Prospect Theory

Prospect theory is a descriptive mathematical model of human decision-making under risk and uncertainty formulated by psychologists Daniel Kahneman and Amos Tversky in their landmark 1979 Econometrica paper (and refined into Cumulative Prospect Theory in 1992), for which Kahneman was awarded the 2002 Nobel Memorial Prize in Economic Sciences. Within business and economics theory, prospect theory replaced the normative monopoly of Von Neumann–Morgenstern Expected Utility Theory (EUT) as an empirical account of how individuals, investors, and managers actually evaluate risky prospects—such as financial portfolios in modern portfolio theory, insurance deductibles, medical treatments, and strategic gambles.

Whereas Expected Utility Theory assumes that rational agents evaluate lotteries over final total wealth states W + x_i using a globally concave marginal utility function u(W + x_i) weighted linearly by objective probabilities p_i, Kahneman and Tversky demonstrated that human subjects systematically violate the independence and invariance axioms of EUT (as in the Allais paradox and framing effects). Prospect theory accounts for these empirical regularities through four core structural principles: (1) reference dependence, where carriers of subjective value are changes in wealth Δx = x − r relative to a neutral reference point r rather than terminal wealth levels; (2) diminishing sensitivity, where the value function v(Δx) is concave for gains (Δx > 0) and convex for losses (Δx < 0); (3) loss aversion, where the value function is substantially steeper for losses than for equivalent gains; and (4) nonlinear probability weighting, where decision weights π(p) overweight small probabilities and underweight moderate-to-high probabilities.

Two-Phase Architecture: Editing (Framing) and Evaluation

In Kahneman and Tversky's original 1979 formulation, choosing among risky prospects proceeds through two sequential cognitive stages reflecting bounded rationality:

1. The Editing (Framing) Phase: Before numerical evaluation, the decision-maker performs mental heuristics to simplify the offered gambles. These operations include coding outcomes as gains or losses relative to a reference point r (typically the current status quo or an aspirational target), combination (summing probabilities associated with identical outcomes), segregation (separating the riskless component of a gamble from its risky component), cancellation (discarding shared outcome-probability components across competing options, which produces the isolation effect), and simplification (rounding probabilities or discarding negligible outcomes).
2. The Evaluation Phase: For a regular two-outcome prospect (x, p; y, q) where p + q ≤ 1 and either x ≥ 0 ≥ y or y ≥ 0 ≥ x, the agent computes the overall subjective prospect value V as:

where v(x) is the subjective value of change x (with v(0) = 0) and π(p) is the nonlinear decision weight associated with stated probability p (with π(0) = 0 and π(1) = 1, but generally π(p) + π(1 − p) < 1, a property termed subcertainty).

Functional Form of the Value Function and Cumulative Prospect Theory (1992)

Because the 1979 model could violate first-order stochastic dominance when applied directly to multi-outcome lotteries (n > 2), Tversky and Kahneman (1992) developed Cumulative Prospect Theory (CPT) by applying rank-dependent cumulative decision weights (inspired by Quiggin, 1982) separately to the cumulative probability distribution of gains and the decumulative distribution of losses. In CPT, the two-part power value function v(x) for outcome x relative to reference point r = 0 takes the parametric form:

Here α ∈ (0, 1) and β ∈ (0, 1) govern the psychophysical diminishing sensitivity (Weber-Fechner law) to larger magnitudes of gains and losses: the subjective difference between a gain of 100 USD and 200 USD feels much larger than the difference between 1,100 USD and 1,200 USD, producing an asymmetric S-shaped value curve that is concave above the origin (v′′(x) < 0 for x > 0, inducing risk aversion over moderate gains) and convex below the origin (v′′(x) > 0 for x < 0, inducing risk-seeking behavior over moderate losses). The coefficient λ > 1 is the loss aversion parameter, capturing the kink at the origin x = 0. Across experimental calibrations in Tversky and Kahneman (1992), the median parameter estimates are:

Thus, losing 100 USD inflicts roughly 2.25 times as much psychological disutility as the utility gained from winning 100 USD.

Simultaneously, CPT transforms objective probability p ∈ [0, 1] via an inverse-S-shaped probability weighting function w⁺(p) for gains and w⁻(p) for losses:

with median experimental curvature parameters γ = 0.61 (for gains) and δ = 0.69 (for losses), or under Drazen Prelec's (1998) axiomatically derived compound-invariance form, w(p) = exp( −(−ln p)^γ ). Because γ, δ < 1, the weighting function is steeply sloped near the endpoints p = 0 and p = 1—reflecting the possibility effect (overweighting small tail probabilities p < 0.15, such as w(0.01) ≈ 0.055) and the certainty effect (disproportionately valuing a jump from p = 0.99 to guaranteed certainty p = 1.00)—while remaining relatively flat across intermediate probabilities p ∈ (0.20, 0.80).

