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I wanted to check the definition of Discounted Cumulative Gain (DCG) measure in the original paper Jarvelin and it seems it differs from the one given in the later literature Wang. Originally, for $n$ documents ranked from $r = 1, \ldots, p$, the $\text{DCG}_p$ is defined as $$\text{DCG}_p = \sum\limits_{r=1}^{b} G_r + \sum\limits_{r=b}^{p}\frac{G_r}{\log_br},$$ where $G_i$ is the relevance (or gain) of the $i$-th document. So the measure depends on the logarithm base $b$. For ranks below $b$, i.e. $r<b$, gains are not penalized. If $b=2$, then we can write: $$\text{DCG}_p = G_1 + \sum\limits_{r=2}^{p}\frac{G_r}{\log_2 r}.$$ It does not look the same as the one given on wikipedia, where the argument of the logarithm is shifted by $1$: $$\text{DCG}_p = G_1 + \sum\limits_{r=2}^{p}\frac{G_r}{\log_2(r+1)}.$$

Where does this change come from? Why others use different metric?

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I believe you are correct, that the paper and Wikipedia disagree. The paper's formula suggests you apply no discount at $r <= b$, which means both of the first two elements are not discounted.

The Wikipedia formula discounts the second element onward.

There's an impassioned statement in Talk about why the Wikipedia formula is right: https://en.wikipedia.org/wiki/Talk:Discounted_cumulative_gain

But I can't see why; it offers no reference other than observing that "it seems plainly wrong to not discount from the second element." I'll comment there.

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  • $\begingroup$ Thanks @Sean Owen. In a book "Learning to Rank for information retrieval" by Tie-Yan Liu, he defines the discounted cumulative gain differently, probably following Burges et al.. It is not only about lack of penalization but also about the amount. In the first formula the last terms is $G_p/\log_2p$, whereas in the second: $G_p/\log_2(p+1)$. Generally, I don't see the point of citing a paper while using different formula (maybe the concept counts). $\endgroup$ – WoofDoggy Aug 10 '18 at 19:21

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