|
Article Excerpt
1. Introduction
This study is based upon an important paper by Wu and Markle (2004). It not only replicates and extends their results, it also provides new tests of the model they proposed to account for their data. In addition, this paper provides a different theoretical interpretation of their findings, and it tests that alternative model. Wu and Markle (2004) reported systematic violations of a behavioral property known as gain-loss separability (GLS), which is implied by many descriptive decision models.
1.1. Gain-Loss Separability (GLS)
GLS is a behavioral property defined on choices between mixed gambles (gambles containing both gains and losses). Let G = ([y.sub.1], [p.sub.1]; [y.sub.2], [p.sub.2];...; [y.sub.n], [p.sub.n]; [x.sub.m], [q.sub.m];...; [x.sub.2], [q.sub.2]; [x.sub.1], [q.sub.1]) represent a mixed gamble in which the outcomes are ranked such that [y.sub.1] < [y.sub.2] < ... < [y.sub.n] < < [x.sub.m] < ... < [x.sub.2] < [x.sub.1] and [[summation].sub.i=1.sup.n] [p.sub.i] + [[summation].sub.j=1.sup.m] [q.sub.j] = 1. Now break G into positive and negative subgambles, as follows: [G.sup.+] = (0, [[summation].sub.i=1.sup.n] [p.sub.i]; [x.sub.m], [q.sub.m];...; [x.sub.2], [q.sub.2]; [x.sub.1], [q.sub.1]) and [G.sup.-] = ([y.sub.1], [p.sub.1]; [y.sub.2], [p.sub.2];...; [y.sub.n], [p.sub.n]; 0, [[summation].sub.j=1.sup.m] [q.sub.j]). Let F represent another mixed gamble that can also be broken into positive and negative sub-gambles.
GLS is the assumption that if a person prefers the positive subgamble [G.sup.+] to the positive subgamble [F.sup.+] and prefers the negative subgamble [G.sup.-] to the negative subgamble [F.sup.-], then that person should prefer G to F. That is, with [>] denoting preference, if [G.sup.+] [>] [F.sup.+] and [G.sup.-] [>] [F.sup.-], then G [>] F. As shown in the next section, GLS is implied by any theory in which the utility of G can be written as a strictly increasing function of the utility of [G.sup.+] and the utility of [G.sup.-]. The violations of GLS observed by Wu and Markle (2004) therefore refute this class of theories, which includes cumulative prospect theory (CPT).
1.2. Cumulative Prospect Theory (CPT)
Let G = ([y.sub.1], [p.sub.1]; [y.sub.2], [p.sub.2];...; [y.sub.n], [p.sub.n]; [x.sub.m], [q.sub.m];...; [x.sub.2], [q.sub.2]; [x.sub.1], [q.sub.1]) represent a mixed gamble with outcomes ranked such that [y.sub.1] < [y.sub.2] < ... < [y.sub.n] < [less than or equal to] [x.sub.m] < ... < [x.sub.2] < [x.sub.1]. Define cumulative probabilities of losses as [P.sub.i] = [[summation].sub.k=1.sup.i] [p.sub.k], and define decumulative probabilities of gains as [Q.sub.j] = [[summation].sub.k=1.sup.j] [q.sub.k]. CPT (Tversky and Kahneman 1992) can be written as follows:
CPU(G) = [n.summation over (i=1)][[W.sup.-]([P.sub.i]) - [W.sup.-]([P.sub.i-1])]u([y.sub.i]) + [m.summation over (j=1)][[W.sup.+]([Q.sub.j]) - [W.sup.+] ([Q.sub.j-1])]u([x.sub.j]), (1)
where [P.sub.i] and [P.sub.i-1] are the probabilities of a loss being equal to or worse (lower) than [y.sub.i] and strictly lower than [y.sub.i], respectively; [Q.sub.j] and [Q.sub.j-1] are the probabilities of winning a positive prize of [x.sub.j] or more, and strictly more than [x.sub.j], respectively ([P.sub.0] = [Q.sub.0] = 0). Utility is defined with respect to changes from the status quo (gains or loses), where u(0) = 0. CPU(G) is the utility ("subjective value") of the gamble; the representation assumes that G [>] F [left and right arrow] CPU(G) [>] CPU(F). The functions [W.sup.+](Q) and [W.sup.-](P) are strictly increasing probability-weighting functions, [W.sup.+](0) = [W.sup.-](0) = 0, and [W.sup.+](1) = [W.sup.-](1) = 1.
