One Percent Feels Like Five

Eight articles in this series have mentioned prospect theory. All of them meant the value function, which is the part about losses and gains. This one is about the decision weights, which is the part with an equation you can evaluate on a calculator and which generates predictions about your own commercial decisions.

The Verdict, Stated First

Five claims, in descending order of confidence.

One. This is the most quantitatively specified finding in the series. Not an effect size but a functional form with fitted parameters, which means it makes point predictions rather than directional ones.

Two. The fourfold pattern explains four common business behaviours that look inconsistent and are not. Buying insurance, buying lottery-like exposure, settling a case you would probably win, and fighting one you would probably lose are all predicted by the same curve.

Three. The weighting function alone drastically overstates willingness to pay. On our own computation it implies someone would pay 742 percent above the fair premium to insure a 0.1 percent loss. Real insurance loadings are nowhere close, which tells you the weighting function is one input among several and not a model of behaviour by itself.

Four. There is a second fourfold pattern, identified in 1952, which neither version of prospect theory addresses. It concerns outcome size rather than probability, and a 2014 paper says its authors "speculate about why Tversky and Kahneman did not address" it.

Five. A 2017 test confirmed the probability pattern and failed to replicate the outcome pattern in losses. That is a mixed result reported precisely, and it is more informative than either a clean confirmation or a clean failure.

Our Grades For These Claims

Applying the scheme from the first article in this series.

Grade A for the fourfold pattern over probabilities. Stated in the 1992 abstract, which we obtained verbatim from four independent sources, and confirmed in a 2017 experimental paper whose abstract we also obtained.

Grade B for the specific parameter values. The figures of 0.61 and 0.69 reach us through a citing paper rather than from the original, though they are the values universally reported.

Grade A for our own computation of the function, which is arithmetic on a published formula and reproducible by anyone.

Grade A that a second fourfold pattern exists and is not addressed by prospect theory, from a peer-reviewed abstract obtained verbatim.

Grade C for anything about magnitudes of behaviour, because the weighting function in isolation produces figures that plainly do not match observed markets.

Our position: this is the best-specified thing in the series and the one where the gap between a model fitting choices and a model predicting behaviour is most visible.

A Note On Method

Everything here is verified to August 2026.

We obtained the 1992 paper's abstract verbatim from four independent sources, including the publisher[1][2]. We did not obtain the paper, its experiments, its estimation procedure or its confidence intervals on the fitted parameters.

The parameter values of 0.61 and 0.69 come from a paper citing them, not from the original[3], and we report no standard errors on them because we obtained none.

We obtained the 2014 paper's abstract verbatim from two sources[4] and the 2017 paper's abstract and a discussion passage[5]. We obtained neither paper.

We did not obtain the 1979 original, the 1952 Markowitz paper, or any of the weighting-function estimation literature, and report all from citation records.

All computation is ours, applies a published formula to published parameters, and uses invented amounts and probabilities throughout.

This article discusses a decision model. It is not insurance, litigation, investment or risk management advice, and nothing here is a recommendation about any policy, case or exposure.

The Half Nobody Covers

Why this article exists, stated plainly. Ours.

Prospect theory has two components. The value function describes how outcomes are valued relative to a reference point, and is the source of loss aversion. The weighting function describes how probabilities are converted into decision weights, and is the source of everything in this article.

Four observations.

The 1992 abstract names both: "Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting functions."[1] Note the plural on the second, since the paper allows different weighting functions for gains and losses.

The value-function half has travelled into general circulation and the weighting half has not, which we think is because the first can be stated as a sentence and the second cannot.

That asymmetry has a cost. The fortieth article in this series withdrew a claim about loss aversion after a published dispute about its magnitude, and the part of prospect theory with the least contested quantitative content is the part that never left the journals.

And the weighting half is the commercially relevant one. Insurance, warranties, litigation and catastrophic risk are all decisions about probabilities, and the value function says nothing about them.

The 1992 Paper

The source.

Tversky, A., and Kahneman, D. (1992), Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, 5(4), 297–323, October, DOI 10.1007/BF00122574[1][6].

Its abstract: "We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses."[1]

Three observations, ours.

This is a revision of the 1979 original, not the original itself, and the revision was motivated by technical failures. A reference source records that the earlier version permitted violations of stochastic dominance, where a prospect could be evaluated as less attractive despite strictly dominating another[7]. This is a reference encyclopedia and not an academic source, flagged here and at every use.

That the authors revised their own theory to remove a defect is creditable and rarely mentioned when the theory is invoked.

