Fourfold pattern of risk attitudes
Also known as: Fourfold pattern
Risk attitudes flip across four cells defined by gains vs. losses and high vs. low probability.
What it means
The fourfold pattern is prospect theory's summary of when people are risk-averse versus risk-seeking, organized in a two-by-two of outcome sign (gain or loss) and probability (high or low). For high-probability gains people are risk-averse (taking the sure thing) and for high-probability losses risk-seeking (gambling to avoid a certain loss); for low-probability gains they are risk-seeking (buying lottery tickets) and for low-probability losses risk-averse (buying insurance). The pattern falls out of combining the S-shaped value function with the inverse-S decision-weight function: overweighting of rare events drives the lottery and insurance cells, while diminishing sensitivity drives the high-probability cells. It is among the theory's better-documented signatures in aggregate data, though it describes median tendencies rather than every individual and can weaken or reverse depending on how preferences are elicited; it still neatly explains why the same person both gambles and insures. It also rationalizes desperate risk-taking when facing a near-certain large loss, as in litigation or failing firms.
How it works
The pattern is a compact prediction that comes from two ingredients of prospect theory working together. The first is the value function, which is concave for gains and convex for losses and steeper for losses than for gains; this diminishing sensitivity means that moving from a sure outcome to a risky one feels different in the two domains. The second is the probability weighting function, which converts stated probabilities into decision weights and is not linear: rare events are overweighted, while moderate and high probabilities are underweighted, producing the characteristic inverse-S shape. In the two cells where a small probability is at stake, the weighting function dominates. A tiny chance of a large gain is treated as if it were larger than it is, which makes the gamble attractive and produces risk seeking; a tiny chance of a large loss is likewise inflated, which makes paying to avoid it attractive and produces risk aversion. In the two cells where a high probability is at stake, diminishing sensitivity dominates. A near-certain gain is preferred as a sure thing because the gap between very likely and certain looms large, producing risk aversion; a near-certain loss is resisted for the same reason, so people accept a gamble that offers a slim chance of avoiding it, producing risk seeking. The sign of the outcome and the size of the probability jointly decide which force wins.
The original demonstration
The two forces were named in Kahneman and Tversky's 1979 statement of prospect theory, which already documented the reflection effect, the observation that risk preferences tend to flip when a problem is restated from gains to losses. The full two-by-two was set out most cleanly in Tversky and Kahneman's 1992 paper on cumulative prospect theory, where they measured median certainty equivalents for simple gambles with a single non-zero outcome of one hundred units. For a five percent chance of gaining, the median participant valued the gamble at about fourteen units, well above its expected value of five, revealing risk seeking; for a ninety-five percent chance of the same gain, the certainty equivalent fell to about seventy-eight against an expected value of ninety-five, revealing risk aversion. The loss side mirrored this. A five percent chance of losing was treated as worse than its expected value, with a certainty equivalent near negative eight against negative five, the signature of insurance-like risk aversion, while a ninety-five percent chance of losing was treated as less bad than its expected value, about negative eighty-four against negative ninety-five, the signature of a last-ditch gamble. Read across the table, those four numbers are the fourfold pattern in its original quantitative form, and they cannot be reproduced by curvature of the utility function alone; the overweighting of the five percent cells is what forces the diagonal.
What the evidence shows
The aggregate pattern has been reproduced many times, and the shape of the probability weighting function that underlies it is one of the more stable estimates in the field. Parametric and nonparametric studies of the weighting function, for example the curvature estimates of Wu and Gonzalez in 1996 and the shape analysis of Gonzalez and Wu in 1999, consistently recover an inverse-S curve that crosses the identity line somewhere around a probability of one third, which is exactly what the fourfold pattern requires. Reviews of probability-dependent risk preferences, such as Fehr-Duda and Epper's 2012 survey, treat the pattern as a settled qualitative regularity while stressing that its magnitude varies with context, stakes and individual differences. A large multinational replication by Ruggeri and colleagues in 2020, spanning nineteen countries and more than four thousand participants, recovered the core prospect-theory effects that generate the pattern, including the reflection of risk attitudes across gains and losses, though the size of the effects differed noticeably from one country to another. The pattern is nonetheless not a fixed property of individuals, and it is sensitive to how choices are elicited. Harbaugh, Krause and Vesterlund reported in 2010 that the fourfold pattern appeared clearly when participants priced gambles but essentially vanished when the same options were presented as direct choices, where decisions were close to indistinguishable from chance. Because pricing and choosing can point to opposite conclusions, the pattern is best read as a robust description of valuation behavior in aggregate rather than a universal law of how any given person chooses.
