  We continue to investigate the race between prime numbers in many residue
classes modulo $q$, assuming the standard conjectures GRH and LI.
  We show that provided $n/\log q \rightarrow \infty$ as $q \rightarrow
\infty$, we can find $n$ competitor classes modulo $q$ so that the
corresponding $n$-way prime number race is extremely biased. This improves on
the previous range $n \geq \varphi(q)^{\epsilon}$, and (together with an
existing result of Harper and Lamzouri) establishes that the transition from
all $n$-way races being asymptotically unbiased, to biased races existing,
occurs when $n = \log^{1+o(1)}q$.
  The proofs involve finding biases in certain auxiliary races that are easier
to analyse than a full $n$-way race. An important ingredient is a quantitative,
moderate deviation, multi-dimensional Gaussian approximation theorem, which we
prove using a Lindeberg type method.
