Calculates Preference Patterns in Tests of Restricted Branch Independence (RBI)

This program calculates predicted preference patterns in 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 and C to Copy and Control and V to paste (e.g., into Excel). Click here for more instructions

Calculates Response Patterns in Tests of RBI step size (/range) w =
beta Lower limit = beta Upper limit = In this design, parameter gamma plays no role (p1 = p2 = p3 = 1/3)
delta Lower limit = delta Upper limit =
Safe Gambles: xi, yi z = z' =
1. (x1, y1) = r11 = r12 = pat1 =
2. (x2, y2) = r21 = r22 = pat2 =
3. (x3, y3) = r31 = r32 = pat3 =
4. (x4, y4) = r41 = r42 = pat4 =
5. (x5, y5) = r51 = r52 = pat5 =
6. (x6, y6) = r61 = r62 = pat6 =
7. (x7, y7) = r71 = r72 = pat7 =
8. (x8, y8) = r81 = r82 = pat8 =
Risky (x', y') = r91 = r92 = totpat =

Instructions

This program calculates predictions for a series of tests of Restricted Branch Independence (RBI).

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')

Many Models Imply No Violations of RBI

Expected utility, Subjective Expected Utility, Subjectively weighted utility, Prospective Reference Theory, "Stripped" Original Prospect Theory, and others imply no violations of RBI.

Lexicographic semiorder models, additive difference models, Regret theory, and similarity theories also imply no violations of RBi, because in these theories, any component that is the same in both gambles has no effect on the choice. The editing rule of cancellation used in original prospect theory also implies no violations of RBI.

Configural Weight Models Allow Violations of RBI

For three-branch, equally likely outcomes, ranked x < y < z, the generic configural weight model can be written:
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.

CPT and TAX 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.

Parameters in the Grid Search

The parameter w is the increment factor as a proportion of the range of parameter values; that is, larger values mean fewer steps will be calculated; e.g., when this parameter is 0.05, there will be about 20 values between the lower and upper value.


betaL is the lower limit of beta
betaH is the upper limit of beta
gammaL is the lower limit of gamma
gammaH is the upper limit of gamma
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

References