COOK-SEIFORD (Distance-Based Consensus Method)

COOK-SEIFORD is an aggregation method that reduces several ranking sources to a single complete order. Where CONDORCET, COPELAND, DODGSON and KEMENY-YOUNG all rest on pairwise majority comparisons, COOK-SEIFORD uses an entirely different logic: it looks directly at the position numbers (first, second, third) the alternatives receive across the ranking sources, and assigns each alternative to the target rank closest to its positions. This assignment is a matching problem, solved with the Hungarian algorithm. The method was proposed by Wade D. Cook and Lawrence M. Seiford in 1978.

The question COOK-SEIFORD asks is this: "whichever final rank I place each alternative into, which arrangement moves it least, in total, from its positions across the different sources?" The method works directly with position numbers rather than pairwise contests. Whether an alternative is first or second is read from which rank it holds, not from whom it beat. For every alternative, the total cost of assigning it to each possible target rank (its absolute distance from that rank across all sources) is calculated. The assignment that minimises this total cost across all alternatives is then found.

This differs fundamentally from the CONDORCET family's "who beats whom" logic; COOK-SEIFORD never invokes the concept of a majority at all. The consequence is that the method is unaffected by a Condorcet cycle, because it never carries out pairwise comparison in the first place. But there is a cost: the assignment problem can have several equal-cost solutions; in that case the method produces not one single "correct" rank but several equally valid candidate ranks (Armstrong, Cook and Seiford, 1982).

The method proceeds through two steps.

First, building the distance matrix. For every alternative and every possible target rank, the absolute difference between that alternative's positions across all sources and that target rank is summed. This produces a cost figure between every alternative and every possible rank.

Second, finding the lowest-total-cost assignment. With each alternative assigned to exactly one target rank, and each target rank to exactly one alternative, the matching that minimises the total cost is found with the Hungarian algorithm. This assignment gives the alternatives' final rank.

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

The output is a target rank number assigned to each alternative; the total cost shows how far this assignment sits from the alternatives' actual positions. If the total cost is small, the assigned rank sits fairly close to the positions given by all the sources; if it is large, the sources disagree substantially on position. This figure is not a percentage or a probability, it is only a unit of total absolute difference.

The most critical point is that the lowest-total-cost assignment is not always unique. Several assignments can give the same lowest total. In that case, which alternative is placed at which rank is chosen among several mathematically equally valid options. This choice must be made by an external rule (alphabetical order, or an additional criterion, for instance) (Armstrong, Cook and Seiford, 1982).

For that reason, instead of writing:

"COOK-SEIFORD found this rank definitively"

it is correct to write:

"This assignment is the arrangement that moves least, in total, from the alternatives' positions; if several equal-cost arrangements exist, the report must state which one was chosen"

COOK-SEIFORD works with crisp data: the position given by each source is a whole number. DecisionMind has no fuzzy, grey or similar data-type extension of this method; it works only with crisp rank data.

You need the following: a table in which every source ranks all the alternatives completely. A minimum of two alternatives and two ranking sources is required. COOK-SEIFORD neither asks for nor produces weights; every source's position is counted equally.

COOK-SEIFORD is suitable if you have several ranking sources and your aim is to reconcile the position numbers they give directly, without needing pairwise majority logic. Because it is unaffected by a Condorcet cycle, it is a practical alternative for small alternative sets where the risk of a cycle is high.

The case where it should not be used is one that requires a single definitive result, where several equal-cost assignments cannot be accepted. Also, if the "who beats whom" information itself matters (for instance, if you need to show that a winner has majority support), moving to a method based on pairwise comparison is more appropriate.

I want to reconcile position numbers directly, and stay unaffected by a cycle → COOK-SEIFORD

I want to see the winner's pairwise majority support → CONDORCET, COPELAND

I want a rank that globally minimises total disagreement → KEMENY-YOUNG

I only want to find the winner, on a least-change logic → DODGSON

COOK-SEIFORD's greatest strength is its independence from pairwise majority logic; a Condorcet cycle never affects it at all, because it never looks at who beats whom in the first place. Because the method works directly with position numbers, it is intuitively understandable: it gives an answer close to the question "roughly where does this alternative sit across the sources on average?" The Hungarian algorithm is computationally light and stays predictable as the number of alternatives grows.

COOK-SEIFORD's fundamental limitation is that the lowest-cost assignment is frequently not unique; several equal-cost solutions can arise, and the method itself makes no choice among them (Armstrong, Cook and Seiford, 1982). Second, the method looks only at position numbers; an alternative being in second place does not show whether it missed first place by a small or a large margin, because the sources give only a rank number, not a score. Third, because the position-based approach does not use pairwise majority information, it does not directly show whether an alternative is clearly preferred by a majority.

The most common mistake is assuming the lowest-cost assignment is unique; if several equal-cost assignments exist, showing only one of them and concealing the others is misleading. A second mistake is reading the total cost as though it were a percentage or an error rate; this figure is only the total absolute position difference. A third is presenting a COOK-SEIFORD result as though it were a majority winner; the method never looks at pairwise majority at all, it only reconciles position numbers. A fourth is allowing some sources to rank the alternatives incompletely; every source must rank all the alternatives.

The governing principle is this:

The rank COOK-SEIFORD gives is the assignment that moves least, in total, from the alternatives' positions across the sources; this assignment may not be unique, and the report must state this clearly.