15. Reciprocal Rank Fusion (RRF)

When you retrieve from more than one source (multi-query variants, or dense + BM25), you get several ranked lists with incompatible score scales. RRF merges them fairly by ignoring scores and using only rank.

The problem it solves

 dense scores:  0.81, 0.79, 0.77   (cosine)
 BM25 scores:   14.2, 9.6, 7.1     (totally different scale)
 → you can't just add these together

The formula

 RRF(d) = Σ   1 / (k + rank_R(d))
         lists R

• rank_R(d) = the position of document d in list R (1 = best).
• k = a small constant (commonly 60) that softens the gap between top ranks so no single list dominates.
• A document ranked highly in multiple lists accumulates a large score — RRF rewards consensus.

Code

def reciprocal_rank_fusion(result_lists, k=60):
    """result_lists: list of ranked lists of doc ids (best first)."""
    scores = {}
    for ranked in result_lists:
        for rank, doc_id in enumerate(ranked, start=1):
            scores[doc_id] = scores.get(doc_id, 0) + 1 / (k + rank)
    return sorted(scores, key=scores.get, reverse=True)

dense = ["d3", "d1", "d7", "d2"]
bm25  = ["d1", "d9", "d3", "d5"]
fused = reciprocal_rank_fusion([dense, bm25])   # d1, d3 rise — they're in both

Where it fits

 list A ─┐
 list B ─┼─▶ RRF (keep ranks, drop scores) ─▶ one fused ranking
 list C ─┘

RRF = keep ranks, drop scores, reward consensus. It's the standard fusion step in Elasticsearch, OpenSearch, Weaviate, Qdrant, and Azure AI Search.

Next: Hybrid Search →