In their 1992 paper, Kahneman and Tversky developed an updated form of prospect theory, which they termed Cumulative Prospect Theory. The theory incorporates rank-dependent functionals which transform cumulative, rather than individual probabilities, in response to a growing literature, and satisfies stochastic dominance, which the original form of prospect theory does not. It retains the most crucial features, though, viz.:

1. Gains and losses, i.e. income is the carriers of value, not final assets or wealth.
2. The value of each outcome is multiplied by a decision weight, not an additive probability.

A risky prospect f is a function that maps states of the world s S into consequences x X. Outcomes of prospects are arranged in increasing order, so that a prospect f is represented by a sequence of pairs (x_i, A_i), and yields x_i if A_i occurs, where A_i is a partition of S.

A prospect is called strictly positive or strictly negative if all its outcomes are positive or negative respectively. It is labeled positive or negative if all its outcomes are non-negative or non-positive, respectively. All other prospects are called mixed. We have the positive part of f, f⁺(s) = f(s) if f(s) > 0, and f⁺(s) = 0 if f(s) 0. The negative part of f, f^-(s) = f(s) if f(s) < 0, and f^-(s) = 0 if f(s) 0.

A number V is assigned to each prospect, so that f g if & only if V(f) V(g).

A capacity W is a function that assigns to each A S a number W(A) satisfying W() = 0, W(S) = 1, & W(A) A(B) when ever A B.

There exists a strictly increasing value function v:X , satisfying v(x₀) = v(0) = 0, and capacities W⁺ & W^-, such that for f = (x_i, A_i), -m i n:

V(f) = V(f⁺) + V(f^-)

V(f⁺) = ⁿ_v=0⁺_i

V(f^-) = ⁿ_v=-m^-_i

If the prospect f is defined by a probability distribution p(A_i) = p_i, then the decision weights ⁺(f⁺) = (⁺₀...⁺_n) and ^-(f^-) = (^-_-m...^-₀) are defined by:

⁺_n = w⁺(p_n)
^-_-m = w^-(p_-m)
⁺_i = w⁺(p_i + ... + p_n) - w⁺(p_i+1 + ... + p_n), 0 i 1;
^-_i = w^-(p_-m + ... + p_i) - w^-(p_-m + ... + p_i-1), 1-m i 0.

where w⁺ & w^- are strictly increasing functions from the unit interval into itself, & they satisfy w⁺(0) = w^-(0) = 0, & w⁺(1) = w^-(1) = 1.

The decision weight ⁺_i is the difference between the capacities of the events "the outcome is at least as good as x_i," and "the outcome is strictly better than x_i."
The decision weight ^-_i is the difference between the capacities of the events "the outcome is at least as bad as x_i," and "the outcome is strictly worse than x_i."
In this manner, the decision weight associated with each outcome is the marginal contribution of the event defined in terms of the capacities. If each capacity W is additive, it is a probability measure & so _i is simply the probability of A_i.

For both positive and negative prospects, the decision weights add to 1, but the sum can be greater or less than 1 for mixed prospects. This is because the decision weights for gains & losses are defined by different capacities.

As in the original version of prospect theory, v is assumed concave above the reference point 0 -- the boundary between gains and losses -- and convex below it, so that v"(x) 0 for x 0, and v"(x) 0 for x 0.

Also, v is steeper for losses than gains, so that v'(x) < v'(-x) for x 0. This principle is called diminishing sensitivity -- the further the distance from the reference point, the smaller the impact of a change. So the impact of a change in probability diminshes with its distance from the boundary -- a change in probability from 0.9 to 1 has a greater impact than a change from 0.6 to 0.7. The weighting function is therefore concave near 0 and convex near 1.

Tversky, Amos & Kahneman, Daniel (1992). "Advances in Prospect Theory: Cumulative Representation of Uncertainty." Journal of Risk & Uncertainty. 5:297-323.