Koczkodaj Inconsistency Index (K)

In AHP and similar methods, the decision-maker compares criteria or alternatives two at a time: "how many times more important is A than B?" These comparisons are gathered into a table (a pairwise comparison matrix), and weights are derived from it. The problem is this: the decision-maker can give inconsistent judgements. If A is 3 times as important as B, and B 2 times as important as C, then A would be expected to come out roughly 6 times as important as C; if the decision-maker instead enters 8 or 2 there, the table is contradictory. The Koczkodaj inconsistency index is a diagnostic tool that measures this contradiction; it produces no ranking or weights, it gives only a numerical answer to the question "can this table be trusted?" Waldemar W. Koczkodaj proposed it in 1993, and it works on a different logic from Saaty's classical consistency ratio (CR).

Saaty's CR takes an average of the deviations across the whole table; if the overall picture is good, a single bad comparison can slip through unnoticed. Koczkodaj's idea is the opposite: look not at the average but at the worst triad, because even a single contradictory judgement from the decision-maker can silently distort the derived weights. The method audits every possible group of three comparisons (a triad) one by one and reports the largest contradiction in the table. This is a consistency measure, not a compensatory ranking method; it shows not the table's overall picture but its weakest point. The philosophical consequence is this: a good average does not excuse a bad triad.

The method proceeds through three steps.

First, extracting the triads. For every group of three criteria (a triad) in the table, the three relevant comparison values are taken: the first against the second, the first against the third, and the second against the third. As n, the number of criteria, grows, the number of possible triads increases rapidly; three criteria give a single triad, four criteria give four triads.

Second, measuring each triad's contradiction. If the three comparisons are consistent, the value between the first and the third should equal the product of the first-to-second and second-to-third values. Koczkodaj calculates the deviation from this expected value in both directions: how much larger the actual value is than the expected one, and how much larger the expected value is than the actual one; the smaller of the two is that triad's contradiction score. The smaller value is taken because inconsistency should not be counted on the same scale in both directions.

Third, selecting the worst. Once the contradiction scores of all the triads in the table have been calculated, the largest of them, that is the most contradictory triad, is reported as the Koczkodaj index K. The customary threshold is 0.10: if K falls below this threshold, the table is considered acceptable; if it exceeds it, revisiting the relevant triad is recommended.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The closer K sits to zero, the more consistent the table; zero means there is no contradiction in any triad. As K grows, it means at least one triad carries a serious contradiction. K is not a percentage, it does not mean "this percentage of the judgements is wrong"; it shows only the worst triad's rate of deviation from its expected value. K falling below 0.10 does not mean every part of the table is flawless, only that its weakest point stays within acceptable bounds. In a table with more than five criteria, because K shows only the worst triad, the other triads need to be examined individually as well.

For that reason, instead of writing:

"K comes out below 0.10, so Saaty's Consistency Ratio (CR) is automatically acceptable too"

it is correct to write:

"K and CR are different measures: K looks at the worst triad, CR at the average deviation. K being below 0.10 says nothing about CR; the two must be calculated and read separately"

The Koczkodaj index works with crisp data: the input is a square pairwise comparison table whose diagonal is 1 and whose reciprocal cells are each other's inverse (if A's value relative to B is 3, then B's value relative to A is 1/3). DecisionMind currently has no extension of this method for fuzzy, grey or another data type; it works only in its base form. You need pairwise comparisons made between at least three criteria or alternatives; with fewer than three items no triad can form and K remains undefined. The recommended size is between three and nine items; as the number of items grows, the number of triads to audit multiplies rapidly. The method produces no weights, it only audits the reliability of the input table.

The Koczkodaj index is suitable if you are scoring criteria or alternatives through pairwise comparison and want to see the worst contradiction before deriving weights from that table; because it points specifically to which comparison the decision-maker should revisit, you are left with a single, concrete piece of "worst triad" information. If you have no pairwise comparison table, and instead have direct ranking or scoring data, this method is not suitable; the consistency check is specific to pairwise comparison. If you want to summarise the table's overall average consistency in one number, this method alone is not sufficient either; K shows only the worst case.

A pairwise comparison table exists, and the worst contradiction needs finding and fixing → Koczkodaj K

The same table's overall average consistency is wanted → Saaty's Consistency Ratio, or the geometric/harmonic consistency indices

No pairwise comparison exists, only direct ranking or scoring data → move from a consistency check to a ranking or weighting method instead

Fewer than three criteria or alternatives exist → no triad can form, K cannot be calculated

The Koczkodaj index's greatest strength is its simplicity: unlike Saaty's CR, it needs no eigenvalue calculation or random consistency tables, and is calculated directly from the triads. Because it points specifically to the worst triad, it gives the decision-maker a concrete direction, "reconsider this comparison"; CR's single average number does not offer this level of detail. It works on the same logic at every dimension of three or more, requiring no additional assumptions.

Its limitations also stem from this structure. Because K reports only the worst triad, if many triads carry moderate but numerous contradictions, it does not show their combined effect; a user wanting to see the whole table must examine every triad individually (Brunelli, 2017). As the number of criteria grows, the number of triads to audit grows rapidly, which increases the computational and interpretive burden on large tables. The acceptance threshold of 0.10 has been adopted as a convention, much like Saaty's CR threshold; different consistency indices propose different thresholds, and which one is "correct" remains a matter of debate in the literature (Brunelli, 2018).

The most common mistake is using K and Saaty's CR interchangeably; the two rest on different calculation logic, and even where the 0.10 threshold is used for both, they do not measure the same thing. A second mistake is looking only at K in tables of four or more criteria and never examining the other triads; K shows the worst, but the second- and third-worst triads may also need revision. A third is calculating the index without first checking that the table's diagonal is 1 and that reciprocal cells are each other's inverse when preparing the table; if this assumption is broken, the resulting K is meaningless.

The governing principle is this:

K shows not the table but the table's weakest point; a K close to zero does not mean every part of the matrix is flawless, only that the worst triad stays within acceptable bounds.