What is the fourfold pattern of risk?
Gains and losses crossed with high and low probability give four cells and two attitudes to risk. Tversky and Kahneman's 1992 experiment found the full set in 22 of 25 students. A later study found it only under one elicitation method.
Answer to the question
What is the fourfold pattern of risk?
The fourfold pattern is the finding in prospect theory that people are risk averse for likely gains and unlikely losses, and risk seeking for unlikely gains and likely losses. Tversky and Kahneman (1992) reported all four cells in 22 of 25 graduate students. Harbaugh and colleagues (2010) found it when people priced gambles, not when they chose.
Key takeaways
- Tversky and Kahneman (1992) report the complete fourfold pattern in 22 of 25 Berkeley and Stanford graduate students, who priced 28 gain and 28 loss prospects.
- In their aggregate figures, 78 per cent of departures from expected value on low-probability gains were risk seeking and 87 per cent on high-probability losses.
- Kahneman's book says experiments with law students and judges confirmed the pattern's prediction about defendants, and names none of them.
- Gilliland and Dunn (2008) found framing and perceived chance of winning, not legal role alone, predicted settlement.
- Harbaugh and colleagues (2010) found the pattern when people priced gambles and random-looking choices when they chose against expected value.
The same person can buy a lottery ticket and an insurance policy, and the fourfold pattern of risk is the table that holds both. In it, people are risk averse in two of its cells and risk seeking in the other two. Daniel Kahneman sets it out in the chapter called The Fourfold Pattern in Thinking, Fast and Slow, as a prediction of prospect theory, the account of choice under risk.
| High probability | Low probability | |
|---|---|---|
| Gains | Risk averse: take the sure sum | Risk seeking: buy the lottery ticket |
| Losses | Risk seeking: gamble to avoid the loss | Risk averse: buy the insurance |
Why the table has four cells instead of two
Two ingredients of prospect theory, which Kahneman developed with Amos Tversky, produce the table. Cross gains and losses with high and low probability and the cells follow. The first is a reference point: an outcome is felt as a gain or a loss against where a person stands, so the same sum can sit in either row. The second is probability weighting, carried by decision weights. A chance of 5 per cent is felt as larger than it is, and a chance of 95 per cent as smaller, so the step to or from certainty looms large.
Kahneman gives each cell a motive in his chapter on the pattern. A near-certain gain brings fear of disappointment, and the sure sum wins. A near-certain loss brings hope of escape, and the gamble wins. A long shot at a large gain brings hope, and the ticket sells. A long shot at a large loss brings fear, and the policy sells.
The fourfold pattern is one of the theory's most direct challenges to expected utility. A single curve of attitude to risk cannot produce both the lottery ticket and the insurance policy, and it cannot produce a gambler who refuses a small sure loss. The fourfold pattern does.
The fullest test is in the cumulative prospect theory paper by Amos Tversky and Daniel Kahneman, published in 1992 in the Journal of Risk and Uncertainty. Twenty-five graduate students from Berkeley and Stanford, with no special training in decision theory, each took part in three one-hour sessions and were paid 25 dollars. They priced 28 gain prospects and 28 loss prospects, each with two outcomes.
A computer derived each price from choices. For every prospect the student chose between the gamble and a descending series of sure amounts, and a second, finer series of sure amounts narrowed the answer. The midpoint between the lowest sure amount accepted and the highest rejected became the certainty equivalent.
The paper reports the complete fourfold pattern for 22 of the 25 students. In aggregate, 78 per cent of the departures from expected value on low-probability gains were risk seeking, and 88 per cent on high-probability gains were risk averse. For losses the signs reversed: 87 per cent risk seeking at high probability and 80 per cent risk averse at low probability. Low probability meant 10 per cent or less, and high meant 50 per cent or more.
Courtrooms, settlements and the bill a city runs up
The legal scholar Chris Guthrie applied the pattern to choices about settling civil claims, in a 2003 article in the Northwestern University Law Review on prospect theory, risk preference and the law. On the logic of the fourfold pattern, a plaintiff with a 95 per cent chance of winning a claim is offered a settlement at 90 per cent of its value, which gives up expected value for certainty. A defendant who is just as likely to lose is in the risk-seeking cell, refuses the offer and gambles. A plaintiff with a long shot holds out for more than the odds justify, because the defendant is buying insurance.
Kahneman writes that the prediction from the fourfold pattern, that the risk-seeking defendant holds the stronger hand, was confirmed by experiments with law students and practising judges, and by analyses of actual negotiations. The book does not name those studies.
One published experiment tests the neighbouring idea. Gilliland and Dunn, in a 2008 paper in Judgment and Decision Making, put simulated litigants in plaintiff and defendant roles with positively or negatively framed offers. Framing drove settlement most consistently: in the first experiment the acceptance rate was 0.74 under positive framing and 0.46 under negative. Role had no significant overall effect there. In the second experiment role mattered, until the participants' own estimate of their chance of losing was held constant, and then it disappeared.
