  We consider the problem of ranking $n$ players from partial pairwise
comparison data under the Bradley-Terry-Luce model. For the first time in the
literature, the minimax rate of this ranking problem is derived with respect to
the Kendall's tau distance that measures the difference between two rank
vectors by counting the number of inversions. The minimax rate of ranking
exhibits a transition between an exponential rate and a polynomial rate
depending on the magnitude of the signal-to-noise ratio of the problem. To the
best of our knowledge, this phenomenon is unique to full ranking and has not
been seen in any other statistical estimation problem. To achieve the minimax
rate, we propose a divide-and-conquer ranking algorithm that first divides the
$n$ players into groups of similar skills and then computes local MLE within
each group. The optimality of the proposed algorithm is established by a
careful approximate independence argument between the two steps.
