# Why isn't ROUGE-N normalized by the number of N-grams in the reference summary?

Note: I'll focus on $$ROUGE-1$$, but the same holds for $$ROUGE-N$$.

For a machine-produced summary $$M$$ and a bunch of reference summaries $$RefSummaries$$, I believe $$ROUGE-1$$ can be calculated in the following manner:

$$ROUGE\textrm{-}1 = \frac{\sum\limits_{R \in \{RefSummaries\}} \sum\limits_{unigram \: i \in R} min(count(i, M), count(i, R))}{\sum\limits_{R \in \{RefSummaries\}} \sum\limits_{unigram \: i \in R} count(i, R)}$$

For simplicity, let's suppose that there's only one reference summary with 10 unigrams (i.e. words). For some $$i$$, the maximum possible value of $$ROUGE-1$$ is 1, which makes perfect sense to me, because the count is normalized by the denominator. If we define the total value of $$ROUGE-1$$ as the sum of $$ROUGE-1$$ for all unigrams in $$R$$, it then follows that the maximum possible total value of $$ROUGE-1$$ is 10, because there are 10 unigrams, and the maximum value for each of them is 1. Why doesn't this final value get normalized by the number of unigrams, i.e. $$\frac{10}{10}=1.0$$?

When reading research papers, I've noticed that they never normalize that value either, so that's why I'm asking. For example, in the paper titled A Neural Attention Model for Abstractive Sentence Summarization by Alexander M. Rush et al., a value of 26.55 for $$ROUGE-1$$ using the ABS model is reported. The goal of that paper is to perform sentence-level summarization, and in chapter 7.1, which describes the dataset, they note that the average sentence length for the dataset is 31.3 words. If the aforementioned $$ROUGE-1$$ value were to be normalized by the average sentence length, the result would be $$\frac{26.55}{31.3} = 0.8482$$. To me, that value is much more meaningful than 26.55, especially when comparing results from different papers with different datasets.

Lest you think it's a one-off, ProphetNet: Predicting Future N-gram for Sequence-to-Sequence Pre-training by Weizhen Qi et al. didn't the normalize the value either, and reported a $$ROUGE-1$$ score of 43.68 on the CNN/DM dataset.

I'm sure that I'm missing something and that plenty of other people have thought of this before me, so why doesn't $$ROUGE-N$$ get normalized by the number of $$N-grams$$ present in the reference summary?