Plate iii in print

The Reference

Kahneman and Tversky; prospect theory, choices weighed as gains and losses from a reference point, loss aversion, and the distortion of probability.

People do not weigh fortunes by what they will own when all is done, but by the gain or loss measured from where they now stand; a loss cuts deeper than a like gain heals; and the same soul that will not gamble to make more will gamble freely to lose less — so that whether a prospect is taken or refused turns less on the prospect than on the point from which it is seen.

— the argument of this plate, after Daniel Kahneman and Amos Tversky, 1979

The last plate ended with a demand: if expected utility fails as a description of how people choose under risk, then describe what they do instead. The answer that organized the entire field came from a psychologist and a mathematician, Daniel Kahneman and Amos Tversky, who in 1979 set down a theory they called, with deliberate modesty, prospect theory — not a rule for how one ought to choose but a map of how people actually do. Its genius was to replace the smooth, single curve of the old theory, which measured the worth of money by the total a person would end up holding, with a small machinery of psychologically real parts. At the center of that machinery is one idea, simple to state and bottomless in its consequences: that the mind does not register wealth as a level but as a CHANGE — a gain or a loss reckoned from some reference point — so that the very same final circumstance can be lived as a windfall or as a wound, depending entirely on where the counting begins.

Consider what reference-dependence overturns. The old theory held that a rational chooser cares only about final states: offered a prospect, he asks what total wealth each outcome leaves him with and values the prospect accordingly, his history irrelevant, his starting point forgotten. Prospect theory says no one actually thinks this way. A clerk who expected a large bonus and receives a merely good one experiences a loss, though he is richer than yesterday; a gambler who is down for the evening and claws back to even feels triumph, though he has gained nothing. What the mind tracks is the distance and direction from a reference point — usually the status quo, but sometimes an expectation, an aspiration, a level recently held and lost — and it is over these gains and losses, not over final fortunes, that the whole apparatus of valuation operates. The reference point is the origin of the psychological coordinate system, and because it can sit in different places, the same objective outcome occupies different places on the map of feeling.1

1. Reference-dependence overturns the old theory's claim that a rational chooser cares only about final states — asking what total wealth each outcome leaves him, his starting point forgotten. Prospect theory says no one thinks this way. A clerk expecting a large bonus who gets a good one feels a loss though richer than yesterday; a gambler down for the evening who claws back to even feels triumph though he has gained nothing. The mind tracks distance and direction from a reference point — the status quo, an expectation, an aspiration — and values gains and losses, not final fortunes. The same objective outcome can sit in different places on the map of feeling.

From this origin the value function runs outward in two directions, and its first great asymmetry is that the two directions are not mirror images. The curve falls more steeply into loss than it rises into gain: a loss is felt more sharply than a gain of the very same size is enjoyed, by something like a factor of two in Kahneman and Tversky's reckoning. This is LOSS AVERSION, and once seen it is everywhere. It is why a fair coin-toss for equal stakes — heads you win a hundred, tails you lose a hundred — is refused by almost everyone, though its expected value is exactly zero and the old theory near the status quo would call the two outcomes a wash; the threatened loss simply outweighs the equal possible gain. It is why people cling to what they have, why a possession surrendered hurts more than the same possession never acquired would have pleased, why the prospect of giving something up colors a choice far more than the prospect of getting something. The reference point divides the world into the region of having-more and the region of having-less, and the second region is psychologically the heavier; the mind leans, always, against loss.2

2. The value function's first asymmetry: it falls more steeply into loss than it rises into gain — a loss felt more sharply than an equal gain is enjoyed, by something like a factor of two. This is LOSS AVERSION. It is why a fair coin-toss for equal stakes, heads win a hundred, tails lose a hundred, is refused by almost everyone though its expected value is zero; the threatened loss outweighs the equal gain. It is why people cling to what they have, a possession surrendered hurting more than one never acquired would have pleased. The mind leans, always, against loss.
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The second asymmetry concerns the curve's shape, and it produces a reversal stranger still. The value function bends so that each further dollar of gain adds a little less satisfaction than the last — the familiar diminishing returns, which make a person risk-averse among gains, preferring a sure gain to a gamble of the same average value. But on the loss side the curve bends the other way: each further dollar of loss stings a little less than the one before, so that the prospect of a large loss is not as much worse than a small one as their difference in size would suggest. And this convexity in the domain of losses makes people risk-SEEKING there. Faced with a sure loss or a gamble that might lose more but might lose nothing, they reach for the gamble — they will take a chance to avoid a certain loss that they would never take to chase an equal gain. Kahneman and Tversky called this mirror-imaging the reflection effect, and it dissolves the old picture of a person who has one settled attitude toward risk. There is no such single attitude; there are two, and which one governs depends on whether the chooser stands in the territory of gains or of losses — cautious above the reference point, bold below it. The same person, the same stakes, the same odds: relabel the prospect so that its outcomes read as losses rather than gains, and risk-aversion turns into its opposite. This is why the mere framing of a decision — a public-health program described as saving four hundred of six hundred lives, or as letting two hundred die — can flip the public's choice between a safe option and a gamble, though the two descriptions name the identical fate. The choice tracks the frame because the frame sets the reference point, and the reference point decides which limb of the value function the outcomes fall on.3

