Overview

This note is mostly based on three books below. When necessary, I provide additional references in the last section.

1. Li, Hang. “Learning to Rank for Information Retrieval and Natural Language Processing, Second Edition.” Learning to Rank for Information Retrieval and Natural Language Processing, Second Edition (2014).
2. Liu, Tie-Yan. “Learning to rank for information retrieval.” Proceedings of the 33rd international ACM SIGIR conference on Research and development in information retrieval (2009): n. pag.
3. [2010.06467] Pretrained Transformers for Text Ranking: BERT and Beyond (Lin et al.)

Rank Aggregation

Suppose there are M queries and N documents, there will be a ranking list for each of n queries. The goal is to aggregate these n ranking lists into one ranking list.

The simplest rank aggregation method is called Borda count. The Borda count algorithm operates on the ranking lists by the following steps:

• Step 1: Aligning ranking list of ranks by document indexes.
• Step 2: Using the total document number N to subtract each entry in the aligned ranking list.
• Step 3: Summing up the transformed ranking lists and generating a ranking based on this summed ranking list.

For example, the lists A, B, C, A, C, B and B, A, C:

• Step 1: After alignment by index A, B, and C, the ranking lists of ranks become 1, 2, 3, 1, 3, 2, and 2, 1, 3.
• Step 2: Using N=3 to subtract each entry gives us 2, 1, 0, 2, 0, 1, and 1, 2, 0.
• Step 3: The summed ranking list of ranks is 5, 3, 1. Therefore, the initial 3 ranking lists is converted to one single ranking list: A, B, C.

This could be easily implemented in Python as following:

from collections import defaultdict

def borda_count(votes):
    N = len(votes[0])
    score_dict = defaultdict(int)

    for vote in votes:
        for rank, candidate in enumerate(vote):
            score_dict[candidate] += N - rank

    aggregated_ranks = sorted(score_dict.keys(), key=score_dict.get, reverse=True)
    return aggregated_ranks


votes = [["A", "B", "C"], ["A", "C", "B"], ["B", "A", "C"]]
print(borda_count(votes))

Reference