필사 모드: Losses Hurt Twice as Much as Gains — The Prospect Theory Original and the 45 Years Since

Introduction — The Economist Who Turned Down One Coin Flip

In 1963, Paul Samuelson offered a colleague a bet over lunch. Flip a coin: heads, Samuelson pays 200 dollars; tails, he collects 100. The expected value is plainly positive, and the colleague still declined. What he said instead was this: he would not take it once, but he would take it a hundred times. Samuelson left behind a short paper showing that this answer is not logically consistent. It contained no explanation of why a person would answer that way.

The explanation arrived sixteen years later. It is prospect theory, usually compressed into the line "losses hurt twice as much as gains." That line now appears like a physical constant in negotiation textbooks, pricing experiment write-ups, and product specs. This installment of Psychology, Straight from the Papers opens the source of that constant. What is actually in the original, where the famous number came from, and which parts survived 45 years — we take them one at a time.

What the 1979 Paper Actually Did

The 1979 Econometrica paper by Daniel Kahneman and Amos Tversky is provocative from the title onward: "Prospect Theory: An Analysis of Decision under Risk". The standard of the day was expected utility theory. People multiply the utility of each outcome by its probability, add them up, and choose the largest total. One of the core axioms of that theory is independence: if you scale down a probability shared by two options by the same factor, the preference ordering should not change.

The material Kahneman and Tversky brought to bear was not laboratory equipment but a questionnaire. They handed hypothetical choice items to students and faculty at the Hebrew University in Israel, the Stockholm School of Economics in Sweden, and the University of Michigan in the United States, and counted the answers. The currency unit was the Israeli pound, and the paper notes that the median monthly net income of an Israeli family at the time was about 3,000 Israeli pounds. In other words, the sums in the items were by no means trivial to the respondents.

Here are the results for the four key items.

The table deserves a second reading. Problem 4 simply scales both probabilities of Problem 3 down by a factor of four. Under the independence axiom the preference should hold, yet the majority view flipped from 20 percent to 65 percent. The certain 3,000 lost its appeal the moment it lost its certainty. This is the certainty effect.

The bottom two rows are more interesting still. Only the sign changed, and the answers reflect as if in a mirror. The people who chose the sure thing in the domain of gains (80 percent) choose the gamble in the domain of losses (92 percent). Rather than lose 3,000 for certain, they will run the risk of losing 4,000. This is the reflection effect.

Here is something to state honestly. This data comes from hypothetical choices, not from an experiment in which real money changed hands. The samples are university members, and each item has fewer than 100 respondents. The authors noted this limitation in the paper themselves. The standing of prospect theory came not from the scale of the data but from the fact that it exposed, with minimal tools, a pattern the incumbent theory could not explain in principle.

The Three Features of the Value Function

The conclusion of the paper is a value function with three lines.

First, reference dependence. Expected utility theory evaluates final states of wealth. Prospect theory evaluates changes relative to a reference point. Whether a salary of 80,000 dollars is a joy or a sorrow is settled not by the number itself but by what last year was and what the person at the next desk gets. This one line is the root of everything else.

Second, concave over gains and convex over losses. Sensitivity dulls as you move away from the reference point. The difference between zero and 100 feels large; the difference between 1,100 and 1,200 feels small, even though it is the same 100. This curvature produces risk aversion in gains and risk seeking in losses at the same time.

Third, a steeper slope on the loss side. The curve kinks at the origin, and the left side is steeper than the right. That is loss aversion.

What deserves attention is that the 1979 paper stated this third feature only qualitatively. There are no numbers attached to the graph. The coefficient we know as "about two" arrives thirteen years later. The 1992 cumulative prospect theory paper by Tversky and Kahneman fit a model to the choices of 25 graduate students and obtained a median loss aversion coefficient of 2.25 and a value function exponent of 0.88. That textbook number, in other words, is a median from 25 people.

Understanding It in Code — One Coefficient Turns Down a Bet

The fastest way to see what the coefficient 2.25 actually does is to compute it yourself. Take the situation Samuelson's colleague faced — a coin flip with positive expected value — and watch the verdict change as the coefficient changes.

