<P> CPT incorporates these observations in a modification of Expected Utility Theory by replacing final wealth with payoffs relative to the reference point, replacing the utility function with a value function that depends on relative payoff, and replacing cumulative probabilities with weighted cumulative probabilities . In the general case, this leads to the following formula for subjective utility of a risky outcome described by probability measure p (\ displaystyle p): </P> <P> U (p): = ∫ − ∞ 0 v (x) d d x (w (F (x))) d x + ∫ 0 + ∞ v (x) d d x (− w (1 − F (x))) d x, (\ displaystyle U (p): = \ int _ (- \ infty) ^ (0) v (x) (\ frac (d) (dx)) (w (F (x))) \, dx+ \ int _ (0) ^ (+ \ infty) v (x) (\ frac (d) (dx)) (- w (1 - F (x))) \, dx,) </P> <P> where v (\ displaystyle v) is the value function (typical form shown in Figure 1), w (\ displaystyle w) is the weighting function (as sketched in Figure 2) and F (x): = ∫ − ∞ x d p (\ displaystyle F (x): = \ int _ (- \ infty) ^ (x) \, dp), i.e. the integral of the probability measure over all values up to x (\ displaystyle x), is the cumulative probability . This generalizes the original formulation by Tversky and Kahneman from finitely many distinct outcomes to infinite (i.e., continuous) outcomes . </P> <P> The main modification to Prospect Theory is that, as in rank - dependent expected utility theory, cumulative probabilities are transformed, rather than the probabilities themselves . This leads to the aforementioned overweighting of extreme events which occur with small probability, rather than to an overweighting of all small probability events . The modification helps to avoid a violation of first order stochastic dominance and makes the generalization to arbitrary outcome distributions easier . CPT is therefore on theoretical grounds an improvement over Prospect Theory . </P>

Advances in prospect theory cumulative representation of uncertainty pdf