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Edited by: Joshua Oon Soo Goh, National Taiwan University, Taiwan

Reviewed by: Tobias Kalenscher, Heinrich-Heine University Duesseldorf, Germany

*Correspondence: Nicholas Brown,

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Kalenscher et al. (

Similar to the seminal study of Tversky (

Economists and psychologists have studied the conceptual challenges of taking a deterministic axiom like transitivity and expanding it into a probability model to incorporate the inherent uncertainty in human behavior (Luce,

They promoted random preference models as a compelling probabilistic specification of transitivity.

They offered the first proper statistical test of the random preference model using “order-constrained inference” methods, and one of the first proper direct statistical tests of weak stochastic transitivity.

They discussed conceptual, mathematical, and statistical problems of commonly used descriptive indices similar to K. Index (see, e.g., Regenwetter et al.,

According to the _{≻}. The binary choice probability,

According to this model, a decision maker has probabilistic transitive preference states and responds in an error-free fashion. Understanding the mathematical and statistical properties of this model has been the subject of a sophisticated technical literature.

According to

According to this model, a decision maker has one deterministic transitive preference state and responds in a noisy fashion. Despite appearances, this model is mathematically nontrivial. For five choice alternatives, it forms the disjoint union of 120 different hypercubes in a 10-dimensional parameter space (Regenwetter et al.,

We also consider two models that permit, but do not require, intransitive preference states. Respondents with

Table

1 | 0.21 | 19.41 | 8.15 | 3.10 | RP Transitive | |

2 | Unconstrained | |||||

3 | < |
< |
9.01 | RP Lexicographic semiorder | ||

4 | 0.065 | 0.82 | 8.52 | Weak stochastic transitivity | ||

5 | < |
< |
0.89 | Unconstrained | ||

6 | 2.21 | 0.92 | < |
- | ||

7 | 9.01 | RP Lexicographic semiorder | ||||

8 | 0.005 | 3.57 | 8.53 | 2.49 | Weak stochastic transitivity | |

9 | 0.26 | 14.77 | 2.44 | 56.28 | 0.40 | RP lexicographic semiorder |

10 | 2.11 | < |
- | |||

11 | 0.26 | 1.90 | 3.97 | < |
Weak stochastic transitivity | |

12 | 0.15 | 5.49 | 8.37 | 3.35 | Weak stochastic transitivity | |

13 | < |
2.33 | - | |||

14 | 0.050 | 1.31 | 8.53 | Weak stochastic transitivity | ||

15 | < |
< |
1.07 | Unconstrained | ||

16 | < |
< |
0.66 | - | ||

17 | 1.67 | 0.42 | < |
- | ||

18 | 0.050 | 7.76 | 8.53 | 1.01 | Weak stochastic transitivity | |

19 | 2.62 | 0.78 | < |
- | ||

20 | 1.26 | 0.32 | - | |||

21 | 1.19 | 0.45 | < |
- | ||

22 | < |
< |
33.32 | RP lexicographic semiorder | ||

23 | 0.20 | 3.46 | < |
Weak stochastic transitivity | ||

24 | 11.21 | 1.44 | RP transitive | |||

25 | 3.32 | RP transitive | ||||

26 | 0.29 | 19.54 | 6.84 | RP transitive | ||

27 | 6.73 | 1.84 | RP transitive | |||

28 | 1.30 | 0.48 | - | |||

29 | 0.25 | 0.35 | 5.29 | < |
Weak stochastic transitivity | |

30 | 0.0067 | 13.65 | 8.53 | 8.91 | RP transitive |

Seven participants were classified according to a lexicographic model or the unconstrained model (i.e., allow intransitivity), compared to the K. Index which favored intransitivity for 18 participants. We selected RPT for six and WST for eight participants. The remaining nine cases produced insufficient evidence for classification. All participants we classified as unconstrained were also classified as intransitive by the K. Index. Notice the nonmonotonic relationship between the K. Index and the Bayes factors. Compare Participants 5 and 24. Participant 5 barely made it to be classified as intransitive by the K. Index while the Bayesian analysis found very strong evidence against both RPT and WST. Participant 24 had a much larger K. Index, while the Bayesian analysis strongly favored RPT and slightly favored WST. A frequentist test of each model, where applicable, yielded good agreement with the Bayesian approach (see Supplementary Table).

We recommend three refinements to the approach of Kalenscher et al. (

The first author communicated with Dr. Kalenscher, carried out the data analyses, and wrote the first draft of the paper as part of the requirements for a Masters in Statistics at The University of Missouri. The second and third authors assisted with conceptual, mathematical, and statistical approaches and contributed to the writing.

This work was supported by National Science Foundation grants (SES-1062045, SES-1459699, PI: Regenwetter) and (SES-1459866, PI: Davis-Stober).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We are thankful to Dr. Tobias Kalenscher for his very helpful communications and for making the data available. We are also thankful to Aron K. Barbey for his helpful comments on a previous draft of this manuscript. We are also thankful to the action editor and our reviewer for their helpful comments on the manuscript. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of their colleagues, their funding agencies, or their universities.

The Supplementary Material for this article can be found online at: