  Approaches to recommendation are typically evaluated in one of two ways: (1)
via a (simulated) online experiment, often seen as the gold standard, or (2)
via some offline evaluation procedure, where the goal is to approximate the
outcome of an online experiment. Several offline evaluation metrics have been
adopted in the literature, inspired by ranking metrics prevalent in the field
of Information Retrieval. (Normalised) Discounted Cumulative Gain (nDCG) is one
such metric that has seen widespread adoption in empirical studies, and higher
(n)DCG values have been used to present new methods as the state-of-the-art in
top-$n$ recommendation for many years.
  Our work takes a critical look at this approach, and investigates when we can
expect such metrics to approximate the gold standard outcome of an online
experiment. We formally present the assumptions that are necessary to consider
DCG an unbiased estimator of online reward and provide a derivation for this
metric from first principles, highlighting where we deviate from its
traditional uses in IR. Importantly, we show that normalising the metric
renders it inconsistent, in that even when DCG is unbiased, ranking competing
methods by their normalised DCG can invert their relative order. Through a
correlation analysis between off- and on-line experiments conducted on a
large-scale recommendation platform, we show that our unbiased DCG estimates
strongly correlate with online reward, even when some of the metric's inherent
assumptions are violated. This statement no longer holds for its normalised
variant, suggesting that nDCG's practical utility may be limited.
