The quasispecies regime for the simple genetic algorithm with roulette-wheel selection

We introduce a new parameter to discuss the behavior of a genetic algorithm. This parameter is the mean number of exact copies of the best fit chromosomes from one generation to the next. We argue that the genetic algorithm should operate efficiently when this parameter is slightly larger than $1$. We consider the case of the simple genetic algorithm with the roulette--wheel selection mechanism. 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 start the genetic algorithm with an initial population whose maximal fitness is equal to $f_0^*$ and whose mean fitness is equal to ${\overline{f_0}}$. We show that, in the limit of large populations, the dynamics of the genetic algorithm depends in a critical way on the parameter $\pi \,=\,\big({f_0^*}/{\overline{f_0}}\big) (1-p_C)(1-p_M)^\ell\,.$ Our results suggest that the mutation and crossover probabilities should be tuned so that, at each generation, $\text{maximal fitness} \times (1-p_C) (1-p_M)^\ell > \text{mean fitness}$.