The perceived chance of winning was the strongest predictor of settling, which the authors describe as consistent with prospect theory. A difference between plaintiffs and defendants by role alone did not hold up in their data.
Kahneman's worked example is a policy question set in New York. The city faces 200 frivolous suits a year. Each has a 5 per cent chance of costing 1 million dollars and can be settled for 100,000. Fighting all 200 means losing about ten, for a total of 10 million dollars. Settling all 200 costs 20 million, twice as much.
Each settlement, examined alone, looks sensible: 100,000 dollars removes a one-in-twenty chance of a seven-figure bill, which is the low-probability loss cell where people buy insurance. Across 200 suits those purchases add up to a premium double the expected loss. The example is Kahneman's own illustration, not an experiment.
Whether the fourfold pattern holds when the method changes
The economist William Harbaugh and two co-authors tested the fourfold pattern under two elicitation methods and published the result in the Economic Journal in 2010. When participants priced the gambles, they were on average risk seeking over low-probability gains and high-probability losses, and risk averse over high-probability gains and low-probability losses. The fourfold pattern appeared.
When participants instead chose between a gamble and its expected value, their decisions were indistinguishable from random choice. The sensitivity to method held between and within participants, and it remained when people could review and change their answers. The authors suggest that pricing is the more complex task and that heavier cognitive load may exacerbate behavioural anomalies.
The 1992 procedure derived cash values from a series of choices against sure amounts, a method that differs from the direct pricing in Harbaugh's design. Neither paper settles how far the fourfold pattern describes everyday choices that arrive as simple accept-or-refuse offers.
Where the four cells show up in daily choices
Insurance and lottery tickets are the standard pair, and the book places them in the two low-probability cells. The high-probability loss cell is where Kahneman locates the failing project: a company that has spent 50 million dollars on a venture, faces forecasts that are worse, and is asked for 60 million more. Declining to stop is a gamble to avoid a loss that has already become near-certain. Our articles on Why You Finish the Series You Stopped Enjoying and Loss Aversion and the Bet You Turn Down follow neighbouring effects, and Why a 1 Per Cent Chance Feels Bigger covers the weighting that produces the bottom row.
Kahneman does not treat the four cells as errors to be stamped out: he writes that the decisions in them are not obviously unreasonable, since a person can empathise with the plaintiff and the defendant alike, and he adds that over the long run the deviations from expected value are likely to be costly.
Common questions
What is the fourfold pattern of risk?
It is the pattern of risk attitudes in prospect theory: people are risk averse for likely gains and unlikely losses, and risk seeking for unlikely gains and likely losses. Tversky and Kahneman (1992) reported the full pattern in 22 of 25 graduate students.
Who discovered the fourfold pattern?
Amos Tversky and Daniel Kahneman, as part of prospect theory. The 1992 paper in the Journal of Risk and Uncertainty reports the pattern with new data. Kahneman's 2011 book, Thinking, Fast and Slow, presents the table in his chapter on the pattern.
Is the fourfold pattern always found?
It appears to depend on the method. Harbaugh and colleagues (2010) report it when participants priced gambles, but choices between a gamble and its expected value were indistinguishable from random.
Sources
Kahneman, D. (2011). Thinking, Fast and Slow. New York: Farrar, Straus and Giroux. Seed book; vault hub Books/Thinking Fast and Slow.
Tversky, A., & Kahneman, D. (1992). 'Advances in Prospect Theory: Cumulative Representation of Uncertainty.' Journal of Risk and Uncertainty, 5(4), 297-323. doi:10.1007/BF00122574 https://doi.org/10.1007/BF00122574
Harbaugh, W. T., Krause, K., & Vesterlund, L. (2010). 'The Fourfold Pattern of Risk Attitudes in Choice and Pricing Tasks.' The Economic Journal, 120(545), 595-611. doi:10.1111/j.1468-0297.2009.02312.x https://doi.org/10.1111/j.1468-0297.2009.02312.x
Gilliland, V., & Dunn, J. C. (2008). 'Decision making in civil disputes: The effects of legal role, frame, and perceived chance of winning.' Judgment and Decision Making, 3(7), 512-527. https://cambridge.org/core/services/aop-cambridge-core/content/view/8A024EE909E55606A184A784200322F5/S1930297500000784a.pdf/decision-making-in-civil-disputes-the-effects-of-legal-role-frame-and-perceived-chance-of-winning.pdf
Guthrie, C. (2003). 'Prospect Theory, Risk Preference, and the Law.' Northwestern University Law Review, 97, 1115. https://ir.vanderbilt.edu/handle/1803/5906