# prospect theory value function, with the exponent Tversky & Kahneman
# fitted in 1992: v(x) = x**0.88 for gains, -lam * (-x)**0.88 for losses
ALPHA = 0.88

def value(x, lam):
    return x**ALPHA if x >= 0 else -lam * (-x) ** ALPHA

WIN, LOSE = 110, -100                     # a coin flip worth +5 in money
print(f"expected value (money): {0.5 * WIN + 0.5 * LOSE:+.2f}")

for lam in (1.0, 1.5, 2.0, 2.25, 3.0):
    eu = 0.5 * value(WIN, lam) + 0.5 * value(LOSE, lam)
    print(f"  lambda {lam:4.2f} -> expected utility {eu:+7.2f}"
          f"  {'accept' if eu > 0 else 'reject'}")

# how big must the upside be before lambda = 2.25 accepts a 100 loss?
print(f"break-even win at lambda 2.25: {(2.25 ** (1 / ALPHA)) * 100:.0f}")

# output:
# expected value (money): +5.00
#   lambda 1.00 -> expected utility   +2.52  accept
#   lambda 1.50 -> expected utility  -11.87  reject
#   lambda 2.00 -> expected utility  -26.25  reject
#   lambda 2.25 -> expected utility  -33.45  reject
#   lambda 3.00 -> expected utility  -55.03  reject
# break-even win at lambda 2.25: 251

Two things to read here. One: at a coefficient of 1.0 the bet is accepted even though the curve is still bent. What produces the refusal is not the curvature but the slope on the loss side. Two: the last line is the most useful in practice. To get a person with a coefficient of 2.25 to accept the risk of losing 100, the upside has to be about 251. A 5 percent edge does not come close; you need two and a half times. It is a computable answer to why so many reasonable proposals get turned down inside organizations.

Is Loss Aversion Really Twofold — the Record of Reexamination

As the rules of this series require, we look at the contrary evidence too. This is the most important passage in this installment.

Start with what replicates. The large-scale replication by Ruggeri and colleagues, published in Nature Human Behaviour in 2020, put the item structure of the 1979 paper to 4,098 people in 19 countries. The core choice patterns — the certainty effect, the reflection effect — replicated robustly across countries and cultures. Unlike ego depletion, which this series has covered, the skeleton of prospect theory passed through the replication crisis.

What wobbles is not the skeleton but the numbers and the interpretation.

First, the coefficient is not a constant. Studies estimating loss aversion coefficients already number in the hundreds, and recent meta-analyses that pool them report an average around two while noting at the same time that the variation across studies is enormous. As stakes get smaller the coefficient approaches one, and it shrinks in familiar domains and with accumulated experience. "The human loss aversion coefficient is 2.25" is not the same kind of sentence as "human body temperature is 36.5 degrees."

Second, Gal and Rucker raised a more fundamental problem in a 2018 paper. The phenomena cited as flagship evidence for loss aversion — the endowment effect, status quo bias — can be explained without loss aversion at all. People may hold on to what they have not because losing hurts but simply because of the inertia of not moving, or because of question-order effects about which side comes to mind first. They also catalogue a fair number of studies in which equally sized gains and losses produced symmetric reactions.

Third, there is a case that actually collapsed. The 1990 mug experiment by Kahneman, Knetsch, and Thaler, in which the selling price ran a little over twice the buying price, was the emblem of the endowment effect. Then in 2005 Plott and Zeiler reported that when you give enough practice trials, guarantee anonymity, and explain the procedure in fine detail, most of that gap disappears. A substantial part of what had been observed may have been an artifact of experimental procedure rather than preference.

The honest summary is this. The structure — reference point, curvature, asymmetry — survived. The habit of nailing the size of the asymmetry to a single number did not.

The Framing Effect and the Probability Weighting Function

The other half of prospect theory deals not with outcomes but with probabilities.