And the phrase "different weighting functions for gains and for losses" matters for everything below, because it means the two domains have separate parameters and the curve is not symmetric.

The Fourfold Pattern

The empirical claim, in the abstract's own words.

"A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability."[1]

A citing source sets it out as four cases and records the authors' own assessment of its importance: the pattern is "regarded as 'the most distinctive implication of prospect theory' by Tversky and Kahneman", comprising "(a) Risk averse over high probability gains. (b) Risk seeking over high probability losses. (c) Risk seeking over low probability gains. (d) Risk averse over low probability losses."[8]

Four observations, ours.

The pattern is a single curve producing four behaviours, which is what makes it a theory rather than a list. One weighting function, applied to gains and losses at high and low probabilities, generates all four cells.

Two of the four are risk seeking, which is worth noting because behavioural finance is usually summarised as a catalogue of risk aversion. The theory predicts risk seeking in half of its cells.

The abstract says the evidence confirms the pattern, combining a review with a new experiment, which is a stronger evidential structure than either alone.

And the same citing source names the applications directly: these attitudes "are often used to justify subjective decision making of individuals for problems such as settlements of civil lawsuits, desperate treatments of terminal illnesses, playing lotteries, and getting insurance coverage."[8]

Four Business Situations

Translating the four cells into decisions a Canadian business owner will recognise. Ours, and they are illustrations of the pattern rather than findings.

Low probability of loss, risk averse. You buy insurance against a fire that will almost certainly not happen, at a price above the expected loss. The pattern predicts this and so does ordinary prudence, which is why it is the least surprising cell.

Low probability of gain, risk seeking. You take a small position in something with a remote chance of a very large payoff, accepting a negative expected value. Lottery tickets are the canonical case and speculative ventures are the commercial one.

High probability of gain, risk averse. You settle a claim you would probably win, accepting less than the expected value, because the certainty of the settlement is worth a premium.

High probability of loss, risk seeking. You fight a case you will probably lose, or continue a venture that is probably failing, because the small chance of avoiding the loss entirely outweighs the certain smaller loss of settling now.

Two observations.

The third and fourth cells together describe litigation behaviour on both sides of a dispute, and they predict that the party likely to win settles cheaply while the party likely to lose fights on. Those are opposite behaviours from the same curve.

And the fourth cell is the one that connects to the fifty-second article. Continuing a probably-failing venture is predicted here by probability weighting alone, without any appeal to sunk costs, which is a third distinct explanation of the same behaviour.

The Function, Computed

The formula and its output. Our own computation, applying a published functional form to published parameters.

The weighting function takes the form w(p) equals p to the power gamma, divided by the quantity p to the gamma plus one-minus-p to the gamma, all raised to the power one over gamma.

A citing paper gives the fitted parameters as 0.61 for gains and 0.69 for losses[3]. We did not obtain the original estimation and report no standard errors.

Evaluating it, with the gain weight first and the loss weight second:

At p = 0.001: 0.0145 and 0.0084. At 0.01: 0.0553 and 0.0397. At 0.05: 0.1316 and 0.1114. At 0.10: 0.1863 and 0.1701.

At 0.25: 0.2907 and 0.2935. At 0.50: 0.4206 and 0.4540. At 0.75: 0.5683 and 0.6264.

At 0.90: 0.7117 and 0.7749. At 0.95: 0.7932 and 0.8499. At 0.99: 0.9116 and 0.9451. At 0.999: 0.9766 and 0.9881.

What Happens To Small Probabilities

Reading the top of that table. Ours.

Four observations.

A 1 percent chance of a gain is weighted at 5.5 percent, which is 5.5 times its stated size. A one-in-a-hundred prospect is acted on as roughly one-in-eighteen.

A one-in-a-thousand chance is weighted at 1.45 percent, which is 14.5 times its stated size. The multiplier grows as the probability shrinks.

That relationship is the whole explanation of the lottery and of catastrophe insurance. The rarer the event, the larger the distortion in proportional terms, which is exactly the region where business decisions about tail risk live.

And note that the loss weights are lower than the gain weights at small probabilities, at 0.0397 against 0.0553 at one percent. Small chances of gain are inflated more than small chances of loss, on these parameters, which is not what most summaries of the theory would lead you to expect.

And To Near-Certainties

Reading the bottom of the table, which is the less-discussed and more commercially useful half. Ours.

Four observations.

A 99 percent chance of a gain is weighted at 91.2 percent. A near-certainty is discounted by about eight percentage points.

A 90 percent chance is weighted at 71.2 percent, a discount of nearly nineteen points. The distortion at the top end is larger than most people assume.

