Ranking patterns of the unfolding model and arrangements

TL;DR

This paper investigates the variety and probabilities of ranking patterns generated by the unidimensional unfolding model using mid-hyperplane arrangements to count the possible configurations.

Contribution

It introduces mid-hyperplane arrangements to analyze the number and likelihood of ranking patterns in the unfolding model, providing a new mathematical framework.

Findings

Derived formulas for the number of ranking patterns.

Calculated probabilities of different ranking patterns.

Connected arrangements to the combinatorial structure of rankings.

Abstract

In the unidimensional unfolding model, given m objects in general position there arise 1+m(m-1)/2 rankings. The set of rankings is called the ranking pattern of the m given objects. By changing these m objects, we can generate various ranking patterns. It is natural to ask how many ranking patterns can be generated and what is the probability of each ranking pattern when the objects are randomly chosen? These problems are studied by introducing a new type of arrangement called mid-hyperplane arrangement and by counting cells in its complement.