Framing first. The Asian disease problem, which Tversky and Kahneman published in Science in 1981, wrote the same situation two ways. A disease is expected to kill 600 people, and there are two programs available. In the condition written in terms of lives saved (152 respondents), 72 percent chose "200 people will be saved for certain." In the condition written in terms of lives lost (155 respondents), 78 percent chose "with probability one third nobody dies, and with probability two thirds 600 people die." The outcomes of the two conditions are arithmetically identical. Only the reference point moved, from survivors to fatalities, and the majority view went the other way.

The probability weighting function is quieter but wider in application. People do not use probabilities as they are; they convert them into decision weights. The signature of this curve is overweighting of small probabilities and underweighting of moderate to large ones. One chance in a million feels far bigger than it is, and 95 percent feels smaller than it is.

That one line solves an old puzzle: the same person buys lottery tickets and insurance. Under expected utility theory this combination is awkward, because you have to claim simultaneously that the person likes risk and dislikes it. Under prospect theory it is a single line. A lottery is a large gain at small probability and insurance is a large loss at small probability, and both sit in the range where that small probability is overweighted. On the gain side the overweighting shows up as risk seeking; on the loss side, as risk aversion.

What to Discard and What to Keep

Discard 1. The habit of treating the coefficient as a constant. "Losses are twice gains" is a number that started as a median from 25 people and, passing through the textbooks, came to be treated like a constant of physics. What the value is in your own domain is something you ultimately have to measure yourself.

Discard 2. Attaching loss aversion automatically to every phenomenon. People fail to switch because of inertia, missing information, switching costs, or plain hassle. Take the point from Gal and Rucker seriously and loss aversion becomes a hypothesis to be handled as one candidate among several, not a default.

Discard 3. Transplanting hypothetical items straight into high-stakes real decisions. The original data was choices on paper.

Keep 1. Look at the reference point before the amount. In a negotiation the effective question is usually not "how much" but "compared with what." Salary negotiations, pricing policy, schedule renegotiation — all the same. Fixing the comparison object is frequently stronger than changing the number. It also meshes with the conclusion of the debate over salary and happiness, that the effect of absolute amounts of money on life runs on a log scale.

Keep 2. Write the same fact in two frames. This is worth something when used as a self-check tool rather than a persuasion technique. If your judgment changes between the version written in lives saved and the version written in lives lost, that judgment responded to a sentence, not to a fact.

Keep 3. Distrust your own instincts in the face of small probabilities. Lotteries, insurance, disaster preparedness, and security investment all live in this range. Here it is better to compute expected values by hand than to rely on intuition.

One last boundary. Framing and reference point manipulation are a persuasion technique and a manipulation technique at once. The same knowledge produces dark pattern copy along the lines of "you are about to miss out on this benefit." The dividing line is simple. If the other person would make the same choice after noticing the frame, it is persuasion; if noticing would make them angry, it is manipulation.

Reading Guide

• The original paper: Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2), 263-291.
• The follow-up with the coefficient: Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323.
• Framing: Tversky, A., & Kahneman, D. (1981). The framing of decisions and the psychology of choice. Science, 211(4481), 453-458.
• Large-scale replication: Ruggeri, K., et al. (2020). Replicating patterns of prospect theory for decision under risk. Nature Human Behaviour, 4(6), 622-633.
• Criticism: Gal, D., & Rucker, D. D. (2018). The loss of loss aversion: Will it loom larger than its gain? Journal of Consumer Psychology, 28(3), 497-516.
• Procedure dependence of the endowment effect: Plott, C. R., & Zeiler, K. (2005). The willingness to pay-willingness to accept gap, the "endowment effect," subject misconceptions, and experimental procedures for eliciting valuations. American Economic Review, 95(3), 530-545.

Reading tip: the 1979 paper looks equation-heavy, but the first half is survey items and response rates, so it simply reads. If you have only five minutes, look at the single graph of the value function. The whole paper is compressed into that S-shaped curve that kinks at the origin and runs steeper on the left. Next is the graph of the probability weighting function. The stretch at the far left where the curve rises above the diagonal is the place where lottery tickets and insurance are sold together. The next installment is cognitive dissonance, the business of rewriting beliefs to match your own behavior.