper handled by assuming such gambles were spotted and discarded first. In "Advances in prospect theory: cumulative representation of uncertainty", published in the Journal of Risk and Uncertainty in 1992, Tversky and Kahneman moved the weights from the probability of each separate outcome to cumulative probabilities, such as the chance of gaining at least 100. The new version handles any number of outcomes, applies to uncertain events as well as stated probabilities, and allows different weighting functions for gains and for losses.

The paper also reported a new experiment. Tversky and Kahneman recruited 25 graduate students from Berkeley and Stanford, 12 men and 13 women with no training in decision theory, for three one-hour sessions several days apart, and paid each of them 25 dollars. On a computer, subjects compared each gamble with a series of sure amounts, and the certainty equivalent was taken from the point where they switched. There were 28 gambles over gains and 28 over losses, with outcomes up to 400 dollars. Fitting a power value function and a one-parameter weighting function to each subject gave these median estimates:

With these values, a gain of x is worth x to the power 0.88 and a loss of x is worth minus 2.25 times x to the power 0.88. The authors presented the estimates as a compact summary of the data and put more weight on the qualitative pattern than on the exact figures. Because gamma is below delta, the weighting for gains is slightly more curved, which matches a finding in the data: at moderate and high probabilities, risk aversion for gains was stronger than risk seeking for losses. In a separate test with mixed gambles, the median subject was indifferent between getting nothing and a 50-50 gamble to lose 100 dollars or win 202 dollars.

The fourfold pattern of risk attitudes

The 1992 paper called the fourfold pattern the most distinctive implication of prospect theory. The value function produces risk aversion for gains and risk seeking for losses. The weighting function reverses both when the probability is small. The median certainty equivalents from the 25 subjects show all four cells:

For probabilities of 50% or more, all 25 subjects were mainly risk averse for gains and risk seeking for losses. The full fourfold pattern appeared in 22 of the 25. Correlations between the same person's answers across the four types of gamble were low, and the authors concluded that there is no general trait of risk aversion or risk seeking.

The low-probability row describes the conditions under which lottery tickets and insurance policies are sold. The 1979 paper was cautious on this point: people also insure against risks of medium probability and sometimes ignore small probabilities of disaster altogether, so prospect theory covers part of insurance behavior and leaves the rest to other factors.

Prospect theory in financial markets

Finance applications of prospect theory cover the pricing of individual securities, the return on the stock market as a whole and the way investors trade. The same tools have been applied to insurance and betting.

Lottery-like stocks and positive skewness

Probability weighting inflates the tails of a return distribution. Applied to markets, it predicts that investors will overpay for securities with a long right tail, a small chance of a very large gain, and that those securities will then earn low average returns. Barberis and Ming Huang derived this in 2008 from cumulative prospect theory: a security's own skewness is priced, even when it is unrelated to the market, which standard expected utility models rule out.

Later data fit that prediction:

• Across different measures of expected skewness, more positively skewed stocks earn lower average returns.
• Stocks that go public are highly skewed, since most perform modestly and a few become very large companies. The higher the predicted skewness of an initial public offering, the lower its long-term return.
• Individual investors favored lottery-type stocks, marked by a low share price, high idiosyncratic volatility and high idiosyncratic skewness, and their demand for them rose in economic downturns, as demand for state lottery tickets does. The stocks underperformed, and low-income investors, who overweighted them most, lost the most to that underperformance.
• Stocks whose past return distribution had a high prospect theory value earned low subsequent returns in the US and in a majority of 46 other national markets.

The same mechanism has been used for the low average returns of distressed and bankrupt stocks, of stocks traded over the counter and of out-of-the-money options, and for the concentrated portfolios some households hold in a few skewed stocks.

Myopic loss aversion and the equity premium

Stocks have paid far more than Treasury bills over long periods, more than standard models can explain with plausible risk aversion. Shlomo Benartzi and Richard Thaler built a model that combines loss aversion with frequent evaluation of the portfolio. In it, the observed premium is consistent with prospect theory parameters if investors assess their holdings about once a year. Probability weighting adds to the effect, because the stock market as a whole is negatively skewed by occasional crashes. The horizon simulations and the feedback experiments that followed are covered in our loss aversion article.

The disposition effect

Investors sell stocks that have risen since purchase more readily than stocks that have fallen. Hersh Shefrin and Meir Statman traced this in 1985 to the convex loss side of the value function: below the purchase price the investor becomes risk seeking and holds on in the hope of breaking even. The link is harder to establish in formal models: to produce the effect, the value function would need to be far more convex over losses than experiments show, and an alternative account, realization utility, places the pleasure or pain at the moment of sale. Both accounts rest on gains and losses measured from the purchase price.

Insurance deductibles

Among 50,000 customers of a large home insurer choosing among deductibles of 100, 250, 500 and 1,000 dollars, households that chose the 500 dollar deductible paid an average premium of 715 dollars a year, and in doing so turned down a 1,000 dollar deductible at an average of 615 dollars. With a claim rate of about 5% a year, they paid 100 dollars a year to avoid a 5% chance of paying an extra 500 dollars, an expected cost of 25 dollars. Expected utility theory needs implausibly high risk aversion to explain that. Overweighting the small chance of a claim explains part of it, and possibly all of it if the reference point is what the household expects to pay, as Kőszegi and Rabin propose.

