This program calculates predicted preference patterns of special TAX model for tests of RBI.
The output window contains calculated preference patterns for each combination of parameters in CSV (comma separated values) format. These can be copied and pasted for further analysis. The data are already selected: use Control & C to Copy and Control & V to paste (e.g., into Excel). Click here for more instructions
For three-branch gambles with equally likely outcomes, (x, y, z), RBI requires that
S = (z, x, y) preferred to R = (z, x', y')
if and only if
S' = (x, y, z') preferred to R' = (x', y', z')
Additive difference models, Regret theory, and certain similarity theories also imply no violations of RBi, because in these theories, any component that is the same in both gambles is cancelled and has no effect on the choice. The editing rule of cancellation used in original prospect theory also implies no violations of RBI.
U(x, y, z) = wLu(x)+wMu(y)+wHu(z)where wL, wM, and wH are the configural weights of the lowestk middle, and highest consequences of the gamble, x, y, and z, which have utilities of u(x), u(y), and u(z), respectively.
Configurally Weighted Utility implies violations of RBI for an experiment with outcomes chosen such that z < x' < x < y < y' < z', with
S = (z, x, y),
R = (z, x', y'),
S' = (x, y, z'), and
R' = (x', y', z'),
as follows:
SR' pattern of violation: S preferred to R and R' preferred to S' occurs iff
wM/wH > [u(y')-u(y)]/[u(x)-u(x')] > wL/wM
RS' pattern of violation: R preferred to S and S' preferred to R' occurs iff
wM/wH < [u(y')-u(y)]/[u(x)-u(x')] < wL/wM
The special TAX model (Birnbaum & Stegner, 1979; Birnbaum, 2008) and CPT (Tversky & Kahneman, 1992) are both special cases of Configurally Weighted Utility in this study, but they make opposite predictions.
The special TAX model implies violations of RBI can only be of the the SR' pattern, for any parameters. Given the "prior" parameters of TAX, delta = -1, beta = 1, it implies that RBI will be violated in this Subdesign 1 of the experiment only for S=(2, 35, 40) preferred to R = (2, 5, 95) and R' = (5, 95, 98) preferred to S' = (35, 40, 98); i.e., in Row 5.
The CPT model with any inverse-S weighting function (in which wL > wM and wH > wM can imply only the RS' pattern of violation. For the prior values of Tversky & Wakker (1995), CPT implies RR' in Rows 1-4 and RS' violations in rows 5-8.
U(z, x, y) = [u(z) + u(x) + u(y)]/3 - (delta/6)*|u(y) - u(z)|,where u(y) and u(z) are the maximal and minimal utilities of consequences in the gamble. Note: In an earlier system of notation, the range term was added to the average (rather than subtracted), but in a revision of notation, the sign of delta has been reversed. Thus, whereas previously the range term was added and a typical value of delta was -1, now the term is subtracted and delta = 1. The weight of the range term was called omega. In the special TAX model, each weight transfer depends on the number of consequences in the gamble and is assumed to be t(p)*delta/(n + 1). In the case of three, equally likely branches, there are two transfers from the highest to lowest branches, so the total relative weight transferred is 2t(p)*delta/(n+1). Since the t(p) are all equal, when divided by the sum of the weights, they will have relative weight = 1/3. Therefore, the weight transferred works out to 2*(1/3)(1/4)*delta = delta/6. See Birnbaum (2008).
Therefore, the weights of lowest, middle, and highest branch utilities are as follows:
wL= 1/3 + delta/6
wM= 1/3
wL= 1/3 - delta/6
In the special TAX model, utility is approximated as a power function of money, as follows:
u(x) = xbeta, for x >= 0.
betaL is the lower limit of beta
betaH is the upper limit of beta
deltaL is the lower limit of delta
deltaH is the upper limit of delta
In this study, gamma plays no role because all values of p are equal. This parameter in the TAX model represents the psychophysical function for probability, t(p) = pgamma.
The display will calculate the certainty equivalents (CE) of the gambles and predicted response patterns for each test using the MIDPOINTS of the range of parameter values given. For the default ranges, these "prior parameters" are approximately, beta = 1 and gamma = 1. Note that for the default values (prior parameters) only one test (Row 5) is predicted to show a violation of RBI.