This is the source of the certainty effect: the gap between 99 percent and 100 percent is worth far more than the gap between 89 and 90, because only the first buys you out of the weighting function entirely.

And it is why the third cell of the fourfold pattern exists. A 90 percent chance of winning a case is experienced as roughly a 71 percent chance, which makes a certain settlement look considerably better than the arithmetic says it is.

Where It Crosses

The single number a practitioner should carry. Our own computation, solved numerically from the published formula.

The weighting function crosses the diagonal, meaning the weight equals the stated probability, at p = 0.339 for gains and p = 0.378 for losses.

Three observations.

Below about a third, probabilities are overweighted. Above it, underweighted. That is the entire function reduced to one threshold, and it is a usable heuristic.

The crossover is not at one half, which is the intuitive guess and is wrong. Most of the probability range is in the underweighted region.

And the two domains cross at slightly different points, 0.339 against 0.378, which is a consequence of the separate parameters and is small enough that a practitioner can treat both as roughly one third.

What It Implies For An Insurance Premium

The commercial computation, and we are going to immediately argue against our own result. Our own arithmetic, invented amounts, and it deliberately isolates the weighting function while ignoring the value function entirely.

Take a loss of $100,000 occurring with probability p. The actuarially fair premium is p times the loss. What does the decision weight alone imply?

At 0.1 percent: fair premium $100, weighted $842, implying a loading of 742 percent.

At 0.5 percent: $500 against $2,502, a loading of 400 percent.

At 1 percent: $1,000 against $3,967, 297 percent.

At 2 percent: $2,000 against $6,237, 212 percent.

At 5 percent: $5,000 against $11,143, 123 percent.

At 10 percent: $10,000 against $17,015, 70 percent.

And Why That Number Is Too Big

The section that matters more than the table above it. Ours.

Four observations.

Real insurance loadings are nowhere near 742 percent. A market in which people would routinely pay eight times the fair price for low-probability cover would look nothing like the one that exists.

So the computation above does not predict what anyone would pay, and we would rather say so at length than let a striking number stand unqualified.

The reason is structural and we stated it at the top. We isolated the weighting function and switched off the value function, which in prospect theory works in the opposite direction here by making a small certain premium feel relatively more painful. The full model contains both and this arithmetic contains one.

What the table does show is the direction and the shape: that the distortion is largest where the probability is smallest, that it declines steeply as probability rises, and that it is trivial by the time you reach ten percent. Those three facts are usable and the specific percentages are not.

There Is A Second Fourfold Pattern

The finding that surprised us most in researching this article.

Scholten, M., and Read, D. (2014), Prospect theory and the "forgotten" fourfold pattern of risk preferences, Journal of Risk and Uncertainty, 48(1), 67–83, DOI 10.1007/s11166-014-9183-2[4].

Its abstract: "Markowitz (Journal of Political Economy 60:151–158, 1952) identified a fourfold pattern of risk preferences in outcome magnitude: When outcomes are large, people are risk averse in gains and risk seeking in losses, but risk preferences reverse when the outcomes are small, with people exhibiting risk seeking in gains and risk aversion in losses."[4]

Four observations, ours.

This is a completely different pattern from the one everybody knows. The famous one varies with probability. This one varies with outcome size, holding probability fixed.

It was identified in 1952, twenty-seven years before the first version of prospect theory.

The practical claim is specific and testable: the same person is risk seeking on small gains and risk averse on large ones. Which describes buying a lottery ticket and insuring a building, in one individual, without contradiction.

And we did not obtain the 1952 paper and report it entirely through the 2014 abstract's description.

Which Prospect Theory Does Not Address

The claim in the same abstract, and it is unusually direct about a canonical theory.

"This fourfold pattern was not addressed by either version of prospect theory."[4]

The authors then show it can be accommodated: "We show how prospect theory can accommodate the pattern by combining an overweighting of low probabilities with a decreasingly elastic value function. We then examine the performance of prospect theory with two decreasingly elastic value functions: Prospect theory performs better, both quantitatively and qualitatively, with a normalized logarithmic value function than with a normalized exponential value function."[4]

And a repository copy records the paper's closing move: "We discuss several issues, and speculate about why Tversky and Kahneman did not address Markowitz's fourfold pattern."[9]

Four observations, ours.

"Not addressed by either version" is a strong statement about a theory that has two Nobel-associated authors and fifteen thousand citations, and it appears in a peer-reviewed abstract without hedging.

