Weighted Voting (Combining Several Rankings by Borda Score)

Weighted voting is a combination (consensus) method used when what you hold is not a decision matrix but several rankings given by several people or sources. Every source that provides a ranking, whether an expert, a jury member, a criterion, or the output of another method, is given a reliability weight. Every alternative within every ranking receives a score based on its position: the top-ranked alternative gets the highest score, the last-ranked gets zero. These scores are multiplied by the source's weight and summed, and the alternatives are re-ranked by this total score. The method is a weighted form of the Borda count, drawn from voting theory and social-choice theory.

The idea behind weighted voting is to use richer information than the question "who received the most first-place votes." Counting only first-place votes (plurality voting) ignores an alternative's wide support at second or third place. The Borda count takes this information into account by converting position at every rank into a score. Weighting, in turn, acknowledges that not every source is equally reliable: one expert's view may carry more weight than another's, depending on experience or relevance to the subject. Its philosophical consequence is that the method carries a consensus logic; it does not look for a single "most correct" view but for the weighted average of several views.

The method proceeds through four steps.

First, gather the rankings and the weights. Every source's complete ranking and the weight given to that source are brought together; it is checked that the weights sum to 1 and are not negative.

Second, assign Borda scores. In every ranking, if there are m alternatives, the top-ranked alternative gets m minus 1 points, the second-ranked gets m minus 2 points, and so on down to zero for the last.

Third, the weighted sum. Every alternative's Borda scores from all sources are multiplied by that source's weight and summed.

Fourth, the final ranking. Alternatives are ordered by this weighted total score from highest to lowest. In the event of a tie, whichever received more first-place votes is placed ahead.

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 weighted Borda score summarises an alternative's relative support across all the sources; this score is a summary only of this particular group of sources and these weights. A high score does not mean "everyone's first preference"; an alternative placed near the top by many sources but first by none of them can score higher in total than an alternative that is strong on one point but weak on another. This is the nature of the method: it favours broad, consistent support over narrow but intense support. If one source gives a ranking very different from the others, this difference dissolves into the weighted average and should be noted separately in the report.

Thus instead of writing:

"Weighted voting revealed the group's true preference"

the report should read:

"The weighted Borda average of the rankings given by these sources at these weights is as follows; if the sources disagree, this average reflects none of them exactly"

Weighted voting works with crisp data, but it asks not for a decision matrix, but for ranking data: a complete list from every source, ordering all the alternatives, and a weight given to every source. Sources can be several experts, the members of a jury, or the individual rankings produced by different criteria. DecisionMind's engine, given a decision matrix and criterion weights, can treat the ranking within each criterion's own column as a "source's vote" and combine these votes using the criterion weight; this means the same mathematics can also be applied to the rankings of several real experts. Weighted voting does not itself produce weights, but it also requires no mandatory external weight source; here, the weights are the sources' reliability and are set directly by the decision-maker. A minimum of two sources and two alternatives is required, and every source is expected to have ranked all the alternatives.

Weighted voting is appropriate when you want to reduce the individual rankings of several experts, jury members, or criteria into a single shared ranking, and want to reflect these sources' reliability with different weights. Typical uses include combining election results, reducing several jury members' final rankings to a single list, and bringing together (ensembling) the rankings produced by several methods.

If what you hold is a raw numerical decision matrix and you want to rank it with a single, consistent measurement logic, a direct ranking method (TOPSIS, SAW, VIKOR) is more suitable; weighted voting is for combining rankings that already exist. If one source gives a ranking that is very different from, and internally inconsistent with, the others, that source needs to be questioned separately before being included in the average.

I hold several complete rankings and want to reduce them to a single shared ranking → weighted voting

I hold a raw numerical decision matrix and want to rank it with one consistent logic → TOPSIS, SAW, VIKOR

No agreement can be reached among the sources, a consensus process is needed → consensus-building methods (Delphi-like)

The source of the weights is unclear, weights are needed first → AHP, BWM, SWARA (subjective)

Weighted voting's greatest strength is that it can reduce several sources' views into a single list without losing information; unlike methods that count only first-place votes, it takes support at every rank into account. Weighting means sources of differing reliability need not be treated as equal. The method is mathematically simple, its computational burden is low, and the result can be followed at every step.

Borda-type scoring carries a known limitation: adding or removing an alternative from the ranking can change the relative scores of the remaining alternatives (Young, 1974). The method tends to favour broad, moderate support over narrow but strong support; this can be an undesirable outcome in some decision contexts. How the sources' weights were determined lies outside the method itself; a correctly computed combination built on poorly set weights still produces a biased result. If one source is using a systematically different ranking logic (for instance, has not assessed some alternatives at all), this may go unnoticed in the weighted average.

The most common mistake is giving the sources equal weight without justification; equal weight is a claim that all sources are equally reliable, and this claim must be defended in the report. A second mistake is submitting a source's ranking incomplete or partial; every source is expected to rank all the alternatives, and an incomplete ranking distorts the Borda scores. A third is reading the weighted Borda score as a "total vote count" and confusing it with an actual election result. A fourth is including a source that clearly gives a deviant ranking in the average at the same weight as the others without questioning it.

The governing principle is this:

The result of weighted voting is a summary only of the sources, rankings and weights you supplied; if one source is contested, the shared ranking is contested too.