From Equation (1), it can be seen that the overall utility of a mixed gamble in CPT is just the sum of the utilities of its positive and negative subgambles. This additivity of the favorable (positive) and unfavorable (negative) parts of a gamble implies GLS. Equation (1) implies CPU(G) = CPU([G.sup.+]) + CPU([G.sup.-]) for all mixed gambles, G. CPT assumes that [G.sup.+] [>] [F.sup.+] [left and right arrow] CPU([G.sup.+]) > CPU([F.sup.+]) and [G.sup.-] [>] [F.sup.-] [left and right arrow] CPU([G.sup.-]) > CPU([F.sup.-]). If both conditions hold, then it follows that CPU([G.sup.+]) + CPU([G.sup.-]) > CPU([F.sup.+]) + CPU([F.sup.-]) [left and right arrow] G [>] F. The same implication follows for any such additive theory, including original prospect theory. GLS is also implied by rank- and sign-dependent utility theory (Luce and Fishburn 1991, 1995), which used the same additive representation later used in CPT (see Luce 2000, Chaps. 6 and 7 for contrasting approaches).
The weighting functions in CPT have been further specified as follows:
[W.sup.+](Q) = [Q.sup.[gamma]]/[[Q.sup.[gamma]] + (1 - Q)[.sup.[gamma]]][.sup.1/[gamma]] and
[W.sup.-](P) = [P.sup.[delta]]/[[P.sup.[delta]] + (1 - P)[.sup.[delta]]][.sup.1/[delta]], (2)
where the constants [gamma] and [delta] were estimated by Tver-sky and Kahneman (1992) to be 0.61 and 0.69, respectively; u(x) was approximated by u(x) = [x.sup.[beta]], where [beta] = 0.88, x > 0, and u(-x) = -[lambda]u(x); x [greater than or equal to] 0. The constant [lambda] is sometimes referred to as the "loss aversion" coefficient; [lambda] was estimated to be 2.25. This paper will evaluate both the general model (Equation (1)), which implies GLS, as well as the above parameterized version of Tversky and Kahneman (1992) in order to relate new findings to previous results and to show where the CPT model goes wrong.
1.3. Violations of GLS
Wu and Markle (2004) reported systematic violations of GLS, illustrated by the three choices in Table 1. A majority of respondents preferred F to G in Choice 3, replicating a pattern found by Levy and Levy (2002), which is consistent with CPT as fit by Tversky and Kahneman (1992) to previous data (Wakker 2003). To test GLS, Wu and Markle (2004) decomposed G and F into their gain and loss subgambles. These choices are shown with their respective choice percentages. Contrary to GLS, the majority (72%) preferred [G.sup.+] over [F.sup.+] in Choice 1; 60% preferred [G.sup.-] over [F.sup.-] in Choice 2; but 62% chose F over G in Choice 3 (Wu and Markle 2004).
