Simulation of Stochastic Parameters in Additive Difference Model (ADM)

This program simulates drifting parameters, according to a random walk in which two parameters, alpha and gamma, each 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 session to session when w is close to 1. The window contains stochastic values of parameters alpha and gamma, and simulated preference patterns, using the notation system in Birnbaum and Wan (2020). 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 (ADM) with power functions, as applied for choices between DEPENDENT gambles by Birnbaum and Diecidue (2015). According to this ADM model,
X = (x1,x2,x3) is preferred over Y = (y1,y2,y3)
iff
σ(x1 - y1)(|x1α - y1α|)γ + σ(x2 - y2)(|x2α - y2α|)γ + σ(x3 - y3)(|x3α - y3α|)γ > 0
Where σ(v) is the augmented sign function that returns -1, 0, or 1, if v < 0, v = 0, or v > 0, respectively.

Stochastic Model of Parameters

The parameters of the additive difference model (α and γ) are allowed to change from session to session according to a stochastic model that is described as follows:
α(t+1) = (w)α(t) + (1 - w)[(αH - αL)ran + αL]
γ(t+1) = (w)γ(t) + (1 - w)[(γH - γL)ran + γL]
Where:
α(t) is the value of parameter α in session t
γ(t) is the value of parameter γ in session t
w is the index of consistency; that is, larger values mean parameters tend to persist (w = 1 is a fixed parameter model; w = 0 is a random parameter model)
ran is a computer-generated random number, uniformly distributed between 0 and 1.
αL is the lower limit of α
αH is the upper limit of α
γL is the lower limit of γ
γH is the upper limit of γ

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 easily modify this 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 directly 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 does NOT use the notation in their paper for preference patterns. Instead, it uses the system of Birnbaum and Wan (2020). For three gambles, X, Y, and Z, the three choice problems are XY, YZ, and ZX. The preference patterns are coded as follows (In XY choice: 1 = choose X, 2 = choose Y; in YZ choice, 1 = choose Y, 2 = choose Z; in ZX choice: 1 = choose Z, 2 = choose X). The preference pattern 111 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.

The preference pattern 222 is also intransitive, corresponding to preference for Y over X, Z over Y, and X over Z. Such a pattern can occur when α is small and γ 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" or "advantage-seeking", and Most Probable Winner is a limiting case as the exponent γ goes to zero.

References