Efficient weight vectors from pairwise comparison matrices

- Efficiency of weight vectors derived from pairwise comparison matrices.
- Equivalence of efficiency definitions.
- The principal eigenvector is weakly efficient.
- Test of (weak) efficiency and finding efficient dominating weight vectors by linear programs.
- Numerical examples of (strong) inefficiency are provided.

Pairwise comparison matrices are frequently applied in multi-criteria decision making. A weight vector is called efficient if no other weight vector is at least as good in approximating the elements of the pairwise comparison matrix, and strictly better in at least one position. A weight vector is weakly efficient if the pairwise ratios cannot be improved in all non-diagonal positions. We show that the principal eigenvector is always weakly efficient, but numerical examples show that it can be inefficient. The linear programs proposed test whether a given weight vector is (weakly) efficient, and in case of (strong) inefficiency, an efficient (strongly) dominating weight vector is calculated. The proposed algorithms are implemented in Pairwise Comparison Matrix Calculator, available at pcmc.online.