Advances in Prospect Theory: Cumulative Representation of Uncertainty

Abstract

Expected utility theory reigned for several decades as the dominant normative and descriptive model of decision making under uncertainty, but it has come under serious question in recent years. There is now general agreement that the theory does not provide an adequate description of individual choice: a substantial body of evidence shows that decision makers systematically violate its basic tenets. Many alternative models have been proposed in response to this empirical challenge (for reviews, see Camerer J Risk Uncertain 2:61–104, 1989; Fishburn Nonlinear preference and utility theory. The Johns Hopkins University Press, Baltimore, 1988; Machina Econ Perspect 1(1):121–154, 1987). Some time ago we presented a model of choice, called prospect theory, which explained the major violations of expected utility theory in choices between risky prospects with a small number of outcomes (Kahneman and Tversky Econometrica 47:263–291, 1979; Tversky and Kahneman J Bus 59(4):S251–S278, 1986). The key elements of this theory are (1) a value function that is concave for gains, convex for losses, and steeper for losses than for gains, and (2) a nonlinear transformation of the probability scale, which overweights small probabilities and underweights moderate and high probabilities. In an important later development, several authors (Quiggin J Econ Behav Organ 3, 323–343; Schmeidler Econometrica 57:571–587, 1989; Yaari Econometrica 55:95–115, 1987; Weymark Math Soc Sci 1:409–430, 1981) have advanced a new representation, called the rank-dependent or the cumulative functional, that transforms cumulative rather than individual probabilities. This article presents a new version of prospect theory that incorporates the cumulative functional and extends the theory to uncertain as well to risky prospects with any number of outcomes. The resulting model, called cumulative prospect theory, combines some of the attractive features of both developments (see also Luce and Fishburn J Risk Uncertain 4:29–59, 1991). It gives rise to different evaluations of gains and losses, which are not distinguished in the standard cumulative model, and it provides a unified treatment of both risk and uncertainty.

Amos Tversky was deceased at the time of publication.

Appendix: Axiomatic Analysis

Let $F=\left\{\,f:S\to X\right\}$ be the set of all prospects under study, and let F ⁺ and F ⁻ denote the positive and the negative prospects, respectively. Let ≲ be a binary preference relation on F, and let ≈ and $>$ denote its symmetric and asymmetric parts, respectively. We assume that ≲ is complete, transitive, and strictly monotonic, that is, if f ≠ g and f(s) ≥ g(s) for all s ∈ S, then f $>$ g.

For any f, g ∈ F and A ⊂ S, define h = fAg by: h(s) = f(s) if s ∈ A, and h(s) = g(s) if s ∈ S – A. Thus, fAg coincides with f on A and with g on S − A. A preference relation ≲ on F satisfies independence if for all f, g, f′, g′∈ F and A ⊂ S, fAg ≲ fAg′ iff f′ Ag ≲ f′ Ag′. This axiom, also called the sure thing principle (Savage 1954), is one of the basic qualitative properties underlying expected utility theory, and it is violated by Allais’s common consequence effect. Indeed, the attempt to accommodate Allais’s example has motivated the development of numerous models, including cumulative utility theory. The key concept in the axiomatic analysis of that theory is the relation of comonotonicity, due to Schmeidler (1989). A pair of prospects f, g ∈ F are comonotonic if there are no s, t ∈ S such that f(s) $>$  f(t) and g(t) $>$  g(s). Note that a constant prospect that yields the same outcome in every state is comonotonic with all prospects. Obviously, comonotonicity is symmetric but not transitive.

Cumulative utility theory does not satisfy independence in general, but it implies independence whenever the prospects fAg, fAg′, f′ Ag, and f′ Ag′ above are pairwise comonotonic. This property is called comonotonic independence.⁵ It also holds in cumulative prospect theory, and it plays an important role in the characterization of this theory, as will be shown below. Cumulative prospect theory satisfies an additional property, called double matching: for all f, g ∈ F, if f ⁺ ≈ g ⁺ and f ⁻ ≈ g ⁻, then f ≈ g.

