Stochastic transitivity models
are
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
versions of the
transitivity property of binary relations studied in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. Several models of stochastic transitivity exist and have been used to describe the probabilities involved in experiments of
paired comparisons
Pairwise comparison generally is any process of comparing entities in pairs to judge which of each entity is preferred, or has a greater amount of some quantitative property, or whether or not the two entities are identical. The method of pairwis ...
, 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:
# Weak Stochastic Transitivity (WST):
and
implies
, for all
;
# Strong Stochastic Transitivity (SST):
and
implies
, for all
;
# Linear Stochastic Transitivity (LST):
, for all
, where