# The (logarithmic) least squares optimality of the arithmetic (geometric) mean of weight vectors calculated from all spanning trees for incomplete additive (multiplicative) pairwise comparison matrices

## TL;DR

This paper proves that the arithmetic and geometric means of weight vectors from all spanning trees are optimal solutions to least squares problems in preference matrices, extending previous results to incomplete matrices and linking to Kirchhoff's laws.

## Contribution

It establishes the optimality of spanning tree-based means for incomplete preference matrices, generalizing prior complete-matrix results with new proofs.

## Key findings

- Optimality of spanning tree means for incomplete matrices
- Extension of previous complete-matrix results
- Connection to Kirchhoff's laws in electric circuits

## Abstract

Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the "spanning tree" and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197-208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.