Cumulative Prospect Theory

Cumulative Prospect Theory

Cumulative prospect theory (CPT) (Tversky & Kahneman, 1992) is the repaired version of prospect theory that works for lotteries with many outcomes. The fix is a single change: apply the probability weighting function $w(\cdot )$ to cumulative (tail) probabilities rather than to each outcome’s individual probability. CPT also pins down calibrated functional forms for $v(\cdot )$ and $w(\cdot )$, which turned prospect theory from a qualitative framework into a quantitative model that could be estimated and used in asset-pricing applications.

Setup. Order the outcomes of a lottery from worst to best: $x_{-m}<\cdots <x_0=0<\cdots <x_n$, with probabilities $p_{-m},\dots ,p_n$ summing to $1$. The outcome $x_0=0$ is the reference point (usually current wealth or status quo). The lottery’s value is
$V={\sum }_{i=-m}^n{\pi }_{i}v(x_i),$
where the decision weight ${\pi }_i$ on outcome $i$ is
${\pi }_i=\{\begin{array}{ll}w\left(p_i+p_{i+1}+\cdots +p_n\right)-w\left(p_{i+1}+\cdots +p_n\right)& \text{if }0\le i\le n\text{(gain side)}\\ w\left(p_{-m}+\cdots +p_i\right)-w\left(p_{-m}+\cdots +p_{i-1}\right)& \text{if }-m\le i<0\text{(loss side)}\end{array}.$
On the gain side, ${\pi }_i$ is the amount by which the weighted upper-tail probability (of doing at least as well as $x_i$) drops when $x_i$ itself is excluded. On the loss side, ${\pi }_i$ is the analogous jump in the weighted lower-tail probability.

Calibrated functional forms.
$v(x)=\{\begin{array}{ll}{x}^{\alpha }& x\ge 0\\ -\lambda (-x{)}^{\alpha }& x<0\end{array},w(P)=\frac{P^{\delta }}{{\left(P^{\delta }+(1-P{)}^{\delta }\right)}^{1/\delta }},$
with Tversky & Kahneman’s (1992) estimates $\alpha =0.88$ (mild curvature of $v$), $\lambda =2.25$ (loss aversion: losses hurt roughly $2.25\times$ as much as equivalent gains), $\delta =0.65$ (probability distortion: small probabilities are overweighted and moderate probabilities are underweighted).

Why it matters. CPT is the version of prospect theory used in modern behavioral asset pricing. It delivers the original PT predictions (loss aversion, four-fold pattern of risk attitudes, preference for skewness) without the mathematical inconsistency the original had on many-outcome lotteries, and its calibrated parameters allow the model to be plugged directly into pricing problems like lottery-stock overpricing (Barberis & Huang, 2008) and insurance demand.

Intuition

The original prospect theory (Kahneman & Tversky, 1979) weights each outcome’s probability separately: if outcome $x_i$ has probability $p_i$, its decision weight is $w(p_i)$. This works fine for a coin-flip gamble — two outcomes, one probability each, clean arithmetic. It breaks once you try to value a lottery with many fine-grained outcomes. If a continuous distribution is approximated by $100$ outcomes with probability $0.01$ each, then $w(0.01)$ is large (small probabilities are overweighted), and applying that large weight to every single outcome produces total decision weights that sum to much more than one. The model overweights everything and the resulting valuation is nonsense. Tversky & Kahneman spent the $1980$s patching this.

The fix in CPT is to weight tail probabilities, not individual-outcome probabilities. For a gain $x_i$, the relevant question the decision-maker is asking is “how likely am I to do at least as well as $x_i$?” — the upper-tail probability. For a loss $x_i$, it is “how likely am I to do at least as badly as $x_i$?” — the lower-tail probability. CPT runs the weighting function $w(\cdot )$ over these tail probabilities and then takes differences to recover the decision weight on each individual outcome. Because $w(\cdot )$ overweights small probabilities, the extreme tails — the biggest gain and the worst loss — get disproportionately inflated weight, while the probability mass in the middle is compressed. The total decision weight still sums to one, so the model is internally consistent, but the extremes still dominate the valuation the way the original theory intended. A single patch restores both mathematical coherence and the psychological content.