Long shots at the racetrack

At the racetrack, bettors overbet long shots and underbet favorites, a pattern known as the favorite-longshot bias. In a large data set that included compound bets such as exactas and trifectas, the bets fitted misperception of probabilities, the explanation prospect theory offers, better than a taste for risk.

Does prospect theory replicate? The 2020 study in 19 countries

Several long-accepted findings in behavioral science have failed recent replication attempts. A team led by Kai Ruggeri reran the 1979 questions with 4,098 participants in 19 countries and 13 languages and published the results in Nature Human Behaviour in 2020. The team kept the original items, scaled the amounts to the local median net household income in the way 3,000 pounds matched Israeli incomes in the 1970s, and kept the choices hypothetical as in the original. It dropped the two holiday problems and the probabilistic insurance problem, whose long wording risked testing reading comprehension.

• Pooled across countries, 94% of items replicated, and 12 of the 13 theoretical contrasts between items held.
• Country by country, 247 of 304 item effects, or 81%, were significant in the original direction.
• 77% of the measured effects were smaller than in 1979.
• Replication rates varied more between items, from 15% to 100%, than between countries, from 69% to 94%.

The authors concluded that the empirical foundations of prospect theory "replicate beyond any reasonable thresholds". The replication covers the 1979 choice patterns. It does not test the 1992 parameter values or the finance applications, which rest on their own evidence.

Criticisms of prospect theory

• The reference point. Kahneman and Tversky gave little guidance on how the reference point is set. For an investor, a gain could mean a positive return, a return above the risk-free rate or a return above what was expected, measured on one stock or on the whole portfolio, over a month or over a year, and each definition gives different predictions. Botond Kőszegi and Matthew Rabin proposed in 2006 that the reference point is a person's recent expectations about outcomes, which several later applications use.
• Process. The theory describes which option people choose and makes no claim about how they reach it. The priority heuristic of Eduard Brandstätter, Gerd Gigerenzer and Ralph Hertwig compares options on one reason at a time, starting with the minimum gain, and stops as soon as the difference is large enough. Across 260 problems from four data sets, including the 1979 and 1992 problems, it predicted 87% of majority choices, the best accuracy of the models tested, cumulative prospect theory among them.
• Description and experience. The overweighting of rare events holds when probabilities are stated. When people learn the odds by sampling outcomes, they choose as if they underweight rare events, partly because the small samples people draw often miss the rare event.
• Stakes and samples. The 1979 choices were hypothetical, and the 1992 parameters come from 25 students paid a flat fee. The 2020 replication widened the sample but kept the hypothetical format.
• The size of loss aversion. Recent meta-analyses put the coefficient anywhere from about 1 to about 2 depending on the method, and some psychologists argue many effects credited to loss aversion reflect inertia. The loss aversion article sets out that dispute.

For how prospect theory fits into the wider field, see our introduction to behavioral finance, our article on anchoring bias, another heuristic Tversky and Kahneman described, and our list of behavioral finance biases with documented cases, including Tversky and Kahneman's 1981 framing experiment.

This article is general information, not investment advice.

Frequently asked questions

What is prospect theory in simple terms?

Prospect theory describes how people choose between risky options. They judge each outcome as a gain or a loss from a reference point, usually what they have now, and they feel a loss more than a gain of the same size. They also give small chances more weight than those chances carry and large chances less. Together these explain why the same person can buy a lottery ticket and an insurance policy, and why people take a sure gain but gamble to avoid a sure loss.

Who developed prospect theory?

The psychologists Daniel Kahneman and Amos Tversky, in a paper published in Econometrica in March 1979. They revised it in 1992 as cumulative prospect theory. Kahneman shared the 2002 Nobel memorial prize in economic sciences with Vernon Smith, and the Nobel announcement credited prospect theory to Kahneman and Tversky together. Tversky had died in 1996.

What is the difference between prospect theory and expected utility theory?

Expected utility theory judges a gamble by the final wealth it leaves and weights each outcome by its probability. Prospect theory judges outcomes as gains and losses from a reference point, treats losses as heavier than gains, and replaces probabilities with decision weights that inflate small chances. Expected utility theory is used as a rule for rational choice, while prospect theory describes the choices people make, including those that break that rule.

Is prospect theory still relevant today?

Yes. A 2020 study in Nature Human Behaviour reran the 1979 questions with 4,098 people in 19 countries and 13 languages, and 94% of the items replicated, with smaller effects than in 1979. In finance, prospect theory is used to explain why lottery-like stocks earn low average returns, why stocks have paid a large premium over bonds, and why investors sell winners sooner than losers.

What are the main criticisms of prospect theory?

The theory gives little guidance on where the reference point sits, and different reference points give different predictions. The theory describes choices without modelling the mental process behind them, and a simple rule, the priority heuristic, predicted 87% of majority choices across 260 test problems. People who learn about risk from experience tend to underweight rare events, the opposite of the overweighting found when probabilities are stated. The size of loss aversion is also disputed.