Axiomatizations of inconsistency indices for triads

Pairwise comparison matrices often exhibit inconsistency, therefore, a number of indices has been introduced to measure their deviation from a consistent matrix. Since inconsistency first emerges in the case of three alternatives, several inconsistency indices are based on triads. Recently, a set of axioms has been proposed, and is required to be satisfied by any reasonable inconsistency index. We illustrate by an example that this set seems to be not exhaustive, hence expand it by adding two new properties. We consider all axioms on the set of triads, and choose the logically independent ones. Finally, it is proved that they characterize the inconsistency ranking induced by the Koczkodaj inconsistency index on this domain.


Introduction
Pairwise comparisons play an important role in a number of decision analysis methods such as the AHP (Analytic Hierarchy Process) (Saaty, 1980). They also naturally emerge in sport tournaments (Bozóki et al., 2016;Chao et al., 2018). Theoretically, an appropriate set of − 1 pairwise comparisons would be sufficient to derive a set of weights or to rank the alternatives. However, usually more information is available. For example, the decision makers are asked to make more pairwise comparisons, because it increases the robustness of the result. It is also clear that a round-robin tournament is fairer than a knockout one as a loss does not lead to the elimination of a player.
Nevertheless, the knowledge of further pairwise comparisons has a price. First, it is time consuming for the decision-maker or for the players. Second, the set of comparisons may become inconsistent: if alternative is better than , and is better than , then still might turn out to be preferred over . While consistent preferences do not automatically imply the rationality of the decision maker, it is plausible to assume that strongly inconsistent preferences indicate a problem: perhaps the decision maker has not understand the elicitation phase, or the strength of players varies during the tournament.
Recently, some authors applied an axiomatic approach by defining reasonable properties required from an inconsistency index (Brunelli and Fedrizzi, 2011;Brunelli, 2016Brunelli, , 2017Brunelli and Fedrizzi, 2015;Cavallo and D'Apuzzo, 2012;Koczkodaj and Szwarc, 2014). There is also one characterization in this topic: Csató (2017a) introduces six independent axioms that uniquely determine the Koczkodaj inconsistency ranking, induced by the Koczkodaj inconsistency index (Koczkodaj, 1993;Duszak and Koczkodaj, 1994). Note that in the case of characterizations, the appropriate motivation of the properties is not crucial. The result only says that there remains one choice if one accepts all axioms.
This work aims to connect these two research directions by placing the axioms of Brunelli (2017) -which is itself an extended set of the properties proposed by Brunelli and Fedrizzi (2015) -and Csató (2017a) into a single framework and showing that the Koczkodaj inconsistency ranking is the unique inconsistency ranking satisfying all properties on the set of triads, that is, in the case of pairwise comparison matrices with three alternatives. Consequently, any reasonable triad inconsistency index is closely related to the Koczkodaj inconsistency index.
The paper is structured as follows. Section 2 presents the setting and the properties of inconsistency indices suggested by Brunelli (2017). It will be revealed in Section 3 that this axiomatic system is not exhaustive. Section 4 introduces two new axioms and discusses logical independence. The Koczkodaj inconsistency ranking is characterized in Section 5. Finally, Section 6 summarizes our results.

Preliminaries
Let denote the set of pairwise comparison matrices. Inconsistency index : → R associates a value for each pairwise comparison matrix. Brunelli and Fedrizzi (2015) have suggested and justified five axioms for inconsistency indices. They are only briefly recalled here.  with . An inconsistency index : → R satisfies axiom if A triad is a pairwise comparison matrix with three alternatives, the smallest pairwise comparison matrix which can be inconsistent. Therefore, triads play a prominent role in the measurement of inconsistency. For instance, the Koczkodaj inconsistency index (Koczkodaj, 1993;Duszak and Koczkodaj, 1994), the Peláez-Lamata inconsistency index (Peláez and Lamata, 2003) and the family of inconsistency indices proposed by Ku lakowski and Szybowski (2014) are all based on triads. In the following, we will focus on the set of triads , thus inconsistency will be measured by a triad inconsistency index : → R. Note that a triad T ∈ can be described by its three elements above the diagonal such as T = ( 12 ; 13 ; 23 ) and T is consistent if and only if 13 = 12 23 .

