Two characterizations of the dense rank

TL;DR

This paper provides two axiomatic characterizations of the dense rank, a method for assigning positions in weak orders, by analyzing its invariance properties under various transformations.

Contribution

It introduces a formal framework and two new axiomatic characterizations of the dense rank, comparing it with other ranking methods in weak orders.

Findings

Dense rank assigns the same position to all alternatives in the same tier.

Two axiomatic characterizations uniquely determine the dense rank based on invariance conditions.

The framework clarifies the properties that distinguish dense rank from other position operators.

Abstract

In this paper, we have considered the dense rank for assigning positions to alternatives in weak orders. If we arrange the alternatives in tiers (i.e., indifference classes), the dense rank assigns position 1 to all the alternatives in the top tier, 2 to all the alternatives in the second tier, and so on. We have proposed a formal framework to analyze the dense rank when compared to other well-known position operators such as the standard, modified and fractional ranks. As the main results, we have provided two different axiomatic characterizations which determine the dense rank by considering position invariance conditions along horizontal extensions (duplication), as well as through vertical reductions and movements (truncation, and upwards or downwards independency).