Explaining the Allais Paradox and the Fourfold Pattern

The interaction between the S-shaped value function v(x) and the inverse-S probability weighting function w(p) generates Tversky and Kahneman's Fourfold Pattern of Risk Attitudes (summarized in the table above), resolving a longstanding puzzle in classical economics: why the exact same person buys both lottery tickets (risk-seeking) and insurance policies (risk-averse). Under Expected Utility Theory with a globally concave utility function u(W), an agent is risk-averse everywhere unless utility contains arbitrary inflection points (Friedman and Savage, 1948). Under Prospect Theory, overweighting small probabilities (w(0.001) ≫ 0.001) outweighs the concavity of v(x) for small-probability gains (driving lottery demand) and outweighs the convexity of v(x) for small-probability losses (driving insurance demand), while for moderate-to-high probabilities (p ≥ 0.50), underweighting (w(p) < p) reinforces concavity in gains (risk aversion) and convexity in losses (risk seeking to avoid a sure loss).

This structure also resolves Maurice Allais's (1953) paradox and the reflection effect. When the same objective mortality or financial decision is framed positively in terms of lives saved (gains domain) vs. framed negatively in terms of lives lost (losses domain)—as in Tversky and Kahneman's (1981) classic "Asian Disease Problem"—72% of respondents chose the riskless certain option under the gain frame, whereas 78% switched to the risky all-or-nothing gamble under the loss frame, even though the terminal outcome distributions were identical.

Financial Market Anomalies: Equity Premium Puzzle and the Disposition Effect

Prospect theory serves as the microeconomic engine for behavioral finance alternatives to the efficient-market hypothesis and the capital asset pricing model:

1. Myopic Loss Aversion and the Equity Premium Puzzle: Benartzi and Thaler (1995) resolved Mehra and Prescott's (1985) equity premium puzzle—why historical U.S. real equity returns exceeded risk-free Treasury bills by over 6% annually, which under standard CRRA expected utility requires an implausible relative risk aversion coefficient > 30—by combining prospect theory's λ = 2.25 loss aversion with narrow mental accounting evaluation horizons of approximately 1 year.
2. The Disposition Effect: Shefrin and Statman (1985) and Odean (1998) documented that individual and institutional investors sell stocks that have appreciated above their purchase price (the reference point r = P₀) at a rate ~50% higher than they sell losing stocks (P_t < P₀), because investors sit in the risk-averse concave region for unrealized gains and the risk-seeking convex region for unrealized losses.
3. Labor Supply and Reference-Dependent Effort: Camerer et al. (1997) showed that New York City cabdrivers often set daily income targets r and quit earlier on high-wage rainy days once their daily reference point is reached, while working longer hours on slow days to avoid closing their mental account in the loss domain.

How does Cumulative Prospect Theory (1992) improve upon original Prospect Theory (1979)?

Original 1979 Prospect Theory applied decision weights π(p_i) directly to individual outcome probabilities, which meant that splitting one outcome into two slightly different outcomes could cause a stochastically dominated lottery to be chosen unless stripped out during the ad-hoc editing phase. Cumulative Prospect Theory (1992) transforms the entire cumulative probability distribution rather than individual probabilities, guaranteeing that first-order stochastic dominance is always satisfied and allowing the model to handle continuous probability distributions and arbitrary numbers of outcomes.

What determines the reference point in Prospect Theory?

In most laboratory experiments and retail financial decisions, the reference point r defaults to the current status quo wealth or original purchase price. However, in dynamic economic environments, Kőszegi and Rabin (2006) demonstrated that the reference point is endogenously determined by the agent's recent rational expectations (probabilistic beliefs held shortly before the outcome is realized), explaining why an anticipated wage cut or expected bonus shapes utility differently from an unanticipated shock.

Why do people overpay for low-deductible insurance and extended warranties under Prospect Theory?

Low-probability losses (p ≈ 0.01) fall in the lower-right cell of the Fourfold Pattern, where the inverse-S probability weighting function w⁻(p) overweights the small probability of product failure by a factor of 4 to 5, while loss aversion (λ ≈ 2.25) amplifies the perceived pain of paying an out-of-pocket deductible relative to the status quo. Together, these two forces cause consumers to pay actuarially unfair insurance premiums many times higher than the expected loss.

1. Kahneman, D., & Tversky, A. (1979). "Prospect theory: An analysis of decision under risk." Econometrica, 47(2), 263–291. https://doi.org/10.2307/1914185
2. Tversky, A., & Kahneman, D. (1981). "The framing of decisions and the psychology of choice." Science, 211(4481), 453–458. https://doi.org/10.1126/science.7455683
3. Tversky, A., & Kahneman, D. (1992). "Advances in prospect theory: Cumulative representation of uncertainty." Journal of Risk and Uncertainty, 5(4), 297–323. https://doi.org/10.1007/BF00122574
4. Benartzi, S., & Thaler, R. H. (1995). "Myopic loss aversion and the equity premium puzzle." Quarterly Journal of Economics, 110(1), 73–92. https://doi.org/10.2307/2118511
5. Kőszegi, B., & Rabin, M. (2006). "A model of reference-dependent preferences." Quarterly Journal of Economics, 121(4), 1133–1165. https://doi.org/10.1093/qje/121.4.1133