  We consider the allocation of $m$ balls (jobs) into $n$ bins (servers). In
the standard Two-Choice process, at each step $t=1,2,\ldots,m$ we first sample
two randomly chosen bins, compare their two loads and then place a ball in the
least loaded bin. It is well-known that for any $m\geq n$, this results in a
gap (difference between the maximum and average load) of $\log_2\log
n+\Theta(1)$ (with high probability).
  In this work, we consider Two-Choice in different settings with noisy load
comparisons. One key setting involves an adaptive adversary whose power is
limited by some threshold $g\in\mathbb{N}$. In each step, such adversary can
determine the result of any load comparison between two bins whose loads differ
by at most $g$, while if the load difference is greater than $g$, the
comparison is correct.
  For this adversarial setting, we first prove that for any $m \geq n$ the gap
is $O(g+\log n)$ with high probability. Then through a refined analysis we
prove that if $g\leq\log n$, then for any $m \geq n$ the gap is
$O(\frac{g}{\log g}\cdot\log\log n)$. For constant values of $g$, this
generalizes the heavily loaded analysis of [BCSV06, TW14] for the Two-Choice
process, and establishes that asymptotically the same gap bound holds even if
load comparisons among "similarly loaded" bins are wrong. Finally, we
complement these upper bounds with tight lower bounds, which establish an
interesting phase transition on how the parameter $g$ impacts the gap.
  The analysis also applies to settings with outdated and delayed information.
For example, for the setting of [BCEFN12] where balls are allocated in
consecutive batches of size $b=n$, we present an improved and tight gap bound
of $\Theta(\frac{\log n}{\log\log n})$. This bound also extends for a range of
values of $b$ and applies to a relaxed setting where the reported load of a bin
can be any load value from the last $b$ steps.
