Methods · Aggregation and voting
KEMENY-YOUNG (Minimum Total Disagreement Method)
KEMENY-YOUNG selects, from among all possible complete rankings, the single ranking that falls into the least total disagreement with every ranking source combined.
Base method's data type: Classical
What Is the Method?
KEMENY-YOUNG is an aggregation method that reduces several ranking sources into one complete order. It rests on pairwise comparisons, as CONDORCET and COPELAND do, but unlike them it does not evaluate individual pairwise relationships one at a time; it evaluates every possible complete ranking as a whole. For each possible ranking, it counts how many points it conflicts with every ranking source, how many pairwise preferences it reverses, and the ranking with the lowest total conflict is selected. The idea was proposed by John Kemeny in 1959, given a mathematical foundation by H. Peyton Young and Arnold Levenglick in 1978, and shown to be the only consistent aggregation rule that fully satisfies the Condorcet principle.
The Philosophy Behind It
The question KEMENY-YOUNG asks is this: which single ranking upsets all the ranking sources least, taken together? The method treats every possible complete ranking as a candidate. For each candidate, it counts how many pairwise points agree with each source's own ranking and how many disagree. The candidate with the least total disagreement is declared the consensus ranking.
This differs from DODGSON's question of how much change is needed to make one alternative the winner: KEMENY-YOUNG optimises not a single alternative but the ranking as a whole. As Young and Levenglick (1978) showed, this approach is the only aggregation rule that jointly satisfies certain axioms such as neutrality and consistency, and if a Condorcet winner exists, KEMENY-YOUNG always places it first. This comes at the cost of computational burden: the number of possible complete rankings grows extremely quickly with the number of alternatives.
How It Works
The method proceeds through four steps.
First, gathering the rankings. Each source's ranking is combined into one table.
Second, listing every possible complete ranking. Every possible arrangement of as many elements as there are alternatives is treated as a candidate ranking.
Third, calculating each candidate's total disagreement. For every candidate ranking, the direction it gives on each pair of alternatives is compared against the direction given by each source; a disagreement counts as one unit. This is summed across every pair and every source.
Fourth, selecting the lowest total. The candidate ranking with the lowest total disagreement is declared the consensus ranking. If more than one candidate shares the same lowest total, all are considered equally good.
The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
KEMENY-YOUNG's output is a complete ranking; every alternative's place in it is the position that, in total, least conflicts with every source's pairwise preferences. The ranking itself carries no percentage or probability; it says only that this arrangement upsets the sources, taken together, the least. A small total disagreement means the sources already show a strong consensus; a large one means there are serious differences of opinion among the sources, and it should not be forgotten that a single ranking cannot fully represent that disagreement.
If a Condorcet winner exists, KEMENY-YOUNG always places it first; this complements CONDORCET, which can say "no winner" in some cases, since KEMENY-YOUNG always produces a complete ranking regardless. But more than one ranking can share the same lowest total disagreement; in that case KEMENY-YOUNG has no single result, only several equally good ones.
Thus instead of writing:
"The ranking KEMENY-YOUNG found is objective truth"
the report should read:
"This ranking is the arrangement that upsets all the sources least, taken together; the larger the total disagreement, the more serious the difference of opinion among the sources"
Data Type and Inputs
KEMENY-YOUNG works with crisp data: every source's rank information is an integer. DecisionMind carries no fuzzy, grey or similar extension of this method; it works only with crisp rank data.
You need a table in which every source ranks every alternative completely. A minimum of two alternatives and two ranking sources is required. Because the number of possible complete rankings grows with the factorial of the number of alternatives, over forty thousand at eight alternatives, the method is NP-hard (Hemaspaandra, Spakowski and Vogel, 2005) and can be computed exactly in practice for up to about eight alternatives; larger sets require heuristic search methods. KEMENY-YOUNG neither requires nor produces weights.
When to Use It, When Not To
If you have more than one ranking source, the number of alternatives is small (eight or fewer is ideal), and you want a single, consistent consensus ranking, KEMENY-YOUNG is appropriate. It always gives a complete ranking regardless of whether a Condorcet winner exists, and it builds that ranking so as to upset all the sources least, taken together.
It should not be used where the number of alternatives is large; exact computation becomes practically impossible beyond eight. Also, if only a single winner is needed and a complete ranking is not required, lighter methods may be preferable.
