On two parametric probability distributions on crisp complete pre-orders

Abstract

Mallows and Plackett-Luce parametric probability distributions are the two most used and well-known in the set of complete linear orders of a finite universe. In this paper, we extend those two distributions on the set of complete preorders of the universe. For that purpose, by considering a parametric family of metrics on the set of complete pre-orders generalizing Kemeny Distance on pre-orders and Kendall metric on orders, we determine a parametric probability distribution on pre-orders generalizing Mallows Distribution. By considering pre-orders as orders on blocks of equivalent elements, we generalize the Plackett-Luce distribution on complete pre-orders.

Keywords: Complete pre-orders; Generalized Mallows distribution; Generalized
Plackett-Luce distribution

AMS 2010 Mathematics Subject Classification: 06A06, 97K50, 91B14