Composition Rules for Probabilities from Paired Comparisons

Abstract

Let $P_{ab}$ be the probability that the outcome of a paired comparison involving $a$ and $b$ is favorable to $a$. This paper discusses composition rules that generate $p_{ac}$ given $p_{ab}$ and $p_{bc}$. The basic properties of composition rules are developed via an axiomatic approach. If $p_{ab} = F(w_a - w_b)$, where $F$ is a distribution function for a distribution that is symmetric about zero, then the paired comparison model is a linear model that is based on the distribution function $F$. It is shown that given any composition rule, which obeys certain basic axioms, there exists a linear model that generates an identical composition rule. The behavior of the composition rules are used to place a partial ordering on the paired comparison models and in particular on the linear models. This partial ordering is denoted as the extreme partial ordering. It is shown that linear models based on distributions with short tails tend to be more extreme than those based on distributions with long tails. The resulting partial ordering includes the result that the Thurstone-Mosteller model is more extreme than the Bradley-Terry model. The extreme ordering can also be used to place a partial ordering on distributions according to the lengths of their tails. The relation of this ordering to the $s$-ordering and $r$-ordering is examined.