Small number of alternatives, want a complete and consistent consensus ranking → KEMENY-YOUNG
Large number of alternatives, a fast net score is enough → COPELAND
Only want to find the winner, a complete ranking is not essential → DODGSON
Want to see clearly when there is no winner, without optimisation → CONDORCET
Strengths
KEMENY-YOUNG's greatest strength is its axiomatic foundation; Young and Levenglick (1978) showed that this rule is the only aggregation method that jointly satisfies desirable properties such as neutrality, consistency and the Condorcet principle. If a Condorcet winner exists, KEMENY-YOUNG always finds it; unlike CONDORCET and COPELAND, it always produces a complete and consistent ranking whether or not a cycle exists. Because the method weighs all of the sources' pairwise preferences at once, it draws on richer information than methods that look only at direction, such as CONDORCET and COPELAND.
Weaknesses
KEMENY-YOUNG's core limitation is computational burden; finding the winning ranking exactly is an NP-hard problem, and the number of possible rankings explodes as the number of alternatives grows (Hemaspaandra, Spakowski and Vogel, 2005). Second, more than one complete ranking can share the same lowest total disagreement; in that case the method gives not a single result but several equally good ones, and an additional criterion is needed to choose among them. Third, the total disagreement figure shows the strength of consensus among the sources, but this figure also grows as the number of sources grows, so analyses of different sizes cannot be compared directly.
Common Mistakes
The most common mistake is trying to run KEMENY-YOUNG exactly when the number of alternatives is large; at this scale a switch to heuristic approximate methods is required. A second mistake is reading the total disagreement figure as a percentage or an error rate; this figure only counts how many pairwise preferences were violated. A third mistake is picking one of several equally optimal rankings without justification and presenting it as the one correct result. A fourth mistake is allowing some sources to rank the alternatives incompletely; every source must rank every alternative.
The governing principle is this:
The ranking KEMENY-YOUNG gives is the arrangement that upsets all the ranking sources least, taken together; the larger the total disagreement, the more serious the difference of opinion among the sources, and the report must show this.
Cases
Each case opens with a table of more than one ranking source, describes in words what the method does to it, and shows how to read the result. The first case is drawn from the method's founding source's axiomatic formulation. The third case shows concretely, with real figures, how KEMENY-YOUNG produces a complete ranking in a cyclical case, unlike CONDORCET and COPELAND.
1. E-commerce: Product prioritisation from three ranking signals (Young and Levenglick, 1978)
On an e-commerce platform, three separate ranking signals, sales volume, customer rating and stock turnover rate (S1, S2, S3), have each ranked three products (A1, A2, A3); the three signals carry equal weight.
| Product | S1 | S2 | S3 |
|---|---|---|---|
| A1 | 1 | 2 | 1 |
| A2 | 2 | 1 | 3 |
| A3 | 3 | 3 | 2 |
The method counts, for each of the six possible complete rankings (from A1-A2-A3 to A3-A2-A1), how many total points it conflicts with the pairwise preferences of the three signals. For the ranking A1, A2, A3: on the A1-A2 pair, two signals put A1 ahead and one signal puts A2 ahead, and because this ranking agrees with the majority it carries a low share of conflict. On the A1-A3 pair, all three signals put A1 ahead, so there is no conflict at all. On the A2-A3 pair, two signals put A2 ahead and one puts A3 ahead, again a low share of conflict. The total disagreement is two; none of the other five possible rankings gives as low a total.
| Product | Rank |
|---|---|
| A1 | 1 |
| A2 | 2 |
| A3 | 3 |
This ranking also coincides with the direction the majority prefers in all three pairwise comparisons; that is, a Condorcet winner exists here too, and KEMENY-YOUNG has placed it first.
The platform hesitates here: the difference between A1 and A2 rests on a single signal's opinion (S2) alone. Had a different signal been chosen for S2, return rate, say, the total disagreement calculation could change, and A1 and A2 could swap places.
In the report: "The arrangement that minimises total disagreement across the three signals' rankings is A1, A2, A3; the total disagreement is two, and the difference between A1 and A2 rests on a single signal's opinion."
