We study the simple genetic algorithm with a ranking selection mechanism (linear ranking or tournament). We denote by
ℓ
\ell
the length of the chromosomes, by
m
m
the population size, by
p
C
p_C
the crossover probability and by
p
M
p_M
the mutation probability. We introduce a parameter
σ
\sigma
, called the strength of the ranking selection, which measures the selection intensity of the fittest chromosome. We show that the dynamics of the genetic algorithm depends in a critical way on the parameter
\[
π
=
σ
(
1
−
p
C
)
(
1
−
p
M
)
ℓ
.
\pi \,=\,\sigma (1-p_C)(1-p_M)^\ell \,.
\]
If
π
>
1
\pi >1
, then the genetic algorithm operates in a disordered regime: an advantageous mutant disappears with probability larger than
1
−
1
/
m
β
1-1/m^\beta
, where
β
\beta
is a positive exponent. If
π
>
1
\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
∗
p^*
(which does not depend on
m
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:
∙
\bullet
π
=
σ
(
1
−
p
C
)
(
1
−
p
M
)
ℓ
\pi =\sigma (1-p_C)(1-p_M)^\ell
should be slightly larger than
1
1
;
∙
\bullet
p
M
p_M
should be of order
1
/
ℓ
1/\ell
;
∙
\bullet
m
m
should be larger than
ℓ
ln
ℓ
\ell \ln \ell
;
∙
\bullet
the running time should be at most of exponential order in
m
m
.
The first condition requires that
ℓ
p
M
+
p
C
>
ln
σ
\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.