Risky Decision Making: Testing for Violations of Transitivity Predicted by an Editing Mechanism

Abstract

Transitivity is the assumption that if a person prefers A to B and B to C, then that person should prefer A to C. This article explores a paradigm in which Birnbaum, Patton and Lott (1999) thought people might be systematically intransitive. Many undergraduates choose C = ($96, .85; $90, .05; $12, .10) over A = ($96, .9; $14, .05; $12, .05), violating dominance. Perhaps people would detect dominance in simpler choices, such as A versus B = ($96, .9; $12, .10) and B versus C, and yet continue to violate it in the choice between A and C, which would violate transitivity. In this study we apply a true and error model to test intransitive preferences predicted by a partially effective editing mechanism. The results replicated previous findings quite well; however, the true and error model indicated that very few, if any, participants exhibited true intransitive preferences. In addition, violations of stochastic dominance showed a strong and systematic decrease in prevalence over time and violated response independence, thus violating key assumptions of standard random preference models for analysis of transitivity.

Reanalysis of Butler and Pogrebna (2018) using true and error model

“…Other recent studies using TE models to evaluate violations of transitivity reported significant but small incidences (Birnbaum & Gutierrez, 2007;Birnbaum & Bahra, 2012;Birnbaum & Diecidue, 2015;Birnbaum, et al, 2016;Birnbaum & Schmidt, 2008). The rates of violation of transitivity estimated here in the Butler and Pogrebna (2018) data are small, but they are higher than those reported in previous studies, so the recipe for constructing triples presented by Butler and Pogrebna (2018) may indeed show promise, even if the MPW theory that motivated this design can be rejected.…”

“…Their Online supplement lists the numbers of participants who repeated the 123 pattern on both repetitions: 2, 5, 7, 0, 2, 1, 0, 5, 1, 0, and 7, for Triples # 1−11, respectively. The numbers who repeated the 231 pattern were 8, 0, 5, 26, 7, 8, 12, 5, 0, 17, and (#4 and 11) were found to have significant deviations by both conservative and refit Monte Carlo methods (see Birnbaum, et al, 2016), but Triple 5 (which had the third largest in Table 3) was not significant by either conservative or refit methods.…”

“…Because the data analyses presented in Butler and Pogrebna (2018) were based on these older methods, a skeptic could remain unconvinced that their data actually contained any real evidence against transitivity, once errors and differences in true preferences are allowed. Fortunately, because their study included two presentations of each choice problem, it is possible to apply the true and error (TE) model to reanalyze their data in order to estimate error rates and the incidence of intransitive and transitive behavior (Birnbaum, 2013;Birnbaum, Navarro-Martinez, Ungemach, Stewart, & Quispe-Torreblanca, 2016;Birnbaum & Wan, 2020)…”

“…A program in R (Birnbaum, et al, 2016), 8 2_ . , is available from this journal's website supplement to Birnbaum and Wan (2020).…”

TEMAP2.R: True and Error model analysis program in R

“…The refit procedure uses these same, best-fit parameters to the original data to generate simulated samples; however, it estimates best-fit parameters in each new simulated sample before calculating the index of fit in each sample. The re-fit method always yields the same or better fits in the simulated samples, and therefore leads to smaller estimated p-values; it is thus more likely to reject the null hypothesis (Birnbaum, et al, 2016).3…”