Simulation of Stochastic Parameters in Additive Difference Model (ADM)

This program simulates drifting parameters, according to a random walk in which both alpha and beta drift toward a randomly chosen set of parameters. The variable, 0 < w < 1, makes the random walk more gradual; that is, a person will tend to keep parameters close to their previous values from trial to trial when w is close to 1.

The window contains stochastic values of parameters alpha and gamma, and simulated preference patterns. 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

Simulates Drifting Parameters in ADM Model weight, w = No. Simulations
alpha Lower limit = alpha Upper limit = gamma Lower limit = gamma Upper limit =
Gamble X = x1 = x2 = x3 =
Gamble Y = y1 = y2 = y3 =
Gamble Z = z1 = z2 = z3 =

Instructions

This program simulates true preference patterns generated according to the Additive Difference model with power functions, as applied for DEPENDENT gambles by Birnbaum and Diecidue (2015). According to this model, one prefers X = (x1,x2,x3) over Y = (y1,y2,y3) iff
sign(x1alpha - y1alpha)(|x1alpha - y1alpha|)gamma + sign(x2alpha - y2alpha)(|x2alpha - y2alpha|)gamma + sign(x3alpha - y3alpha)(|x3alpha - y3alpha|)gamma > 0
Where sign(v) is the function that returns the sign of the argument (-1, 0, or 1, if v < 0, v = 0, v > 0, respectively).

Stochastic Model of Parameters

The parameters of the additive difference model (alpha and gamma) are allowed to change from session to session according to a stochastic model that is described as follows:
alpha(t+1) = (w)alpha(t) + (1 - w)(alphaH-alphaL)ran + alphaL)
gamma(t+1) = (w)gamma(t) + (1 - w)(gammaH-gammaL)ran + gammaL)
Where alpha(t) is the value of parameter alpha in block t
gamma(t) is the value of parameter gamma in block t
w is the weight of inertia; that is, larger values mean parameters tend to persist (w = 0 is a random parameter model; w = 1 is a fixed parameter model.
ran is a computer-generated random number, uniformly distributed between 0 and 1.
alphaL is the lower limit of alpha
alphaH is the upper limit of alpha
gammaL is the lower limit of gamma
gammaH is the upper limit of gamma

This stochastic process creates a random walk in two dimensions, in which parameters will tend to resemble those on previous trial but drift on each trial toward a randomly chosen set of new parameters within the range of acceptable values. When w = 0, parameters are chosen randomly on each trial, uniformly from the range; when w = 1, the parameters will stay fixed. JavaScript programmers can modify the program to explore other specifications for the stochastic behavior of the parameters.

This program implements a specific process to generate the kinds of Markov transition matrices that were postulated as stochastic models of changing true preference patterns in Birnbaum and Wan (2020).

Gambles and Preference Patterns

The three gambles listed as default values have been selected from a paper by Butler and Pogrebna (2018) that appeared to show the strongest evidence for systematic violations of transitivity of preference. This program uses the notation in their paper for preference patterns. For three gambles, X, Y, and Z, the three choice problems are XY, YZ, and ZX. The preference patterns are coded as follows (1 = choose X, 2 = choose Y; 3 = choose Z). The preference pattern 123 indicates choosing X in the XY choice, Y in the YZ choice, and Z in the ZX choice; therefore, this pattern is an intransitive pattern.

Note that according to the Most Probable Winner model, one would choose X over Y because the two branches in X with 15 beat the corresponding branches in Y, which are 10. One chooses Y over Z because the two branches with 10 beat the two branches in Z with 5; and one chooses Z over X because 27 beats 15 and 5 beats 3. Therefore, the Most probable winner model predicts the 123 preference pattern. However, the most commonly observed intransitive pattern in Butler and Pogrebna (2018) in this triple of gambles was the opposite, 231.

The preference pattern 231 is also intransitive, corresponding to preference for Y over X, Z over Y, and X over Z. Such a pattern can occur when alpha is small and gamma exceeds 1, producing "regret." The Additive difference model of Birnbaum and Diecidue (2015) can be used to represent both "regret" as in Loomes and Sugden (1982) and the opposite patterns that might be called "rejoicing", including Most Probable Winner. The model implemented here assumes that participants treat the gambles as dependent, which permitted a way to represent the most frequently observed preference patterns in the Butler and Pogrebna (2018) data (Birnbaum, 2020).

References