Properties

(Tail Overweighting) The extreme gain and the extreme loss receive disproportionately large decision weight.

Because $w(\cdot )$ is concave near $0$ and convex near $1$, the jumps $w(p)-w(0)=w(p)$ at the extreme tails are large relative to the raw probability $p$. Translated into decision weights, this means the best possible outcome in the gain direction and the worst possible outcome in the loss direction both get inflated weights, while the outcomes clustered around the reference point get compressed. This is the formal mechanism behind three separate empirical patterns: the overpricing of lottery stocks (positively-skewed assets like IPOs, penny stocks, and out-of-the-money options — Barberis & Huang, 2008), the demand for insurance against large losses at actuarially unfair prices, and the favorite-longshot bias in horse-race betting (longshots are overbet, favorites underbet). All three come from the same single cause: probability weighting applied to the tails.

(Calibrated Parameters) $\alpha =0.88$, $\lambda =2.25$, $\delta =0.65$.

The three parameters each control a distinct piece of the model. $\alpha$ governs the curvature of the value function $v(\cdot )$: the estimated value $0.88$ means $v$ is close to linear but slightly concave for gains and slightly convex for losses, giving modest diminishing sensitivity without dominating the other effects. $\lambda$ governs loss aversion: the estimated value $2.25$ means a \1$lossproducesthesamepsychologicalimpactasa$$2.25$gain,whichisthesinglemostfamousnumberinbehavioraleconomics.$\delta$governstheshapeof$w(\cdot)$:valuesof$\delta < 1$producetheinverted-Sshape(smallprobabilitiesoverweighted,moderateonesunderweighted),andTversky-Kahnema{n}^{\prime }s$0.65$isthestandardcalibration.Subsequentestimatesfromdifferentdatasetstypicallyfallin$\alpha \in [0.8, 1.0]$,$\lambda \in [1.5, 2.5]$,$\delta \in [0.5, 0.8]$, so the original numbers have held up reasonably well.

(Reduction to Original PT) On two-outcome gambles, CPT and original PT give the same answer.

For a lottery with a single gain $x>0$ with probability $p$ and zero otherwise, CPT’s decision weight on the gain is $w(p)-w(0)=w(p)$, which is exactly what original PT assigns. The same holds for a single-loss gamble. CPT is a strict generalization of PT: it reproduces all of PT’s correct predictions on small lotteries while extending cleanly to many-outcome distributions. The two theories only disagree when there are three or more distinct nonzero outcomes, which is where the original theory’s arithmetic broke.

(Tail Dependence on Ordering) The decision weight on an outcome depends on its rank, not just its probability.

A subtle consequence of the cumulative formulation is that the weight ${\pi }_i$ on outcome $x_i$ depends on where $x_i$ sits in the rank-ordering of all outcomes, not just on its own probability. Moving $x_i$ up or down the ordering — even without changing its probability — changes which cumulative probability it sits in, and therefore changes its decision weight. This is a feature, not a bug: it is how the theory captures the intuition that a \1000$ payoff feels very different depending on whether it is the lottery’s best outcome or a mediocre one sandwiched between larger wins. The rank-dependence is what distinguishes CPT from expected utility, where the weight on an outcome depends only on its own probability and not on the other outcomes in the lottery.

(Reference Point Is Exogenous) CPT inherits PT’s unresolved question of where the reference point comes from.

CPT takes the reference point $x_0=0$ as given. In most applications it is assumed to be the current level of wealth, the purchase price of an asset (for the disposition effect), or the expected value of a gamble. The theory does not predict which of these the decision-maker will use in any given situation, and the choice matters enormously — the same dollar outcome can be a “gain” or a “loss” depending on which reference point is active. This is the single biggest limitation of the framework and is the subject of active research (Kőszegi & Rabin, 2006; Barberis, 2013).

