Fix an integer $r\geq 3$. Let $q$ be a large positive integer and
$a_1,...,a_r$ be distinct residue classes modulo $q$ that are relatively prime
to $q$. In this paper, we establish an asymptotic formula for the logarithmic
density $\delta_{q;a_1,...,a_r}$ of the set of real numbers $x$ such that
$\pi(x;q,a_1)>\pi(x;q,a_2)>...>\pi(x;q,a_r),$ as $q\to\infty$; conditionally on
the assumption of the Generalized Riemann Hypothesis GRH and the Grand
Simplicity Hypothesis GSH. Several applications concerning these prime number
races are then deduced. Indeed, comparing with a recent work of D. Fiorilli and
G. Martin for the case $r=2$, we show that these densities behave differently
when $r\geq 3$. Another consequence of our results is the fact that, unlike
two-way races, biases do appear in races involving three of more squares (or
non-squares) to large moduli. Furthermore, we establish a conjecture of M.
Rubinstein and P. Sarnak (on biased races) in certain cases where the $a_i$ are
assumed to be fixed and $q$ is large. We also prove that a conjecture of A.
Feuerverger and G. Martin concerning "bias factors" (which follows from the
work of Rubinstein and Sarnak for $r=2$) does not hold when $r\geq 3$. Finally,
we use a variant of our method to derive Fiorilli and Martin asymptotic formula
for the densities in two-way races.