3. The second asymmetry is the curve's shape. Each further dollar of gain adds less satisfaction than the last — diminishing returns, making people risk-averse among gains. But the loss limb bends the other way: each further dollar of loss stings less than the last, making people risk-SEEKING among losses. Faced with a sure loss or a gamble that might lose more but might lose nothing, they reach for the gamble. The reflection effect: no single attitude toward risk, but two — cautious above the reference point, bold below it. Relabel a prospect so its outcomes read as losses, and risk-aversion flips. Hence framing: a program described as saving four hundred of six hundred, or letting two hundred die, flips the choice though the fate is identical.

One piece remains, and it concerns not the value of outcomes but the weight given to their odds. Expected utility multiplies each outcome by its raw probability; prospect theory finds that people instead pass probabilities through a distorting lens, a weighting that is not linear. Small probabilities are OVERWEIGHTED — a one-in-a-thousand chance is treated as though it were rather more — which is why the same person will buy a lottery ticket and an insurance policy in the same afternoon, overpaying for a tiny chance at a jackpot and overpaying again to be rid of a tiny chance of ruin; both bets fasten on a rare event and magnify it. Probabilities in the broad middle range, by contrast, are underweighted and treated as more alike than they are, so that the difference between a forty- and a fifty-percent chance moves the chooser less than it should. And at the very top of the scale the weighting does something decisive: it gives a special, outsized weight to CERTAINTY, to the difference between a probability of ninety-nine in a hundred and a probability of one — the jump to a sure thing. Here, at last, is the engine of the certainty effect the previous plate could only name. Allais's chooser guarded the certain million because certainty itself carries a weight the merely-probable cannot; prospect theory does not merely catalogue that behavior but generates it, from the shape of the weighting curve. The anomaly that broke the old theory falls out as a prediction of the new one.4

4. The last piece concerns the weight given to odds. People pass probabilities through a distorting, nonlinear lens. Small probabilities are OVERWEIGHTED — why the same person buys a lottery ticket and an insurance policy in one afternoon, each bet fastening on a rare event and magnifying it. Middling probabilities are underweighted and treated as more alike than they are. And certainty gets outsized weight — the jump from ninety-nine in a hundred to a sure thing. Here is the engine of the certainty effect: Allais's chooser guarded the certain million because certainty carries a weight the merely-probable cannot. Prospect theory generates the anomaly that broke the old theory.

Assembled, the machinery is modest in its parts and vast in its reach: a value function defined over gains and losses from a reference point rather than over final wealth; steeper for losses than for gains, so that loss aversion pervades choice; concave above the reference point and convex below it, so that risk-aversion and risk-seeking trade places across the line; and a probability-weighting curve that swells the rare and the certain and flattens the middle. With these four pieces Kahneman and Tversky predicted as regularities — not as scattered errors — the whole catalogue of departures that had embarrassed the old theory: the Allais paradox, the reflection of risk attitudes, the power of a frame, the simultaneous appetite for lotteries and insurance.5 And they were scrupulous about what they were doing. Prospect theory is a DESCRIPTIVE theory, an account of how people choose, and it makes no claim to tell them how they should; the gap between the two, which Allais had pried open, prospect theory mapped without pretending to close. It became the founding edifice of behavioral economics, the work for which Kahneman would receive the Nobel award decades later, his collaborator Tversky by then beyond its reach. But prospect theory is a theory of valuing prospects, of choice among gambles laid out with their odds attached; it says less about the other and prior task, the one of forming the beliefs and estimates on which any choice must rest — of judging how likely, how typical, how large. That task has its own systematic departures, and its own pair of discoverers, who are the same pair. The next plate turns from how people value the odds to how they guess them.

5. Assembled, the machinery is modest in parts and vast in reach: a value function over gains and losses from a reference point, not final wealth; steeper for losses (loss aversion); concave above and convex below the reference point (risk attitudes trading places); and a probability-weighting curve that swells the rare and the certain and flattens the middle. With these four pieces K&T predicted as REGULARITIES the whole catalogue that embarrassed the old theory. And prospect theory is DESCRIPTIVE — how people choose, not how they should; the founding edifice of behavioral economics, Kahneman's later Nobel, Tversky by then beyond its reach.