Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model

B. Haddenhorst, E. Hüllermeier, M. Kolb, International Journal of Approximate Reasoning 119 (2020) 373–407.

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Journal Article | Published | English
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Haddenhorst, Björn; Hüllermeier, Eyke; Kolb, MartinLibreCat
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Abstract
Reciprocal relations are binary relations Q with entries Q(i,j)∈[0,1], and such that Q(i,j)+Q(j,i)=1. Relations of this kind occur quite naturally in various domains, such as preference modeling and preference learning. For example, Q(i,j) could be the fraction of voters in a population who prefer candidate i to candidate j. In the literature, various attempts have been made at generalizing the notion of transitivity to reciprocal relations. In this paper, we compare three important frameworks of generalized transitivity: g-stochastic transitivity, T-transitivity, and cycle-transitivity. To this end, we introduce E-transitivity as an even more general notion. We also use this framework to extend an existing hierarchy of different types of transitivity. As an illustration, we study transitivity properties of probabilities of pairwise preferences, which are induced as marginals of an underlying probability distribution on rankings (strict total orders) of a set of alternatives. In particular, we analyze the interesting case of the so-called Babington Smith model, a parametric family of distributions of that kind.
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Journal Title
International Journal of Approximate Reasoning
Volume
119
Issue
2
Page
373-407
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Haddenhorst B, Hüllermeier E, Kolb M. Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model. International Journal of Approximate Reasoning. 2020;119(2):373-407. doi:https://doi.org/10.1016/j.ijar.2020.01.007
Haddenhorst, B., Hüllermeier, E., & Kolb, M. (2020). Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model. International Journal of Approximate Reasoning, 119(2), 373–407. https://doi.org/10.1016/j.ijar.2020.01.007
@article{Haddenhorst_Hüllermeier_Kolb_2020, title={Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model}, volume={119}, DOI={https://doi.org/10.1016/j.ijar.2020.01.007}, number={2}, journal={International Journal of Approximate Reasoning}, publisher={Elsevier}, author={Haddenhorst, Björn and Hüllermeier, Eyke and Kolb, Martin}, year={2020}, pages={373–407} }
Haddenhorst, Björn, Eyke Hüllermeier, and Martin Kolb. “Generalized Transitivity: A Systematic Comparison of Concepts with an Application to Preferences in the Babington Smith Model.” International Journal of Approximate Reasoning 119, no. 2 (2020): 373–407. https://doi.org/10.1016/j.ijar.2020.01.007.
B. Haddenhorst, E. Hüllermeier, and M. Kolb, “Generalized transitivity: A systematic comparison of concepts with an application to preferences in the Babington Smith model,” International Journal of Approximate Reasoning, vol. 119, no. 2, pp. 373–407, 2020, doi: https://doi.org/10.1016/j.ijar.2020.01.007.
Haddenhorst, Björn, et al. “Generalized Transitivity: A Systematic Comparison of Concepts with an Application to Preferences in the Babington Smith Model.” International Journal of Approximate Reasoning, vol. 119, no. 2, Elsevier, 2020, pp. 373–407, doi:https://doi.org/10.1016/j.ijar.2020.01.007.

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