Source: Young and Levenglick (1978), the axiomatic foundation of Kemeny's (1959) idea. This example is DecisionMind's validation example for the KEMENY-YOUNG engine; it was constructed for illustrative purposes and is not a case tied to a specific page number in the paper.
2. Food Safety: Three inspection units' supplier ranking
At a food safety authority, three inspection units (D1, D2, D3) will rank four suppliers (T1, T2, T3, T4). With four alternatives, the number of possible complete rankings is twenty-four; KEMENY-YOUNG evaluates all of them and selects the one with the lowest total disagreement. Suppose all three units place T1 among the top three but disagree about the order of T2, T3 and T4; the result places T1 first, and the order of the remaining three is settled by whichever arrangement least disturbs the consensus among the units.
The authority runs into a limit here: even at four alternatives the number of possible rankings has already grown quickly; with eight suppliers this number would exceed forty thousand, pushing exact computation past practical limits. This shows that KEMENY-YOUNG is powerful at small scale but costly at large scale.
In the report: "In the three inspection units' rankings, T1 has settled into first place; the order of the remaining three suppliers has been determined by the arrangement that minimises the units' total disagreement."
3. Telecom: Seven regional managers' infrastructure supplier preference (a cyclical example)
A telecom operator's seven regional managers will choose among three infrastructure suppliers (T1, T2, T3). Three managers prefer T1 to T2 and T2 to T3. Two managers prefer T2 to T3 and T3 to T1. Two more prefer T3 to T1 and T1 to T2.
T1 beats T2 5-2. T2 beats T3 5-2. T3, however, beats T1 4-3. In the comparable example on the CONDORCET card, this cycle gives a "no winner" result, and in the comparable example on the COPELAND card all three suppliers receive an equal net score. KEMENY-YOUNG calculates the total disagreement of each of the six possible complete rankings one by one: the total disagreement of the ranking T1, T2, T3 is eight; this is not merely no higher than the other five possible rankings, it is the lowest, and uniquely so.
| Ranking | Total disagreement |
|---|---|
| T1, T2, T3 | 8 |
| T2, T3, T1 | 10 |
| T3, T1, T2 | 10 |
| T1, T3, T2 | 11 |
| T2, T1, T3 | 11 |
| T3, T2, T1 | 13 |
Here KEMENY-YOUNG finds a single, consistent ranking, T1, T2, T3, in a table where CONDORCET says "no winner" and COPELAND says "all three are equal." The reason is that T1's superiority over T2 and T2's superiority over T3 both come by a wide majority (five to two), while T3's superiority over T1 comes by a narrow majority (four to three); KEMENY-YOUNG breaks the cycle by taking this margin difference into account.
The regional managers hesitate here: the difference between the ranking T1, T2, T3 and the second-best ranking (T2, T3, T1) is only two units of total disagreement. This small gap shows that the ranking could change if a few managers changed their opinion.
In the report: "The seven regional managers' preferences are cyclical; nonetheless, the single arrangement that minimises total disagreement is T1, T2, T3, and this arrangement carries only two units less disagreement than the second-best arrangement."
Source: this case was constructed to show how Kemeny's (1959) and Young and Levenglick's (1978) idea behaves in the face of a cycle, using an example with seven sources; the figures were calculated for this card.
4. What Not to Do
Presenting the first case's total disagreement figure, two, as an error percentage is wrong; this figure only counts how many pairwise preferences were violated. A second error is missing the cycle in the third case and declaring that, according to KEMENY-YOUNG, T1 is definitely first, beyond dispute; the difference between the ranking T1, T2, T3 and the second-best ranking is only two units, and the report must state this. A third error is running KEMENY-YOUNG exactly at more than eight alternatives and presenting the result as a fast computation; at this scale the number of possible rankings rises far beyond what is practically computable.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/kemeny-young
Kemeny, J. G. (1959). Mathematics without numbers. Daedalus, 88, 577-591. (no DOI)
Young, H. P., & Levenglick, A. (1978). A consistent extension of Condorcet's election principle. SIAM Journal on Applied Mathematics, 35(2), 285-300. DOI: 10.1137/0135023
Hemaspaandra, E., Spakowski, H., & Vogel, J. (2005). The complexity of Kemeny elections. Theoretical Computer Science, 349(3), 382-391. DOI: 10.1016/j.tcs.2005.08.031