Where it shows up
The most familiar illustration is that the same household buys insurance and lottery tickets at the same time, a combination that looks incoherent under expected-utility theory but falls straight out of the two low-probability cells: rare losses are overweighted, so protection is worth a premium, and rare gains are overweighted, so a long shot is worth a small stake. The high-probability cells surface wherever a near-certain outcome is on the table. In legal disputes, a party with a strong case behaves like the high-probability-gain cell and tends to accept settlements below the expected judgment to lock in the near-certain win, while a party likely to lose behaves like the high-probability-loss cell and prefers the gamble of trial, which is one reason cases with lopsided merits can still fail to settle. The same logic describes gambling for resurrection, where managers of a firm facing an almost-certain collapse take on high-variance bets that a solvent firm would refuse, because a near-certain large loss makes even a poor gamble look attractive.
Limits and caveats
Several qualifications keep the pattern from being oversold. It is a statement about medians and aggregates; individuals differ widely, and a sizeable minority do not display all four cells, with some people treating very small probabilities as effectively zero and others as near-certainties, so behavior in the low-probability corners is especially noisy and can be bimodal. The pattern also depends on the response mode, as the divergence between pricing and choice makes clear, and it can weaken or invert when probabilities are learned from experience rather than described in words, because rare events are often underweighted rather than overweighted when people rely on small samples. Finally, the fourfold pattern is a description rather than an explanation of ultimate causes: it summarizes what decision weights and value curvature jointly imply, but it does not by itself settle competing accounts of why probabilities are weighted nonlinearly. Read with these limits in mind, it remains a useful organizing device, a reminder that risk aversion and risk seeking are not traits of a person but of a situation defined by the sign of the stakes and the size of the odds.
Examples
Someone facing a near-certain large legal loss rejects a modest settlement and gambles on trial, while the same person buys insurance against a tiny chance of a house fire.
A household pays premiums on home and health cover, protecting against rare but large losses, while also buying the occasional lottery ticket; within one budget it is risk-averse toward the rare loss and risk-seeking toward the rare gain.
Offered a guaranteed payout worth slightly less than the expected value of a coin-flip for a bigger prize, a contestant takes the sure money, illustrating the high-probability-gain preference for certainty.
A near-insolvent company, facing an almost-certain orderly wind-down that would wipe out its equity, instead commits its remaining cash to a low-odds turnaround bet, the classic high-probability-loss move sometimes called gambling for resurrection.
In settlement talks the side with a strong case accepts a discount to secure a near-certain win, while the side likely to lose prefers to roll the dice at trial, so disputes with lopsided merits can remain unsettled.
First described in Tversky & Kahneman (1992).
Key references
- Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291. doi.org/10.2307/1914185
- Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323. 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.org/10.1111/j.1468-0297.2009.02312.x
- Ruggeri, K., Ali, S., Berge, M. L., Bertoldo, G., Bjorndal, L. D., Cortijos-Bernabeu, A., ... & Folke, T. (2020). Replicating patterns of prospect theory for decision under risk. Nature Human Behaviour, 4(6), 622-633. doi.org/10.1038/s41562-020-0886-x
- Fehr-Duda, H., & Epper, T. (2012). Probability and risk: Foundations and economic implications of probability-dependent risk preferences. Annual Review of Economics, 4, 567-593. doi.org/10.1146/annurev-economics-080511-110950
- Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighting function. Management Science, 42(12), 1676-1690. doi.org/10.1287/mnsc.42.12.1676
- Gonzalez, R., & Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38(1), 129-166. doi.org/10.1006/cogp.1998.0710