Dominance Theory (Brauers-Zavadskas Dominance Theory)

Dominance theory is an aggregation method that combines several rank lists, such as the ranks produced by several different sub-approaches of a multi-criteria decision method. It was introduced as the final step of the MULTIMOORA method: MULTIMOORA produces three separate ranks through three sub-approaches (the ratio system, the reference point, and the full multiplicative form), and dominance theory reduces these three ranks to a single final rank. The method compares every pair of alternatives: if an alternative is never behind the other on any list and is ahead on at least one list, that alternative "dominates" the other. The final score is the difference between the number of alternatives a given alternative dominates and the number of alternatives that dominate it. Its output is not a mean or a sum, but a rank based on this dominance difference.

The idea behind dominance theory is to seek strict superiority rather than taking an average or summing scores. An alternative is only considered "genuinely better" if it comes out ahead without ever falling behind on any list; between two alternatives, one ahead on one list and behind on another, the method takes no side. This makes dominance theory an eliminative, cautious aggregation rule: unlike compensatory methods such as mean rank or the Borda count, it does not offset a weakness on one list with strength on another. The philosophical consequence is that dominance theory confirms only undisputed superiority; where a genuine disagreement exists between the lists, it leaves this openly as "no distinction can be made," rather than resolving it with a hidden average.

The method proceeds through a single step.

The single step, pairwise dominance count. Every pair of alternatives is compared one by one. If an alternative sits at an equal or better position than the other alternative on every rank list, and is strictly better on at least one list, that alternative is counted as dominating the other. For pairs where one is ahead on one list and behind on another, no dominance is counted; neither side dominates the other. Every alternative's final score is the difference between the number of alternatives it dominates and the number of alternatives that dominate it. Alternatives are ranked from the highest to the lowest of this difference.

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

The dominance score shows how many alternatives an alternative holds strict superiority over; it is not a score or a percentage. Receiving zero does not necessarily mean "middling"; an alternative that dominates no one and is dominated by no one also receives zero, and this means there is a genuine disagreement between the lists over that alternative. It is possible for every alternative's score to come out zero; this does not mean the method failed, it means that on none of the lists does one alternative come out ahead of another across every dimension. In that case, dominance theory alone does not determine a winner, and a complementary rule (such as mean rank or the Borda count) is needed.

Thus instead of writing:

"Alternatives with a dominance score of zero show no distinction between them, they are all equally good"

the report should read:

"A dominance score of zero means that no alternative holds strict superiority over another across every list; this is not a tie, it is information that no distinction could be made"

Dominance theory works with crisp data, and its input is a rank matrix: alternatives in the rows, rank lists (experts or sub-methods) in the columns, and every cell holding an integer rank from 1 to m. DecisionMind currently holds no fuzzy or other-data-type extension of this method; it works only in its base form, and is generally used as the final step of methods with several sub-approaches, such as MULTIMOORA. You must ensure every list ranks all alternatives completely. At least two alternatives and two rank lists are required; where agreement between lists is low, most alternatives' scores can come out close to zero. The method neither produces weights nor requires them.

Dominance theory is suitable when you have several rank lists and your goal is only to confirm undisputed superiorities verified on every list; it is a particularly strong tool for eliminating an alternative, that is, showing that it is outperformed by another alternative on every dimension. If there is strong disagreement between the lists and your goal is still to produce a complete rank, dominance theory alone falls short, because it draws no distinction for most pairs. If a quick reconciliation between just two rank lists is wanted, compensatory methods such as mean rank or the Borda count produce a more usable complete rank.

Undisputed superiorities across rank lists need marking and weak alternatives need eliminating → Dominance Theory

There is disagreement between the lists, yet a complete rank is still needed → Mean Rank or Borda Count

Only two lists exist and a quick reconciliation is wanted → Mean Rank

All dominance scores come out close to zero → a complementary aggregation rule must be added

Dominance theory's greatest advantage is its caution: it brings forward an alternative only where a superiority is verified across every list, and does not present a coincidental balancing between lists as superiority. It is an explicit elimination device; it clearly shows when one alternative is outperformed by another on every dimension. In methods such as MULTIMOORA, where several sub-approaches are expected to confirm one another, it provides a final step that explicitly measures this confirmation (Brauers and Zavadskas, 2011).

Its limitations follow from this same caution. Where there is no strong agreement between the lists (a situation common in real decisions), strict dominance is not found for most pairs of alternatives, and scores cluster close to zero; in that case the method loses its discriminating power and needs a complementary rule (Baležentis and Baležentis, 2014). Dominance theory counts only the direction of being ahead on a list, not its magnitude; being ahead by a large margin on a list is weighted the same as being ahead by a small margin. As the number of alternatives grows, the number of pairs to compare grows rapidly, which makes interpretation harder in large sets of alternatives.

The most common mistake is reporting a result of zero for every alternative as "all equally good"; the correct statement is that none of the lists shows a clear superiority. A second mistake is reading the dominance score as a percentage or a degree of strength and comparing it across analyses; the score is meaningful only for this particular set of alternatives and these particular lists. A third mistake is using dominance theory expecting it to produce a complete rank on its own in situations known to have strong disagreement between the lists; in that case the method cannot distinguish most pairs and a complementary rule is required.

The governing principle is this:

Dominance theory confirms only undisputed superiorities; a score of zero is not a tie but a signal that no clear superiority could be found between the lists, and the report must state it that way.