Binary Probabilities Induced by Rankings

A system $[ p_{ij} : i, j \in \{ 1, 2, \cdots ,n \}, i \ne j,p_{ij} + p_{ji} = 1 ]$ of binary probabilities is said to be induced by rankings if there is a probability distribution P on the set of $n!$ linear orders of $\{ 1,2, \cdots ,n \}$ such that, for all distinct i and $j,p_{ij} $ is the sum of the P values over all linear orders in which i precedes j. It has been known for some time that the triangle inequality $p_{ij} + p_{jk} \geqq p_{ik} $ is necessary for $\{ p_{i j} \} $ to be induced by rankings and that it is also sufficient if $n\leqq 5$. The insufficiency of the triangle inequality when $n\geqq 6$ has been known since about 1970, and other necessary conditions for $n\geqq 6$ have been known since 1978.The present paper generates additional necessary conditions that pertain to $n\geqq 6$ and shows that they are independent of previous necessary conditions. It then observes that the set of conditions on the $p_{ij} $ that are sufficient for $\{ p_{ij} \} $ to be induced by rankings regardles...