Stochastic transitivity models[1][2][3][4] 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).
is called transitive, in the standard non-stochastic sense, if
Stochastic versions of transitivity include: The marble game - Assume two kids, Billy and Gabriela, collect marbles.
Billy collects blue marbles and Gabriela green marbles.
When they get together they play a game where they mix all their marbles in a bag and sample one randomly.
If the sampled marble is green, then Gabriela wins and if it is blue then Billy wins.
is the number of blue marbles and
is the number of green marbles in the bag, then the probability
{\displaystyle \mathbb {P} ({\text{Billy}}\succsim {\text{Gabriela}})}
of Billy winning against Gabriela is
{\displaystyle \mathbb {P} ({\text{Billy}}\succsim {\text{Gabriela}})={\frac {B}{B+G}}={\frac {e^{\ln(B)}}{e^{\ln(B)}+e^{\ln(G)}}}={\frac {1}{1+e^{\ln(G)-\ln(B)}}}}
In this example, the marble game satisfies linear stochastic transitivity, where the comparison function
is the number of marbles of the player.
This game happens to be an example of a Bradley–Terry model.