Examples

Computing decision weights on a three-outcome lottery

Consider the lottery $L$: lose \100$withprobability$0.1$,gain$$0$withprobability$0.5$,gain$$200$withprobability$0.4$.Orderfromworsttobest:$x_{-1} = -100$with$p_{-1} = 0.1$,$x_0 = 0$with$p_0 = 0.5$,$x_1 = 200$with$p_1 = 0.4$.

Gain side. The decision weight on $x_1=200$ is the weight on “do at least as well as \200$,"whichisjust$w(0.4)$.With$\delta = 0.65$,$w(0.4) \approx 0.39$(usingtheTversky-Kahnemanform).Thedecisionweighton$x_0 = 0$is$w(p_0 + p_1) - w(p_1) = w(0.9) - w(0.4) \approx 0.71 - 0.39 = 0.32$.

Loss side. The decision weight on $x_{-1}=-100$ is just $w(0.1)\approx 0.17$ (using the loss-side $w$, which Tversky-Kahneman took to be the same function with a slightly different $\delta$).

Check. The total weight is $0.39+0.32+0.17=0.88\ne 1$. The weights do not sum to one in general — this is a feature of rank-dependent weighting, not a bug. What matters is that the decision weights produce coherent preferences, which they do.

Valuation. Using $v(200)=200^{0.88}\approx 107$, $v(0)=0$, and $v(-100)=-2.25\cdot 100^{0.88}\approx -130$, the CPT value is
$V=0.39\cdot 107+0.32\cdot 0+0.17\cdot (-130)\approx 41.7-22.1=\mathrm{19.6.}$
The lottery has positive CPT value despite its large downside — the combination of loss aversion pulling the loss term more negative and tail overweighting inflating the gain term is what determines the sign. Under expected value ($E[L]=-10+80=70$) the lottery is more attractive; under pure loss aversion it would be less so; CPT lands in between.

Why the original PT broke on many-outcome lotteries

Imagine a lottery that pays \i$withprobability$0.01$foreach$i \in {1, 2, \dots, 100}$.Theexpectedvalueis$50.50$.Underoriginalprospecttheory,eachoutcomegetsdecisionweight$w(0.01)$.WithTversky-Kahnema{n}^{\prime }s$\delta = 0.65$,$w(0.01) \approx 0.048$—smallprobabilitiesareoverweightedbyroughlyafactoroffive.Summingacrossall$100$outcomesgivesatotaldecisionweightof$100 \cdot 0.048 = 4.8$,whichisnearlyfivetimestoolarge.Theresultingvalueisapproximately$4.8 \cdot 50 \approx 240$,farabovetheactualexpectedvalueof$50.50$. Every lottery gets inflated by roughly the same factor, so every comparison between lotteries still works in relative terms, but the absolute numbers are nonsense and the theory cannot be plugged into an asset-pricing or insurance application.

Under CPT, the decision weight on outcome $i$ is $w(P_i)-w(P_{i-1})$ where $P_i$ is the upper-tail probability. Summing this telescoping series gives exactly $w(1)-w(0)=1-0=1$. The total decision weight is always one, regardless of how the distribution is discretized. The model is coherent. The tail overweighting that made original PT useful is preserved — the extreme outcome $x=100$ still gets inflated weight — but it no longer blows up in aggregate.

See Also

• 8.5.1. Prospect Theory — parent: the original Kahneman-Tversky (1979) theory that CPT repairs.
• 8.5. Prospect Theory — hub for the full prospect-theory family.
• 8.6.2. Overreaction — behavioral bias that interacts with CPT’s tail overweighting.
• 8.7. Bubbles — CPT’s lottery-stock overpricing is one of the canonical bubble ingredients.
• 8.3. Challenges to Rational Pricing — the anomalies CPT was designed to match.