Expected value is a clean ledger. Prospect theory is a dirty one.

That is not an insult. It is the point.

People do not usually meet risk as a probability distribution floating in a vacuum. They meet it as a change from a reference point:

I might win this
I might lose that
I can make this uncertainty go away

The sign matters. The tail matters. Certainty matters. A 1% chance is not just 0.01 when it is the only path to a jackpot or the only path to ruin.

Kahneman and Tversky’s 1979 prospect theory paper made this discomfort with expected utility precise enough to fight about.1 The later cumulative version by Tversky and Kahneman repaired important theoretical problems by weighting cumulative ranks rather than isolated probabilities, while preserving the descriptive ingredients that made the theory bite: reference dependence, loss aversion, diminishing sensitivity, and nonlinear probability weighting.2

This post is a small lab for those ingredients. The question is not:

what should a perfectly rational investor do?

The question is:

how can the same dollar lottery acquire two different prices?

A Value Function With a Scar

The canonical prospect-theory value function is drawn around a reference point, usually zero:

v(x) = x^alpha                    for gains
v(x) = -lambda * (-x)^alpha       for losses

When alpha is below 1, the function bends. The second hundred dollars matters less than the first hundred. A second hundred-dollar loss also hurts less than the first, but the loss side is multiplied by lambda. In the 1992 cumulative prospect theory estimates, the common benchmark is alpha = beta = 0.88 and lambda = 2.25.2

That number is often summarized as “losses loom larger than gains.” Fine, but the slogan hides the mechanical consequence:

a positive-EV mixed gamble can have a negative certainty equivalent

A 50/50 gamble to win $100 or lose $60 has expected value +$20. Under the default lab parameters, its prospect-theory certainty equivalent is about -$7.53. The model does not say the gamble is morally bad. It says that, for this descriptive preference functional, the pain side of the ledger is heavy enough to cross zero.

Probability Is Not Used Raw

The second scar is on probability. Cumulative prospect theory uses decision weights derived from transformed cumulative probabilities. The Tversky-Kahneman weighting function used in the lab is:

w(p) = p^gamma / (p^gamma + (1 - p)^gamma)^(1 / gamma)

with separate curvature parameters for gains and losses. The 1992 estimates used gamma = 0.61 for gains and delta = 0.69 for losses.2

For tiny probabilities, this curve can sit far above the diagonal. In the lab’s default lottery case, a 1% gain tail receives decision weight 0.0553, about 5.53x its objective probability. The same 1% loss tail receives decision weight 0.0397, about 3.97x its probability.

This is why the old observation is so stable: small probabilities can make both lottery tickets and insurance policies attractive.1 The two products look opposite in cash-flow terms, but they rhyme psychologically. Each sells a transformation of a tail.

The Lab

The lab below computes three ledgers for a small prospect:

  • expected value in dollars;
  • cumulative prospect-theory value;
  • the certainty equivalent, obtained by inverting the prospect-theory value function back into dollars.

The model is deterministic. The audit uses a neutral parameter setting alpha = 1, lambda = 1, gamma = 1, delta = 1 to verify that the certainty equivalent collapses back to expected value. It also checks the default lottery, insurance, and mixed-gamble cases.

Three Things to Try

First, leave the lab on Lottery ticket. The default ticket costs $2 and has a 1% chance of paying a $200 jackpot, so its expected value is exactly zero:

0.01 * 198 - 0.99 * 2 = 0

The prospect-theory certainty equivalent is about $2.06. That is not a claim that buying fair lottery tickets is a wealth-maximizing strategy. It is a claim about a descriptive ledger where the salient gain tail receives more decision weight than its raw probability.

Second, switch to Insurance quote. The default case is a 1% chance of a $200 loss. Expected loss is $2.00, but the prospect-theory certainty equivalent of carrying the risk is about -$5.11. In this descriptive model, a $2 quote is cheap relative to the pain-weighted risk, and the maximum premium that would make the person indifferent is about $5.11.

Third, switch to Mixed gamble, set the probability to 50%, gain to 100, and loss to 60. Expected value is $20.00, but the default prospect-theory certainty equivalent is about -$7.53. Now reduce Loss aversion from 225% to 100%. The certainty equivalent turns positive.

That last slider is the cleanest way to see why “risk aversion” is not one thing. Probability curvature, value curvature, and loss aversion can point in different directions.

The Fourfold Pattern Is a Local Weather Report

Prospect theory is often taught through the fourfold pattern:

high-probability gains:        risk averse
high-probability losses:       risk seeking
low-probability gains:         risk seeking
low-probability losses:        risk averse

This pattern is not a universal law of every human at every stake. It is a compact way to remember how the value curve and probability-weighting curve interact. In the 1979 paper, the certainty effect helped explain why people often prefer a sure gain to a larger probable gain, while the reflection effect showed a mirror-image appetite for risk over losses.1

The cumulative version adds a rank-dependent correction: for multi-outcome prospects, the model transforms cumulative probabilities, not each isolated probability separately. This matters because independently weighting each outcome can violate stochastic dominance. The rank-dependent construction is part of why cumulative prospect theory became the standard version to use in formal modeling.2

The lab keeps the examples small enough to inspect, but it uses that cumulative decision-weight logic: gains are ranked from best down, losses from worst up, and the decision weight is the difference between two transformed cumulative ranks.

This Is Not a Permission Slip

There is a common bad use of prospect theory:

people overweight small probabilities,
therefore the product is fine

No. A descriptive model is not a welfare theorem.

The fact that a tail is psychologically heavy does not mean the person is making a mistake, and it also does not mean the seller is innocent. Insurance can be prudent risk transfer. Lotteries can be entertainment, exploitation, or a tiny purchased fantasy, depending on the institution around them. A positive prospect-theory value is a statement about a model of choice, not a statement about social value.

This distinction matters in finance too. If an investor dislikes losses relative to a reference point, then path shape, drawdown framing, and statement frequency can change behavior even when terminal wealth distributions look similar. A model that prices only terminal expected wealth will miss that interface. A model that prices only feelings will miss the budget constraint.

The right posture is not to replace expected value with prospect theory. It is to keep both ledgers visible.

What the Audit Checks

The JavaScript audit runs 13,518 deterministic checks across 1,152 parameterized cases:

  • with neutral parameters, certainty equivalent equals expected value;
  • the default lottery has zero expected value and positive prospect certainty equivalent;
  • the 1% gain tail is overweighted;
  • the default insurance case has expected loss $2.00, but a larger maximum premium;
  • the positive-EV mixed gamble is rejected under default loss aversion;
  • lowering loss aversion can make that same mixed gamble acceptable;
  • malformed inputs are clamped to valid parameter ranges;
  • the certainty equivalent really inverts the prospect-theory value function.

Default facts from the shipped lab:

{
  "scenario": "lottery",
  "expectedValue": 0,
  "certaintyEquivalent": 2.06,
  "gainTailWeight": 0.0553,
  "gainTailMultiplier": 5.53,
  "lossTailWeight": 0.0397,
  "lossTailMultiplier": 3.97
}

The one-percent chance has a price. The tricky part is noticing which ledger is quoting it.

  1. Daniel Kahneman and Amos Tversky, “Prospect Theory: An Analysis of Decision under Risk”, Econometrica, 1979. ↩ ↩2 ↩3

  2. Amos Tversky and Daniel Kahneman, “Advances in Prospect Theory: Cumulative Representation of Uncertainty”, Journal of Risk and Uncertainty, 1992. Bibliographic listing: SCI Utah library page. ↩ ↩2 ↩3 ↩4