The paper is constructive rather than destructive. It shows the pattern can be accommodated by changing the value function's form, which extends prospect theory rather than refuting it.

The word "speculate" in that closing sentence is the authors marking their own explanation as unverified, and we did not obtain what they speculate. That is a gap we would like to close and cannot.

And this is the pattern the fiftieth article named as most common in this series, appearing again. A qualification exists in print, has existed for decades in the underlying case, and has not travelled. Markowitz's pattern is older than prospect theory and less known than any part of it.

A Worked Example Of The Second Pattern

The Markowitz pattern stated as a choice, because it is easier to test against your own intuition than to follow in the abstract.

An academic preprint gives the canonical illustration: "people would rather choose a sure 1 million dollar option rather than a (low-probability) risky gamble yielding $10 million dollars with probability 0.1 and nothing otherwise."[10] The same source describes the pattern as risk aversion for gains and risk seeking for losses "in moderate-to-large outcomes", reversing "when outcomes are small, with people being risk-seeking for gains and risk-averse for losses."[10]

Four observations, ours.

Note what that choice does. The gamble and the certainty have identical expected values, at one million each, and almost everyone takes the certainty. That is risk aversion at a large outcome.

Now shrink both by a factor of a million. A sure dollar against a one-in-ten chance of ten dollars, same expected value, and the preference commonly reverses. That reversal is the entire Markowitz claim.

The two patterns are therefore separable and can act together or against each other. A small-probability, small-outcome gain is pushed toward risk seeking by both. A small-probability, large-outcome gain is pushed one way by probability weighting and the other by outcome magnitude.

And that is why the second pattern matters commercially rather than academically. Most business risk decisions involve large outcomes at low probabilities, which is precisely the cell where the two patterns pull in opposite directions and where knowing only one of them predicts the wrong behaviour.

What Replicated And What Did Not

A later experimental test of both patterns, and the result is mixed in an informative way.

Bouchouicha, R., and Vieider, F. M., Accommodating stake effects under prospect theory, Journal of Risk and Uncertainty[5].

Its abstract states the background: "One of the stylized facts underlying prospect theory is a fourfold pattern of risk preferences. People have been shown to be risk seeking for small probability gains and large probability losses, while being risk averse for large probability gains and small probability losses. Another fourfold pattern of risk preferences over outcomes, postulated by Harry Markowitz in 1952, has received much less attention and is currently not integrated into prospect theory."[5]

Its result: "In two experiments, we show that risk preferences may change over outcomes. While we find people to be risk seeking for small outcomes, this turns to risk neutrality and later risk aversion as stakes increase."[5]

And from its discussion: "For the smallest probability of p = 0.1, we find risk aversion to be the prevalent pattern throughout. For p = 0.9, we find risk seeking across the outcome spectrum. For the intermediate probability of p = 0.5, risk neutrality cannot be rejected. This pattern is consistent with the fourfold pattern of risk preferences over probabilities incorporated into prospect theory."[5]

Reading That Result Carefully

Because the same discussion contains a failure, and it applies to only one of the two patterns. Ours.

The passage continues: "When it comes to Markowitz's fourfold pattern over outcomes, however, we cannot replicate the patterns we found for gains. In particular, there does not appear to be a clear pattern to risk preferences as outcomes change for any of the probability levels. This finding is in agreement with previous evidence, with most studies investigating stake effects for losses finding no stake effects."[5]

Four observations, ours, and precision matters here.

The probability-based fourfold pattern was confirmed. The passage explicitly says the observed pattern is consistent with the one incorporated into prospect theory. That is a successful test of the finding this article is mostly about.

The outcome-magnitude pattern held for gains and did not replicate for losses. The authors found stake effects in the gain domain, running from risk seeking through neutrality to aversion as stakes rose, and no clear pattern in losses.

They report that this agrees with previous evidence, noting most studies of stake effects for losses find none. So the asymmetry is established rather than anomalous.

And we want to be exact about what this does to the two patterns. The famous one survives a modern test. The forgotten one survives in half its domain. Neither of those is the headline a summary would reach for and both are what the source says.

What The Model Was Fitted To

A limitation that follows from the theory's own description of its scope, and which bears on every business application in this article. Ours.

The 1992 abstract says the theory "applies to uncertain as well as to risky prospects with any number of outcomes"[1]. In this literature a risky prospect is one whose probabilities are known and stated, and the parameters were fitted to choices between such prospects.

Four observations.

That is not the situation of most commercial decisions. A business owner facing a possible supplier failure has no stated probability, only a judgment, and the weighting function takes a stated probability as its input.