To characterize the present theory, we assume the following structural conditions: S is finite and includes at least three states; X = Re; and the preference order is continuous in the product topology on Re^k, that is, {f ∈ F:f ≥ g} and {f ∈ F : g ≥ f} are closed for any g ∈ F. The latter assumptions can be replaced by restricted solvability and a comonotonic Archimedean axiom (Wakker 1991).

Theorem 24.1

Suppose (F ⁺, ≲) and (F ⁻, ≲) can each be represented by a cumulative functional. Then (F, ≲) satisfies cumulative prospect theory iff it satisfies double matching and comonotonic independence.

The proof of the theorem is given at the end of the appendix. It is based on a theorem of Wakker (1992) regarding the additive representation of lower-diagonal structures. Theorem 24.1 provides a generic procedure for characterizing cumulative prospect theory. Take any axiom system that is sufficient to establish an essentially unique cumulative (i.e., rank-dependent) representation. Apply it separately to the preferences between positive prospects and to the preferences between negative prospects, and construct the value function and the decision weights separately for F ⁺ and for F ⁻. Theorem 24.1 shows that comonotonic independence and double matching ensure that, under the proper rescaling, the sum V(f ⁺) + V(f ⁻) preserves the preference order between mixed prospects. In order to distinguish more sharply between the conditions that give rise to a one-part or a two-part representation, we need to focus on a particular axiomatization of the Choquet functional. We chose Wakker’s (1989a, b) because of its generality and compactness.

For x ∈ X, f ∈ F, and r ∈ S, let x{r}f be the prospect that yields x in state r and coincides with f in all other states. Following Wakker (1989a), we say that a preference relation satisfies tradeoff consistency ⁶ (TC) if for all x, x′, y, y′ ∈ X, f, f′, g, g′ ∈ F, and s, t ∈ S.

$$ x\left\{s\right\}f\lesssim y\left\{s\right\}g,{x}^{\prime}\left\{s\right\}f\gtrsim {y}^{\prime}\left\{s\right\}g\kern0.5em \mathrm{and}\kern0.5em x\left\{t\right\}{f}^{\prime}\gtrsim y\left\{t\right\}{g}^{\prime}\kern0.5em \mathrm{imply}\kern0.5em {x}^{\prime}\left\{t\right\}{f}^{\prime}\gtrsim {y}^{\prime}\left\{t\right\}{g}^{\prime }. $$

To appreciate the import of this condition, suppose its premises hold but the conclusion is reversed, that is, y′{t}g′ $>$ x′{t}f′. It is easy to verify that under expected utility theory, the first two inequalities, involving {s}, imply u(y) − u(y′) ≥ u(x) − u(x′), whereas the other two inequalities, involving {t}, imply the opposite conclusion. Tradeoff consistency, therefore, is needed to ensure that “utility intervals” can be consistently ordered. Essentially the same condition was used by Tversky et al. (1988) in the analysis of preference reversal, and by Tversky and Kahneman (1991) in the characterization of constant loss aversion.

A preference relation satisfies comonotonic tradeoff consistency (CTC) if TC holds whenever the prospects x {s}f, y{s}g, x′{s}f, and y′{s}g are pairwise comonotonic, as are the prospects x{f}f′, y{t}g′, x′{t}f′, and y′{t}g′ (Wakker 1989a). Finally, a preference relation satisfies sign-comonotonic tradeoff consistency (SCTC) if CTC holds whenever the consequences x, x′, y, y′ are either all nonnegative or all nonpositive. Clearly, TC is stronger than CTC, which is stronger than SCTC. Indeed, it is not difficult to show that (1) expected utility theory implies TC, (2) cumulative utility theory implies CTC but not TC, and (3) cumulative prospect theory implies SCTC but not CTC. The following theorem shows that, given our other assumptions, these properties are not only necessary but also sufficient to characterize the respective theories.

Theorem 24.2

Assume the structural conditions described above.

• (a) (Wakker 1989a) Expected utility theory holds iff ≲ satisfies TC.

• (b) (Wakker 1989b) Cumulative utility theory holds iff ≲ satisfies CTC.