Motivation
The axiomatic system suggested by Brunelli (2017) is not necessarily exhaustive in the sense that it may allow for some strange inconsistency indices. Consider the following one.

Definition 1. Scale-dependent triad inconsistency index: Let T = [ ] ∈ be any triad. Its inconsistency according to the scale-dependent triad inconsistency index is
The scale-dependent triad inconsistency index sums the differences of non-diagonal matrix elements from the value exhibiting consistency. ⃒ ⃒ ⃒ ≥ | 13 − 12 23 | for every possible (positive) value of 12 , 13 , and 23 if and only if ≥ 1. It can be assumed without loss of generality that 13 − 12 23 ≥ 0, implying 13 − 12 23 ≥ 0.
Consider . It can be assumed that 13 is the element to be changed because of the axiom .
(T) = 0 if 13 = 12 23 , and all terms in the formula of (T ( )) gradually increase when goes away from 1.
The example below illustrates that the scale-dependent triad inconsistency index may lead to questionable conclusions. Example 1. Take two alternatives and such that the decision maker is indifferent between them. Assume that a third alternative appears in the comparison, and is judged three times better than , while is assessed to be two times better than . Suppose that is a divisible alternative and is exchanged by its half.
The two situations can be described by two triads: Here (S) = 19/6 ≈ 3.167 and (T) = 5. In other words, the scale-dependent inconsistency index suggests that S is less inconsistent than T, contrary to the underlying data since inconsistency is not expected to be influenced by the 'amount' of alternative .
In our view, Example 1 delivers the message that the axioms of Brunelli (2017) should be supplemented even on the set of triads.

An improved axiomatic system
We propose two new axioms of inconsistency indices for triads. Homogeneous treatment of alternatives says that if the first and the second alternatives are equally important on their own, but they are also compared to a third alternative, then the inconsistency of the resulting triad should not be influenced by the relative importance of the new alternative.
Note that Example 1 has shown the violation of by the scale-dependent triad inconsistency index .
Scale invariance implies that inconsistency is independent from the mathematical representation of the preferences. For example, consider three alternatives such that the first is 'moderately more important' than the second and the second is 'moderately more important' than the third. It makes sense to expect the level of inconsistency to be the same if 'moderately more important' is coded by the number 2, 3, or 4, and so on, even allowing for a change in the direction of the two preferences. If the encoding is required to preserve consistency, one arrives at the property . and have been introduced in Csató (2017a) for inconsistency rankings (and has been called homogeneous treatment of entities there). In order to understand the implications of the extended axiomatic system, the logical consistency and independence of the eight axioms should be discussed.

Definition 2. Koczkodaj inconsistency index
be a pairwise comparison matrix. Its inconsistency according to the Koczkodaj inconsistency index is However, some axioms can be implied by a conjoint application of the others on the set of triads.

Lemma 1.
, and imply on the set of triads .

Lemma 2.
, , , and imply on the set of triads .
There are not other direct implication among the remaining six properties.
Theorem 1. Axioms , , , , and are independent on the set of triads .
Proof. Independence of a given axiom can be shown by providing a triad inconsistency index that satisfies all axioms except the one at stake:

2
: The triad inconsistency index 2 : → R such that for all triads T ∈ . 4 is essentially the Koczkodaj inconsistency index , but takes only the elements above the diagonal into account.

5
: The triad inconsistency index 5 : → R such that for all triads T ∈ .

6
: The triad inconsistency index 6 : → R such that Proving that the triad inconsistency index satisfies all axioms but the th is straightforward for 1 ≤ ≤ 4, therefore left to the reader.
To conclude, the axiomatic system consisting of , , , , and satisfies logical consistency and independence on the set of triads.