Which means applying it to business requires a step the theory does not supply: somebody has to produce the number in the first place. The fifty-first through sixtieth articles in this series are largely about how badly that step goes.

So the honest chain is longer than it looks. A judged probability, itself subject to everything the preceding ten articles documented, is then transformed by a function fitted to stated probabilities. Errors at the first stage are not corrected by accuracy at the second.

And the practical consequence is an ordering. Improving the estimate is worth more than correcting the weighting, because the weighting function applies a known and modest distortion to whatever it is given, while the estimate can be wrong by any amount at all.

What Actually Survives

Our reading, stated directly.

Five statements.

The fourfold pattern over probabilities is well supported. Stated in a heavily cited paper, confirmed by review and new experiment there, and consistent with a later independent experimental test.

The functional form and its parameters are usable and specific. One third is the crossover, below it probabilities are inflated and above it discounted, and the distortion is steepest at the extremes.

The weighting function alone does not predict magnitudes. Our own computation implies insurance loadings that no market exhibits, which is a limit of the isolated component rather than of the theory.

A second fourfold pattern exists, is older, and is not integrated into the theory. It concerns outcome size and it replicated for gains and not for losses.

And the theory's own authors revised it once to remove a technical defect, which is worth knowing when it is invoked as settled.

Why Firms Fight Cases They Will Lose

The application we think is most valuable, because it explains a behaviour that looks like stubbornness and is predicted by arithmetic. Ours, and not legal advice.

Four points.

A defendant facing a 90 percent chance of losing is in the high-probability loss cell, where the pattern predicts risk seeking. On our own computation the loss weight at 0.90 is 0.7749, so a near-certain loss is experienced as roughly a three-in-four loss.

That gap makes fighting look better than it is. The one-in-four feeling of escape is worth more than the one-in-ten reality, and settling converts a weighted 77 percent loss into a certain one.

Meanwhile the plaintiff with a 90 percent chance of winning sits in the high-probability gain cell, where the gain weight is 0.7117. Their near-certain win feels like a seven-in-ten win, which makes a certain settlement attractive.

So the model predicts both parties move toward the defendant's preferred outcome: the likely loser fights and the likely winner settles cheaply. That is a prediction about a negotiation, generated by one curve, and it is checkable against any firm's own settlement history.

If You Sell A Warranty

The application on the revenue side. Ours, untested, and not insurance advice.

Four points.

Extended warranties and service plans sell protection against low-probability losses, which is the cell where the weighting distortion is proportionally largest.

The commercial implication is that willingness to pay for such cover exceeds its expected cost, which is why the products are profitable and is not in itself a criticism of them.

But our own computation is a warning against pricing on this basis. The isolated weighting function implies loadings of several hundred percent and real markets do not sustain them, so the distortion is real and much smaller than the naive calculation suggests.

And there is a professional point worth stating for this publication's readers. A firm advising a client on whether to buy such cover is advising against a documented bias, and the correct advice depends on the client's ability to absorb the loss rather than on the probability alone.

Low-Probability Catastrophic Risk

The application where the direction reverses, which is the one most likely to be got wrong. Ours.

Four points.

The pattern says small probabilities are overweighted, which predicts people take tail risk more seriously than the arithmetic warrants, not less.

That contradicts the common business claim that firms systematically ignore tail risk, and we think both can be true for a reason the model supplies. The weighting function applies to probabilities that have been stated. It says nothing about risks nobody has assigned a probability to.

So the prediction is specific and testable. A named, quantified tail risk will be overweighted. An unnamed one will be weighted at zero, because it does not enter the calculation at all.

And that suggests the useful intervention is not adjusting anyone's weighting but getting the risk written down with a number attached, which is the same instrument the last four articles converged on for four different failures.

Using The One-Third Rule

The whole function compressed into something a person can apply in a meeting, with the honest caveat attached. Ours.

Three uses.

When a probability below a third is quoted at you, expect it to be over-weighted in the room. That includes your own reaction. The remedy is to compute the expected value explicitly and compare, since the distortion operates on the felt probability and not on the arithmetic.

When a probability above a third is quoted, expect it to be under-weighted, and expect the discount to grow as the figure approaches certainty. A ninety percent chance is treated as roughly seventy percent in the gain domain, which is a nineteen point gap and is larger than most people would guess.

When you are selling certainty, you are selling into the steepest part of the curve. The move from ninety-five percent to a guarantee is worth about twenty-one points of decision weight for five points of probability, on our own computation.

Two caveats we would attach every time.