Wu and Markle (2004) proposed a configural model of CPT to account for violations of GLS by allowing different weighting functions for the case of mixed gambles from those used for purely positive or purely negative gambles. Their model will be referred to as configural cumulative prospect theory (CCPT) because it retains the equations of CPT but uses different weighting functions for different configurations of consequences: positive, negative, and mixed:
CCPU([G.sup.+]) = [m.summation over (j=1)] [[W.sup.+]([Q.sub.j]) - [W.sup.+]([Q.sub.j-1])]u([x.sub.j]), (3a)
CCPU([G.sup.-]) = [n.summation over (i=1)] [[W.sup.-]([P.sub.i]) - [W.sup.-]([P.sub.i-1])]u([x.sub.i]), (3b)
CCPU(G) = [n.summation over (i=1)] [[W.sup.-+]([P.sub.i]) - [W.sup.-+]([P.sub.i-1])]u([x.sub.i]) + [m.summation over (j=1)] [[W.sup.+-]([Q.sub.j]) - [W.sup.+-]([Q.sub.j-1])]u([x.sub.j]), (3c)
where the terms are as defined in Equation (1), except that [W.sup.-] and [W.sup.+] are the weighting functions for gambles composed of purely nonpositive consequences and purely nonnegative consequences, respectively, and the functions [W.sup.-+] and [W.sup.+-] are the weighting functions for the negative and positive components of mixed gambles, respectively. This approach recalls a suggestion by Edwards (1962), who proposed that different weighting functions may be required for gambles with strictly positive, strictly nonnegative (including zero), strictly negative, strictly nonpositive (including zero), and mixed consequences. Although CCPT violates GLS, it must satisfy the property of coalescing (described next) that is implied by these equations.
1.4. Coalescing
Coalescing is the assumption that if there are two probability-consequence branches in a gamble leading to the same consequence, they can be combined by adding their probabilities. For example, consider gamble A = ($100, 0.25; $100, 0.25; $0, 0.5). Gamble A is a three-branch gamble in which one branch has a probability of 0.5 to win $0, and two other branches of probability 0.25 to win $100.
According to the property of coalescing, this three-branch gamble, A, is equivalent to the two-branch gamble, A' = ($100, 0.5; $0, 0.5). Gamble A' is called the coalesced form of the gamble, and A is one of many possible split forms. Assuming transitivity, coalescing implies that people should make the same choice between A' and B as they do between A and B, apart from random error.
In original prospect theory (Kahneman and Tversky 1979), coalescing was assumed as an editing rule (combination), but is not implied by the equations. Therefore, depending on whether we accept the editing rule or equations, original prospect theory either satisfies or violates coalescing.
In CPT or CCPT, however, coalescing follows from the rank-dependent representation (Birnbaum and Navarrete 1998, pp. 57-58; Luce 2000). A number of studies, however, reported systematic evidence against coalescing (Starmer and Sugden 1993; Humphrey 1995; Birnbaum 2004a, b). Such violations are consistent with the older class of configural weight models (Birnbaum and Stegner 1979) that preceded the class that have come to be known as rank dependent, including CPT.
1.5. Transfer of Attention Exchange (TAX) Model
The TAX model is a type of configurally weighted averaging model in which the weights of branches are affected by the ranks of the consequences on those branches. Such models were developed to account for integrated psychophysical and evaluative judgments in psychology (Birnbaum 1974, Birnbaum and Stegner 1979). Although they have some similarity to models that were later introduced as rank-dependent utility models (Quiggin 1982, 1993), including CPT, they differ in important respects. These models violate coalescing and attribute the Allais paradoxes to violations of coalescing (Birnbaum 1999a, 2004a) rather than to violations of "independence," as is done in CPT.
The basic ideas of the TAX model are as follows: A risky gamble is represented as a tree with probability-consequence branches. Aside from configural effects, the weight of each branch is a function of the branch's probability. However, weight is transferred from branch to branch according to the ranks of the consequences on those branches. Intuitively, these transfers of weight represent shifts of attention among the branches. In a risk-averse person, branches leading to lower consequences end up with more weight. Marley and Luce (2005) have axiomatized a general form of TAX and have shown that it is an idempotent, rank-weighted utility model. In the so-called "special" TAX model (Birnbaum and Chavez 1997), a simpler form of TAX, all transfers of weight between any two branches represent the same proportion of the probability weight of the branch giving up its weight.
Consider gambles of...
|
