Why aggregate preferences
There are n alternatives — options, objects, values — and m “opinions” about them: experts, criteria, measuring instruments. Each opinion orders the alternatives from best to worst. We need one common order — the consensus ranking — that agrees best with all opinions at once.
This is how multi-criteria and group decisions, artificial intelligence and machine learning tasks, and the analysis of heterogeneous data work: properties of different physical nature cannot be added, but each of them can compare objects.
Example: assessing a microclimate
A production site is divided into five zones a1, …, a5. The zones are ranked by four heterogeneous properties — humidity, temperature, air speed and presence of hazardous equipment. Humidity cannot be added to temperature, but zones can be ordered by each property. The Borda rule gives the final ranking a3 ≻ a4 ≻ a5 ≻ a2 ≻ a1. The Kemeny rule has two equally good solutions here — the same ranking and a4 ≻ a3 ≻ a5 ≻ a2 ≻ a1: an example of the multiplicity of solutions discussed below. The example is loaded into the demo.
Rankings and profile
A ranking λk orders the alternatives and allows ties: ai ≻ aj means “ai is preferred”, ai ~ aj means “equivalent” (tolerance). Such a relation is a weak order. A set of m rankings is a preference profile Λ(m, n).
A profile is conveniently condensed into the tournament matrix S = [sij]. Every ranking in which ai is preferred to aj adds 2 to sij, a tie adds 1; so all entries are integers:
sii = 0; sij + sji = 2m for i ≠ j.
Aggregation rules
How to get the consensus ranking is set by an aggregation rule. There are many rules, and they rest on different ideas of what “agreement” means.
Simple majority
The winner is the alternative ranked first most often. Simple, but it looks only at the top of each ranking and ignores the rest.
Borda rule
Each alternative gets the score zi = Σj ≠ i sij — the row sum of the tournament matrix; the ranking is by decreasing score. Polynomial complexity.
Jean-Charles de Borda (1733–1799)
Condorcet rule
The winner beats every other alternative in pairwise comparison: sij > sji, i.e. sij > m, for all j ≠ i. The most axiomatically justified rule, but a winner may not exist — the Condorcet paradox: a1 ≻ a2, a2 ≻ a3, a3 ≻ a1.
Marquis de Condorcet (1743–1794)
Kemeny rule
The consensus ranking is the strict order β closest to the whole profile in Kemeny distance: β = arg minρ D(ρ, Λ), the minimum taken over all strict orders ρ. If a Condorcet winner exists, it comes first in β; but unlike the Condorcet rule, an answer always exists — there is no paradox.
John G. Kemeny (1926–1992)
The Kemeny distance between rankings counts disagreeing pairs. For a ranking λ let σij(λ) = 1 if ai ≻ aj; 0 if ai ~ aj; −1 if aj ≻ ai. Then
A pair in the opposite order adds 2 to the distance, a tie against a strict preference adds 1.
The Kemeny rule has two difficulties:
- NP-hardness. The number of strict orders grows as n!. The team developed RECURSALL, a recursive branch-and-bound algorithm that finds all exact solutions for n < 20.
- Multiple solutions. There may be several exact solutions at the same distance from the profile. A convolution rule merges them into one final ranking by the scores of the output profile — the set of all exact solutions. The result may contain ties and keeps the order of every pair on which all solutions agree.
Demo: vote yourself
Enter rankings, one per line, from best to worst: a₁ > a₂ ~ a₃ > a₄
(> — preferred, ~ — equivalent). A label before a colon is optional. The browser applies all four rules.
From rankings to intervals
IF&PA carries preference aggregation over to measurement results. The range of actual values (RAV) — from the lowest lower bound to the highest upper bound of the intervals — is split into n discrete values; these are the alternatives. Each measurement result xk ± uk — an interval — becomes an “opinion”: values inside the interval are preferred to values outside, and values within each group are tied.
Such a ranking induced by an interval is called an interval-induced ranking. These rankings have a special structure: the preferred values are always consecutive. So with n values there are exactly n(n + 1)/2 of them — as many as there are segments of consecutive values.
The profile of interval-induced rankings is aggregated. The winners are the values “preferred” by the largest number of measurement results, i.e. covered by the most intervals; their median is the preferential median. For such profiles the Kemeny and Borda rules yield the same preferential median, so the NP-hard problem reduces to a polynomial one.
Glossary
All IF&PA notation is collected in a table on the Method page.
| Alternative | an object of comparison: an option, a zone, a discrete value of a quantity |
| Ranking | an order of alternatives from best to worst with possible ties (weak order) |
| Tolerance (~) | equivalence of two alternatives in a ranking |
| Preference profile Λ(m, n) | a set of m rankings of n alternatives |
| Tournament matrix | the matrix of pairwise wins sij over the whole profile |
| Consensus ranking β | the final ranking that agrees best with the profile under the chosen rule |
| Range of actual values (RAV) | the segment from the lowest lower bound to the highest upper bound of the intervals, split into discrete values |
| Partition norm h | the step between neighbouring discrete values of the RAV |
| Interval-induced ranking | a ranking of RAV values induced by an interval: inside ≻ outside |
| Coverage ci | the number of intervals containing the value ai |
| Preferential median (PM) | the result of interval fusion: the median of the winners in the consensus ranking of the interval-induced profile |
References
- Muravyov S.V. Ordinal measurement, preference aggregation and interlaboratory comparisons. Measurement, 2013, 46(8), 2927–2935.
- Muravyov S.V. Dealing with chaotic results of Kemeny ranking determination. Measurement, 2014, 51, 328–334.
- Muravyov S.V., Khudonogova L.I., Emelyanova E.Yu. Interval data fusion with preference aggregation. Measurement, 2018, 116, 621–630. doi:10.1016/j.measurement.2017.08.045
- Muravyov S.V., Emelyanova E.Yu. Kemeny rule for preference aggregation: reducing all exact solutions to a single one. Measurement, 2021, 182, 109403.
- Muravyov S.V. et al. Kemeny and Borda rules in constructing central tendency estimators based on the preferential median. Proc. AISP'25, 2025. DOI: 10.1109/AISP68263.2025.11396113.