The rule describes a documented tendency, not a law, and the parameters come to us at one remove with no standard errors, so treating 0.339 as precise would be a mistake the source material does not support.

And it operates on stated probabilities. If nobody has stated one, there is nothing for the function to distort, and the failure is upstream in a place the previous ten articles in this series describe at length.

One observation to close the section. The rule is unusual in this series for being a correction you apply to a number rather than to a person, which is why we would trust it further than most of the advice these articles have produced. You do not have to persuade anyone they are biased; you have to write down the expected value and compare it with what the room wants to do.

The Certainty Premium

A last implication, and the most directly negotiable. Ours.

Three observations.

Because the function discounts near-certainties, the last few percentage points of probability are worth disproportionately more than the middle ones. Moving a counterparty from 95 percent to 100 percent buys more than moving them from 60 to 65.

On our own computation the weight rises from 0.7932 at 95 percent to 1.0 at certainty, a gain of about 21 points of weight for 5 points of probability. Between 60 and 65 percent the weighted gain is a fraction of that.

The commercial form is a guarantee. Removing the last residual uncertainty from an offer is worth more than the residual uncertainty is, which is why guarantees, fixed fees and caps command a premium out of proportion to the risk they transfer.

What To Do

Learn the one-third crossover. Below about a third, stated probabilities are inflated in decisions; above it, discounted. That single threshold carries most of the function's practical content.

Expect the likely loser to fight and the likely winner to settle. On the published parameters a 90 percent loss is weighted at 0.775 and a 90 percent gain at 0.712, and both distortions push toward the same settlement outcome.

Do not price anything off the weighting function alone. Our own computation implies insurance loadings of several hundred percent that no market sustains, because isolating one component of a two-component model is not a model.

Distinguish named tail risks from unnamed ones. The theory predicts a quantified small probability is overweighted; a risk with no number attached is not weighted at all.

Price the last five points of probability higher than the middle ones. The certainty effect means guarantees and caps are worth more than the risk they remove.

Ask which fourfold pattern someone means. There are two, one over probabilities and one over outcome size, and the second is older and less integrated.

Treat outcome-size effects as established for gains and not for losses. A 2017 test found stake effects running from risk seeking to risk aversion in gains and no clear pattern in losses, agreeing with prior evidence.

Note that the theory was revised by its own authors. The 1992 version exists because the 1979 one permitted violations of stochastic dominance, which is creditable and is rarely mentioned.

The Limits Of This Analysis

Several caveats matter, and one of them concerns a number in this article that should not be used. This article discusses a decision model and is not insurance, litigation, investment or risk management advice; nothing here is a recommendation about any policy, case or exposure, and the applications are our own reasoning and untested. Everything is verified to August 2026. We did not obtain the 1992 paper, only its abstract from four independent sources, so we report nothing of its experiments, its estimation method, or its confidence intervals. The parameter values of 0.61 and 0.69 reach this article through a citing paper, not from the original, and we report no standard errors on them because we obtained none; a different fitted pair would move every figure we compute. We did not obtain the 1979 original, the 1952 Markowitz paper, the 2014 paper, or the 2017 paper, and report all from abstracts, a discussion passage, and citation records. One source is a reference encyclopedia and two are academic preprints citing the literature, all flagged at every use. All computation is ours. It applies a published formula to published parameters, which is arithmetic, but the amounts and probabilities are invented throughout. The insurance loading figures deliberately isolate the weighting function and switch off the value function, which is not what prospect theory does; they show shape and direction and they are not predictions of what anyone would pay, as the body states at length. Our account of the litigation cells applies the published weights to invented probabilities and is an illustration of the pattern, not a model of any dispute. And this article covers one half of a two-part theory, so nothing here should be read as a complete account of decision under risk.

Related Reading

No. 62 The Discount Rate Nobody Can Pin Down A review of three decades found spectacular variation in implied discount rates. Then a paper in the AER found no present bias at all, and showed what... Behavioural Finance The Number That Is Irrelevant Both Ways A third explanation for continuing a venture that is probably failing. Behavioural Finance The Streak That Proves Nothing Why the probability you assign in the first place is usually the weaker link.

Frequently Asked Questions

What is probability weighting?

The half of prospect theory describing how stated probabilities are converted into decision weights. On the published parameters, a one percent chance is weighted at 5.5 percent in the gain domain and a 99 percent chance at 91.2 percent. Small probabilities are inflated and near-certainties discounted.

What is the fourfold pattern?

From the 1992 abstract: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability. One curve producing four behaviours, which the authors regarded as the most distinctive implication of the theory.

Is there a single number to remember?

