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Reciprocal rank fusion formula

AUTHOR · XERJ documentation team · REVIEWED BY · XERJ engineering team · PUBLISHED · 2026-11-13 · UPDATED · 2026-11-14
SHORT ANSWER · UPDATED 2026-11-14

What is the reciprocal rank fusion formula? For each list an item appears in, add 1 / (k + rank), then add those values. The lists do not need a shared score scale. Cormack, Clarke and Buettcher found k = 60 near-optimal in their 2009 pilots. XERJ uses k = 60 by default and accepts another value.

TL;DR Reciprocal rank fusion merges two or more ranked lists by adding 1 / (k + rank) for each list an item appears in. The lists do not need a shared score scale.

Try it: a labeled example

Fusing two short ranked lists

Example only, not a measurement. Two lists of three documents, with k = 60, the constant Cormack, Clarke and Buettcher found near-optimal in their 2009 pilots. Score = 1 / (k + rank), then add.

DocumentRank in list ARank in list BRRF score
Document X121/61 + 1/62 = 0.0325
Document Y211/62 + 1/61 = 0.0325
Document Z3(missing)1/63 = 0.0159

X and Y tie at the top. Z appears in only one list, so it ranks lower. A real corpus will not look like this three-row example.

The formula and the choice k = 60 come from Cormack, Clarke and Buettcher, Reciprocal Rank Fusion outperforms Condorcet and individual Rank Learning Methods, SIGIR 2009. The three documents are a teaching example, not a XERJ run.
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How it works

You start with two or more ranked lists, for example a BM25 list and a vector list. For each document, look up its rank in each list. Add 1 / (k + rank) for every list it appears in. Missing from a list means that list adds nothing. Sort by the total.

The table above is a labeled example with three documents and k = 60. It is not a measurement on a XERJ corpus. In that example, a document that ranks first in one list and second in the other ties with a document that does the reverse. A document that appears in only one list ranks lower.

k is a constant that softens the gap between early and later ranks. The 2009 paper reports k = 60 as near-optimal in the authors' pilots.

The formula

The score for a document is the sum, over every ranked list it appears in, of:

score(d) = 1 / (k + rank_i(d))

rank_i(d) is the 1-based rank of d in list i. If the document is missing from a list, that list adds nothing. k is a constant. A larger k flattens the gap between rank 1 and later ranks. In their 2009 pilots, Cormack, Clarke and Buettcher found k = 60 near-optimal. That value is also the published XERJ default.

The table above is a labeled example, not a measurement on a XERJ corpus. Here is the same example in Python:

# Example only, not a XERJ measurement.
# k = 60 comes from Cormack, Clarke and Buettcher, SIGIR 2009.

def rrf_score(ranks, k=60):
    return sum(1.0 / (k + rank) for rank in ranks)

# Document X ranks 1 in list A and 2 in list B:
print(round(rrf_score([1, 2]), 4))  # 0.0325

How XERJ uses it

XERJ uses reciprocal rank fusion to merge a lexical arm with a vector arm in hybrid search. The product default is k = 60, and the request can set another value. A weighted linear sum is also available when you want to keep raw scores, after normalization.

The practical guide is when to use reciprocal rank fusion. The request shape is in how to run hybrid search in XERJ.

History and research

The method was introduced by Cormack, Clarke and Buettcher, Reciprocal Rank Fusion outperforms Condorcet and individual Rank Learning Methods, SIGIR 2009, pages 758 to 759. An ACM page also lists the paper at doi 10.1145/1571941.1572114.

The k = 60 figure on this page is theirs, from those pilots. It is also the published XERJ default. This page does not add a new fusion experiment.

Guides

See when to use reciprocal rank fusion, how XERJ combines search and the hybrid search recipe.

Related terms

Hybrid search, BM25, and semantic search.

FAQ

What is the reciprocal rank fusion formula?

For each list an item appears in, add 1 / (k + rank). Then add those values. A common choice of k is 60, from the 2009 paper.

Why use ranks instead of raw scores?

BM25 scores and vector similarities sit on different scales. Ranks do not, so you can fuse the lists without a shared unit.

What does the constant k do?

A larger k flattens the gap between rank 1 and later ranks. In their 2009 pilots, Cormack, Clarke and Buettcher found k = 60 near-optimal. XERJ uses k = 60 by default and accepts another value.

How does XERJ use RRF?

As the usual fusion method in hybrid search. The how-to, and when to pick a weighted sum instead, is the hybrid search answer and the when-to-use RRF answer.

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