# On the geometric mean method for incomplete pairwise comparisons

## TL;DR

This paper extends the geometric mean method to handle incomplete pairwise comparison matrices, providing theoretical guarantees for the existence and optimality of solutions in ranking tasks.

## Contribution

The paper introduces an extension of the geometric mean method specifically designed for incomplete pairwise comparison matrices, with theoretical validation.

## Key findings

- Existence of solutions is guaranteed for the extended method.
- The proposed approach is shown to be optimal under certain conditions.
- The method effectively handles incomplete comparison data.

## Abstract

When creating the ranking based on the pairwise comparisons very often, we face difficulties in completing all the results of direct comparisons. In this case, the solution is to use the ranking method based on the incomplete PC matrix. The article presents the extension of the well known geometric mean method for incomplete PC matrices. The description of the methods is accompanied by theoretical considerations showing the existence of the solution and the optimality of the proposed approach.