One third. On our own computation the weighting function crosses the diagonal at 0.339 for gains and 0.378 for losses. Below roughly a third, probabilities are overweighted in decisions. Above it, underweighted. The intuitive guess of one half is wrong.

Does this mean people overpay for insurance by 700 percent?

No, and we argue against our own figure in the body. That computation isolates the weighting function and switches off the value function, which in the full model works the other way here. It shows the shape of the distortion, not a prediction of willingness to pay, and real markets sustain nothing like those loadings.

Why do firms fight cases they will probably lose?

On the published parameters a 90 percent probability of loss carries a decision weight of 0.775, so a near-certain loss is experienced as roughly three-in-four. The one-in-four feeling of escape outweighs the one-in-ten reality, and settling converts that into a certainty. Meanwhile the likely winner discounts their own 90 percent to 0.712 and settles cheaply.

Doesn't this contradict firms ignoring tail risk?

Both can hold, and the model says why. The weighting function applies to probabilities that have been stated. A named, quantified tail risk will be overweighted; an unnamed one is weighted at zero because it never enters the calculation. The intervention is getting a number written down, not adjusting anyone's weighting.

What is the second fourfold pattern?

One identified in 1952 that varies with outcome size rather than probability: risk aversion for large gains and risk seeking for large losses, reversing for small outcomes. A 2014 paper states it was not addressed by either version of prospect theory, and a later test found it holds for gains and does not replicate for losses.

