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Exact bounds for efficient consistent matrices obtained from a reciprocal matrix

Susana Furtado, Charles Johnson

TL;DR

The paper develops a precise, interval-based description of the set of consistent matrices $W=ww^{(-T)}$ obtainable from efficient vectors $w$ for a given reciprocal matrix $A$, by partitioning into a union of at most $\frac{(n-1)!}{2}$ intervals tied to Hamiltonian cycles with cycle product $<1$. It shows that every $W$ in these intervals corresponds to some $w\in\mathcal{E}(A)$ and, conversely, every $w\in\mathcal{E}(A)$ yields a $W$ inside this union, enabling exact characterization of the feasible region and the induced partial order among alternatives. The authors analyze the structure of $\mathcal{E}(A)$ via path matrices $P_{A,\tau}$, maximal attainable bounds, and the possible efficient orders, and prove that $\mathcal{E}(A)=\mathcal{E}(B)$ implies $A=B$ in the simple-perturbed case and for $n=4$, with discussion of necessary conditions for general $n$. These results sharpen prior bounds and provide a constructive, cycle-based framework for recovering or distinguishing decision matrices from their efficient vectors. The findings have implications for decision analysis where ordinal structure induced by efficient rankings is critical and for understanding identifiability in pairwise comparison models.

Abstract

For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The maximal sets of entries in the lower and upper bound matrices of each interval that are attainable by some consistent matrix in the interval are described. This allows us to understand which subsets of the alternatives lie above which other subsets in all efficient orders for each interval. As a result, the partial order on the alternatives dictated by the efficient vectors follows. Then, we use the tools developed to also show that, when the n-by-n reciprocal matrices A,B are simple perturbed consistent matrices, or n=4, the sets of efficient vectors for A and B coincide only if A=B.

Exact bounds for efficient consistent matrices obtained from a reciprocal matrix

TL;DR

The paper develops a precise, interval-based description of the set of consistent matrices obtainable from efficient vectors for a given reciprocal matrix , by partitioning into a union of at most intervals tied to Hamiltonian cycles with cycle product . It shows that every in these intervals corresponds to some and, conversely, every yields a inside this union, enabling exact characterization of the feasible region and the induced partial order among alternatives. The authors analyze the structure of via path matrices , maximal attainable bounds, and the possible efficient orders, and prove that implies in the simple-perturbed case and for , with discussion of necessary conditions for general . These results sharpen prior bounds and provide a constructive, cycle-based framework for recovering or distinguishing decision matrices from their efficient vectors. The findings have implications for decision analysis where ordinal structure induced by efficient rankings is critical and for understanding identifiability in pairwise comparison models.

Abstract

For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The maximal sets of entries in the lower and upper bound matrices of each interval that are attainable by some consistent matrix in the interval are described. This allows us to understand which subsets of the alternatives lie above which other subsets in all efficient orders for each interval. As a result, the partial order on the alternatives dictated by the efficient vectors follows. Then, we use the tools developed to also show that, when the n-by-n reciprocal matrices A,B are simple perturbed consistent matrices, or n=4, the sets of efficient vectors for A and B coincide only if A=B.
Paper Structure (11 sections, 35 theorems, 55 equations)

This paper contains 11 sections, 35 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path matrices
  4. Exact Bounds for $W$
  5. Maximal sets of attainable bounds
  6. Orders occurring among the efficient vectors
  7. $\mathcal{E}(A)=\mathcal{E}(B)$ implies $A=B$: background
  8. $\mathcal{E}(A)=\mathcal{E}(B)$ implies $A=B$: the simple perturbed case
  9. Necessary conditions for $\mathcal{E}(A)=\mathcal{E}(B)$ for general $n$
  10. $\mathcal{E}(A)=\mathcal{E}(B)$ implies $A=B$: the case $n=4$
  11. Conclusions

Key Result

Lemma 1

FJ1 Suppose that $A\in\mathcal{PC}_{n}.$ If $S\in M_{n}$ is a monomial matrix, then $\mathcal{E}(SAS^{-1})=S\mathcal{E}(A)$.

Theorems & Definitions (51)

  • Lemma 1
  • Example 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 7
  • Example 8
  • Lemma 9
  • Theorem 10
  • ...and 41 more