  A central problem in ranking is to design a ranking measure for evaluation of
ranking functions. In this paper we study, from a theoretical perspective, the
widely used Normalized Discounted Cumulative Gain (NDCG)-type ranking measures.
Although there are extensive empirical studies of NDCG, little is known about
its theoretical properties. We first show that, whatever the ranking function
is, the standard NDCG which adopts a logarithmic discount, converges to 1 as
the number of items to rank goes to infinity. On the first sight, this result
is very surprising. It seems to imply that NDCG cannot differentiate good and
bad ranking functions, contradicting to the empirical success of NDCG in many
applications. In order to have a deeper understanding of ranking measures in
general, we propose a notion referred to as consistent distinguishability. This
notion captures the intuition that a ranking measure should have such a
property: For every pair of substantially different ranking functions, the
ranking measure can decide which one is better in a consistent manner on almost
all datasets. We show that NDCG with logarithmic discount has consistent
distinguishability although it converges to the same limit for all ranking
functions. We next characterize the set of all feasible discount functions for
NDCG according to the concept of consistent distinguishability. Specifically we
show that whether NDCG has consistent distinguishability depends on how fast
the discount decays, and 1/r is a critical point. We then turn to the cut-off
version of NDCG, i.e., NDCG@k. We analyze the distinguishability of NDCG@k for
various choices of k and the discount functions. Experimental results on real
Web search datasets agree well with the theory.