References

1. Tversky, A., & Kahneman, D. (1992). Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323, October, DOI 10.1007/BF00122574. Publisher record reproducing the abstract in full: on the authors developing a new version of prospect theory employing cumulative rather than separable decision weights and extending the theory in several respects; on this version, called cumulative prospect theory, applying to uncertain as well as risky prospects with any number of outcomes and allowing different weighting functions for gains and for losses; on two principles, diminishing sensitivity and loss aversion, being invoked to explain the characteristic curvature of the value function and the weighting functions; and on a review of the experimental evidence together with the results of a new experiment confirming a distinctive fourfold pattern of risk attitudes, being risk aversion for gains and risk seeking for losses of high probability, and risk seeking for gains and risk aversion for losses of low probability. Note: the publisher's record. We obtained the abstract only; we did not obtain the paper, its experiments, its estimation procedure or any confidence intervals on its fitted parameters. link.springer.com
2. Bibliographic service record for the same paper, reproducing the abstract identically and recording the DOI as 10.1007/BF00122574; together with a separate academic library citation record giving the citation as A. Tversky and D. Kahneman, Journal of Risk and Uncertainty, volume 5, number 4, pages 297–323, 1992. Note: two further independent reproductions of the abstract and citation, used to confirm the wording of the fourfold pattern verbatim. scispace.com
3. Repository page for the 1992 paper carrying scholarly citing text, recording that the weighting function has curvature parameters constrained between zero and one, and referring to the parameter values given by Tversky and Kahneman as 0.61 and 0.69 in the weighting function. Note: a repository page reproducing text from a citing paper. Our source for the parameter values, which reach this article at one remove from the original; we obtained no standard errors and a different fitted pair would move every figure computed here. researchgate.net
4. Scholten, M., & Read, D. (2014). Prospect theory and the "forgotten" fourfold pattern of risk preferences. Journal of Risk and Uncertainty, 48(1), 67–83, DOI 10.1007/s11166-014-9183-2. Publisher record reproducing the abstract in full: on Markowitz having identified in 1952, in the Journal of Political Economy, a fourfold pattern of risk preferences in outcome magnitude, under which people are risk averse in gains and risk seeking in losses when outcomes are large, with preferences reversing when outcomes are small so that people exhibit risk seeking in gains and risk aversion in losses; on this fourfold pattern not having been addressed by either version of prospect theory, citing the 1979 and 1992 papers; on the authors showing how prospect theory can accommodate the pattern by combining an overweighting of low probabilities with a decreasingly elastic value function; and on prospect theory performing better, both quantitatively and qualitatively, with a normalized logarithmic value function than with a normalized exponential one. Note: the publisher's record. We obtained the abstract in full and not the paper, and we did not obtain the 1952 paper it describes. link.springer.com
5. Bouchouicha, R., & Vieider, F. M. Accommodating stake effects under prospect theory. Journal of Risk and Uncertainty, DOI 10.1007/s11166-017-9266-y. Publisher record reproducing the abstract and a discussion passage: on one of the stylized facts underlying prospect theory being a fourfold pattern of risk preferences, with people shown to be risk seeking for small probability gains and large probability losses while risk averse for large probability gains and small probability losses; on another fourfold pattern over outcomes, postulated by Harry Markowitz in 1952, having received much less attention and not currently being integrated into prospect theory; on the authors showing in two experiments that risk preferences may change over outcomes, with people risk seeking for small outcomes turning to risk neutrality and later risk aversion as stakes increase; on a one-parameter logarithmic utility function fitting such stake effects significantly better than the power or exponential functions mostly used; on the discussion recording that for the smallest probability of 0.1 risk aversion was the prevalent pattern throughout, for 0.9 risk seeking across the outcome spectrum, and for 0.5 risk neutrality could not be rejected, this pattern being consistent with the fourfold pattern over probabilities incorporated into prospect theory; and on the authors being unable to replicate Markowitz's fourfold pattern over outcomes for losses, with no clear pattern to risk preferences as outcomes change at any probability level, in agreement with previous evidence and with most studies investigating stake effects for losses finding none. Note: the publisher's record. We obtained the abstract and one discussion passage and not the paper. link.springer.com
6. Economics working paper database record confirming the citation as Tversky, Amos and Kahneman, Daniel, 1992, Advances in Prospect Theory: Cumulative Representation of Uncertainty, Journal of Risk and Uncertainty, Springer, volume 5, issue 4, pages 297–323, October; carried on a reference list which also identifies Scholten, Marc and Read, Daniel, 2014, Prospect theory and the "forgotten" fourfold pattern of risk preferences, Journal of Risk and Uncertainty, volume 48, issue 1, pages 67–83, February. Note: a bibliographic database record used for independent confirmation of both citations including month of publication. ideas.repec.org
7. Reference encyclopedia entry on cumulative prospect theory, recording that it was formulated by Amos Tversky and Daniel Kahneman in their 1992 paper published in the Journal of Risk and Uncertainty; that the work built on the original prospect theory by introducing cumulative decision weights to address key limitations in handling probabilities, extending the model to prospects with multiple outcomes; that the primary motivations stemmed from violations of stochastic dominance in the original prospect theory, where certain prospects could be evaluated as less attractive despite strictly dominating others in outcomes and probabilities; and that the separable decision weights of the earlier model proved inadequate for generalizing to multi-outcome prospects. Note: a reference encyclopedia, not an academic source, flagged at every use. Our source for the technical motivation behind the 1992 revision. grokipedia.com
8. Academic preprint applying cumulative prospect theory to pricing, section on the fourfold pattern of risk attitudes, recording that the pattern is regarded as the most distinctive implication of prospect theory by Tversky and Kahneman; setting it out as four categories, being risk averse over high probability gains, risk seeking over high probability losses, risk seeking over low probability gains, and risk averse over low probability losses; and recording that these risk attitudes are often used to justify subjective decision making for problems such as settlements of civil lawsuits, desperate treatments of terminal illnesses, playing lotteries, and getting insurance coverage. Note: an academic preprint, not a primary source, flagged at every use. Cited for its four-cell restatement and for the applications it names. arxiv.org
9. Repository page for the 2014 paper carrying its abstract and closing statement, recording that the authors discuss several issues and speculate about why Tversky and Kahneman did not address Markowitz's fourfold pattern; and carrying citing text noting that people have been shown to be risk seeking for small probability gains and large probability losses while being risk averse for large probability gains and small probability losses. Note: a repository page. Our source for the paper's closing move; we did not obtain what the authors speculate. researchgate.net
10. Academic preprint on decision-making under risk, section on the fourfold pattern of risk preferences in outcome magnitude, recording that behavioral economics has documented this pattern citing Markowitz (1952), Hershey and Schoemaker (1980) and Scholten and Read (2014); and describing it as people being risk-averse for gains and risk-seeking for losses at moderate-to-large outcomes, with the pattern reversing when outcomes are small so that people are risk-seeking for gains and risk-averse for losses. Note: a second academic preprint, not a primary source, flagged at every use. Used as an independent restatement of the Markowitz pattern. arxiv.org

This article discusses a decision model and is not insurance, litigation, investment or risk management advice. The 1992 paper was not obtained beyond its abstract, and its parameter values reach this article through a citing paper with no standard errors. The 1979, 1952, 2014 and 2017 papers were not obtained. One source is a reference encyclopedia and two are academic preprints, all flagged at every use. All computation is the authors' own and applies a published formula to published parameters using invented amounts and probabilities. The insurance loading figures deliberately isolate the weighting function and are not predictions of willingness to pay, as the body explains at length.