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- Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of paired comparisons, specifically in scenarios where transitivity is expected, however, empirical observations of the binary relation is probabilistic. For example, players' skills in a sport might be expected to be transitive, i.e. "if player A is better than B and B is better than C, then player A must be better than C"; however, in any given match, a weaker player might still end up winning with a positive probability. Tightly matched players might have a higher chance of observing this inversion while players with large differences in their skills might only see these inversions happen seldom. Stochastic transitivity models formalize such relations between the probabilities (e.g. of an outcome of a match) and the underlying transitive relation (e.g. the skills of the players). A binary relation on a set is called transitive, in the standard non-stochastic sense, if and implies for all members of . Stochastic versions of transitivity include: 1.
* Weak Stochastic Transitivity (WST): and implies , for all ; 2.
* Strong Stochastic Transitivity (SST): and implies , for all ; 3.
* Linear Stochastic Transitivity (LST): , for all , where is some increasing and symmetric function (called a comparison function), and is some mapping from the set of alternatives to the real line (called a merit function). (en)
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- Stochastic transitivity models are stochastic versions of the transitivity property of binary relations studied in mathematics. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of paired comparisons, specifically in scenarios where transitivity is expected, however, empirical observations of the binary relation is probabilistic. For example, players' skills in a sport might be expected to be transitive, i.e. "if player A is better than B and B is better than C, then player A must be better than C"; however, in any given match, a weaker player might still end up winning with a positive probability. Tightly matched players might have a higher chance of observing this inversion while players with large differences in thei (en)
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- Stochastic transitivity (en)
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