• (c) Cumulative prospect theory holds iff ≲ satisfies double matching and SCTC.

A proof of part c of the theorem is given at the end of this section. It shows that, in the presence of our structural assumptions and double matching, the restriction of tradeoff consistency to sign-comonotonic prospects yields a representation with a reference-dependent value function and different decision weights for gains and for losses.

Proof of Theorem 24.1

The necessity of comonotonic independence and double matching is straightforward. To establish sufficiency, recall that, by assumption, there exist functions π ⁺, π ⁻, v ⁺, v ⁻, such that ${V}^{+}={\sum {\pi}^{+}{\it v}^{+}}$ and ${{V}^{-}={\sum {\pi}^{-}{\it v}^{-}}}$ preserve ≲ on F ⁺ and on F ⁻, respectively. Furthermore, by the structural assumptions, π ⁺ and π ⁻ are unique, whereas v ⁺ and v ⁻ are continuous ratio scales. Hence, we can set v ⁺(1) = 1 and v ⁻(−l) = θ $<$ 0, independently of each other.

Let Q be the set of prospects such that for any q ∈ Q, q(s) ≠ q(t) for any distinct s, t ∈ S. Let F _g denote the set of all prospects in F that are comonotonic with G. By comonotonic independence and our structural conditions, it follows readily from a theorem of Wakker (1992) on additive representations for lower-triangular subsets of Re^k that, given any q ∈ Q, there exist intervals scales {U _qi }, with a common unit, such that ${U}_q={{\sum}_i{U}_{qi}}$ preserves ≲ on F _q . With no loss of generality we can set U _qi (0) = 0 for all i and U _q (1) = 1. Since V ⁺ and V ⁻ above are additive representations of ≲ on ${F}_q^{+}$ and ${F}_q^{-}$, respectively, it follows by uniqueness that there exist aq, bq $>$ 0 such that for all i, U _qi equals ${a}_q{\pi}_i^{+}{\it v}^{+}$ on Re⁺, and U _qi equals ${b}_q{\pi}_i^{-}{\it v}^{-}$ on Re⁻.

So far the representations were required to preserve the order only within each F _q . Thus, we can choose scales so that b _q  = 1 for all q. To relate the different representations, select a prospect h ≠ q. Since V ⁺ should preserve the order on F ⁺, and U _q should preserve the order within each F _q , we can multiply V ⁺ by a _h , and replace each a _q by a _q /a _h . In other words, we may set a _h  = 1. For any q ∈ Q, select f ∈ F _q , g ∈ F _h such that f ⁺ ≈ g⁺ $>$ 0, f ⁻ ≈ g ⁻ $>$ 0, and g ≈ 0. By double matching, then, f ≈ g ≈ 0. Thus, a _q V ⁺ (f ⁺) + V ⁻(f ⁻) = 0, since this form preserves the order on F _q . But V ⁺(f ⁺) = V ⁺(g ⁺) and V⁻(f ⁻) = V ⁻(g ⁻), so V ⁺(g ⁺) + V ⁻(g ⁻) = 0 implies V ⁺(f ⁺) + V ⁻ (f ⁻) = 0. Hence, a _q  = 1, and V(f) = V ⁺(f ⁺) + V ⁻(f ⁻) preserves the order within each F _q .

To show that V preserves the order on the entire set, consider any f, g ∈ F and suppose f ≲ g. By transitivity, c(f) ≲ c(g) where c(f) is the certainty equivalent of f. Because c(f) and c(g) are comonotonic, V(f) = V(c(f)) ≥ V(c(g)) = V(g). Analogously, f $>$ g implies V(f) $>$  V(g), which complete the proof of theorem 24.1.

