The quasispecies regime for the simple genetic algorithm with ranking selection

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.