  We study the top-$K$ ranking problem where the goal is to recover the set of
top-$K$ ranked items out of a large collection of items based on partially
revealed preferences. We consider an adversarial crowdsourced setting where
there are two population sets, and pairwise comparison samples drawn from one
of the populations follow the standard Bradley-Terry-Luce model (i.e., the
chance of item $i$ beating item $j$ is proportional to the relative score of
item $i$ to item $j$), while in the other population, the corresponding chance
is inversely proportional to the relative score. When the relative size of the
two populations is known, we characterize the minimax limit on the sample size
required (up to a constant) for reliably identifying the top-$K$ items, and
demonstrate how it scales with the relative size. Moreover, by leveraging a
tensor decomposition method for disambiguating mixture distributions, we extend
our result to the more realistic scenario in which the relative population size
is unknown, thus establishing an upper bound on the fundamental limit of the
sample size for recovering the top-$K$ set.