Proof of Theorem 24.2 (part c)

To establish the necessity of SCTC, apply cumulative prospect theory to the hypotheses of SCTC to obtain the following inequalities:

$$ \begin{array}{ll}V\left(x\left\{s\right\}f\right)\hfill & ={\pi}_s{\it v}(x)+{\displaystyle \sum_{r\in S-s}{\pi}_r{\it v}\left(\,f(r)\right)}\hfill \\ {}\hfill & \le {\pi}_s^{\prime }{\it v}(y)+{\displaystyle \sum_{r\in S-s}{\pi}_r^{\prime }{\it v}\left(g(r)\right)=V\left(y\left\{s\right\}g\right)}\hfill \\ {}V\left({x}^{\prime}\left\{s\right\}f\right)\hfill & ={\pi}_s{\it v}\left({x}^{\prime}\right)+{\displaystyle \sum_{r\in S-s}{\pi}_r{\it v}\left(\,f(r)\right)}\hfill \\ {}\hfill & \ge {\pi}_s^{\prime }{\it v}\left({y}^{\prime}\right)+{\displaystyle \sum_{r\in S-s}{\pi}_r^{\prime }{\it v}\left(g(r)\right)=V\left({y}^{\prime}\left\{s\right\}g\right)}.\hfill \end{array}\vspace*{-3pt} $$

The decision weights above are derived, assuming SCTC, in accord with Eqs. (24.1) and (24.2). We use primes to distinguish the decision weights associated with g from those associated with f. However, all the above prospects belong to the same comonotonic set. Hence, two outcomes that have the same sign and are associated with the same state have the same decision weight. In particular, the weights associated with x{s}f and x′{s}f are identical, as are the weights associated with y{s}g and with y′{s}g. These assumptions are implicit in the present notation. It follows that

$$ {\pi}_s{\it v}(x)-{\pi}_s^{\prime }{\it v}(y)\le {\pi}_s{\it v}\left({x}^{\prime}\right)-{\pi}_s^{\prime }{\it v}\left({y}^{\prime}\right). $$

Because x, y, x′, y′ have the same sign, all the decision weights associated with state s are identical, that is, ${\pi}_s={\pi}_s^{\prime }$. Cancelling this common factor and rearranging terms yields v(y) − v(y′) ≥ v(x) − v(x′).

Suppose SCTC is not valid, that is, x{t}f ≲ y{t}g′ but x′{t}f′ $<$ y′{t}g′. Applying cumulative prospect theory, we obtain

$$ \begin{array}{ll}V\left(x\left\{t\right\}{f}^{\prime}\right)\hfill & ={\pi}_t{\it v}(x)+{\displaystyle \sum_{r\in S-t}{\pi}_r{\it v}\left(\,{f}^{\prime }(r)\right)}\hfill \\ {}\hfill & \ge {\pi}_t{\it v}(y)+{\displaystyle \sum_{r\in S-t}{\pi}_r{\it v}\left({g}^{\prime }(r)\right)}=V\left(y\left\{t\right\}{g}^{\prime}\right)\hfill \\ {}V\left({x}^{\prime}\left\{t\right\}{f}^{\prime}\right)\hfill & ={\pi}_t{\it v}\left({x}^{\prime}\right)+{\displaystyle \sum_{r\in S-t}{\pi}_r{\it v}\left(\,{f}^{\prime }(r)\right)}\hfill \\ {}\hfill & <{\pi}_t{\it v}\left({y}^{\prime}\right)+{\displaystyle \sum_{r\in S-t}{\pi}_t{\it v}\left({g}^{\prime }(r)\right)}=V\left({y}^{\prime}\left\{t\right\}{g}^{\prime}\right).\hfill \end{array} $$

Adding these inequalities yields v(x) – v(x′) $>$ v(y) – v(y′) contrary to the previous conclusion, which establishes the necessity of SCTC. The necessity of double matching is immediate.

To prove sufficiency, note that SCTC implies comonotonic independence. Letting x = y, x′ = y′, and f = g in TC yields x{t}f′ ≲ x{t}g′ implies x′{t}f′ ≲ x′{t}g′, provided all the above prospects are pairwise comonotonic. This condition readily entails comonotonic independence (see Wakker 1989b).

To complete the proof, note that SCTC coincides with CTC on (F ⁺, ≲) and on (F ⁻, ≲). By part b of this theorem, the cumulative functional holds, separately, in the nonnegative and in the nonpositive domains. Hence, by double matching and comonotonic independence, cumulative prospect theory follows from Theorem 24.1.