  We study the simple genetic algorithm with a ranking selection mechanism
(linear ranking or tournament). We denote by $\ell$ the length of the
chromosomes, by $m$ the population size, by $p_C$ the crossover probability and
by $p_M$ the mutation probability. We introduce a parameter $\sigma$, called
the selection drift, which measures the selection intensity of the fittest
chromosome. We show that the dynamics of the genetic algorithm depend in a
critical way on the parameter $$\pi \,=\,\sigma(1-p_C)(1-p_M)^\ell\,.$$ If
$\pi<1$, then the genetic algorithm operates in a disordered regime: an
advantageous mutant disappears with probability larger than $1-1/m^\beta$,
where $\beta$ is a positive exponent. If $\pi>1$, then the genetic algorithm
operates in a quasispecies regime: an advantageous mutant invades a positive
fraction of the population with probability larger than a constant $p^*$ (which
does not depend on $m$). We estimate next the probability of the occurrence of
a catastrophe (the whole population falls below a fitness level which was
previously reached by a positive fraction of the population). The asymptotic
results suggest the following rules: $\pi=\sigma(1-p_C)(1-p_M)^\ell$ should be
slightly larger than $1$; $p_M$ should be of order $1/\ell$; $m$ should be
larger than $\ell\ln\ell$; the running time should be of exponential order in
$m$. The first condition requires that $ \ell p_M +p_C< \ln\sigma$. These
conclusions must be taken with great care: they come from an asymptotic regime,
and it is a formidable task to understand the relevance of this regime for a
real-world problem. At least, we hope that these conclusions provide
interesting guidelines for the practical implementation of the simple genetic
algorithm.