Characterization
It is still a question whether the extended set of properties is exhaustive on the set of triads . We will show that the axioms are closely related to the Koczkodaj inconsistency index: they practically imply that is the only appropriate index for measuring the inconsistency of triads. Proof. Assume that (S) ≥ (T). The idea is to gradually simplify the comparison of the inconsistencies of the two triads by using our axioms.
Remark 1. As Theorem 2 shows, axioms , , and allow for some odd triad inconsistency indices, for example, the flat triad inconsistency index : → R such that (T) = 0 for any triad T ∈ . By attaching properties and , inconsistency index is excluded, but they still allow for the 'discretized' Koczkodaj triad inconsistency index : → R defined as It can be checked that the proof of Theorem 2 does not work in the reverse direction of (S) ≥ (T) ⇒ (S) ≥ (T), because monotonicity on single comparisons was introduced without strict inequalities by Brunelli and Fedrizzi (2015).
Axiom 9. Strong monotonicity on single comparisons ( ): Let A ∈ × be any consistent pairwise comparison matrix, ̸ = 1 a non-diagonal element and ∈ R. Let A ( ) ∈ × be the inconsistent pairwise comparison matrix obtained from A by replacing the entry with and with . An inconsistency index : ℛ → R satisfies axiom if With the introduction of , there is no need for all of the six axioms.
Lemma 3. , , and imply on the set of triads.
As Theorem 1 has already revealed, the weaker property of cannot substitute in the proof of Lemma 3.

Proposition 4. Axioms
, , , and form a logically consistent and independent axiomatic system on the set of triads .
Proof. For consistency, it can be checked that the Koczkodaj inconsistency index satisfies strong monotonicity on single comparisons.
For independence, see the proof of Theorem 1. The inconsistency indices 3 , 4 , 5 and 6 satisfy , too.
With this strengthening of , we are able to characterize the Koczkodaj inconsistency index on the set of triads. On the basis of Proposition 5, our main result of this paper can be formulated on the measurement of triad inconsistency.

Theorem 3. The Koczkodaj inconsistency index is essentially the unique triad inconsistency index satisfying strong monotonicity on single comparisons, invariance under inversion of preferences, homogeneous treatment of alternatives and scale invariance.
The term essentially refers to the fact that the four axioms , , and characterize only the Koczkodaj inconsistency ranking (Csató, 2017a). Nonetheless, Csató (2017a) argues that it does not make much sense to distinguish inconsistency indices that rank pairwise comparison matrices uniformly. One can attach continuity to these four axioms, but it is rather a technical property.
Remark 2. Unfortunately, Remark 1 remains valid in the case of Csató (2017a, Theorem 1), which is true only in the following form: Let A and B two pairwise comparison matrices. If ⪰ is an inconsistency ranking satisfying positive responsiveness, invariance under inversion of preferences, homogeneous treatment of entities, scale invariance, monotonicity and reducibility, then A ⪰ B implies A ⪰ B.
Contrary to Csató (2017a, Theorem 1), the implication does not hold in the other direction, although the problem can be easily solved by introducing the first axiom, positive responsiveness ( ) in a stronger version called with strict inequalities: Consider two triads S = (1; 2 ; 1) and T = (1; 2 ; 1) such that 2 , 2 ≥ 1. Inconsistency ranking ⪰ satisfies if S ≻ T ⇐⇒ 2 < 2 . Then the Koczkodaj inconsistency ranking would be the unique inconsistency ranking satisfying strong positive responsiveness, invariance under inversion of preferences, homogeneous treatment of entities, scale invariance, monotonicity and reducibility.
The contribution of this paper can be shortly summarized as a unification of the two axiomatic approaches. The first aims to justify reasonable properties and analyse indices in their light (Brunelli and Fedrizzi, 2015;Brunelli, 2017). The second concentrates on the exact derivation of certain indices without spending too much time on the motivation of the axioms (Csató, 2017a). Precisely, it has been shown that the axiomatic system of Brunelli (2017) is not exhaustive even if there are only three alternatives. However, by the introduction of two new axioms, a unique inconsistency ranking can be identified on the set of triads. It is a powerful argument against indices which violate some of the axioms on this domain, like the Ambiguity Index Hämäläinen, 1995, 1997) or the Cosine Consistency Index (Kou and Lin, 2014). Our result can also serve as a solid basis for measuring the inconsistency of pairwise comparison matrices with more than three alternatives.