Luce’s Independence from Irrelevant Alternatives (IIA) choice axiom states that the ratio of choice probabilities of two given alternatives out of any set of alternatives remains unchanged. The complete decomposition axiom arises as a natural extension of the IIA axiom when the axiomatization is extended from the simple best choice to ranking probabilities. This axiom states that the event that a ranking
of some alternatives occur is independent from the fact that another alternative will precede all those alternatives. In this case, the choice probabilities also have the IIA property (logit form) and the ranking probabilities can be expressed as a product of logit choice probabilities. This paper deals with the axiomatization of ranking probabilities. We consider instead of the "complete" IIA and decomposition axioms
weaker conditions: "partial" IIA and decomposition axioms. We also consider the "reverse" partial IIA and partial decomposition axioms, that is when the independency deals with the worst instead of the best alternatives. Our theorems provide a new characterization of such logit models of ranking called Independence from Irrelevant Rankins
(IIR) of conditional ranking probabilities between pairewise events and ranking events.