2 Variation of Loss of Lineages

It is not possible to increase the number of genetic lineages during the evolutionary process without explicitly introducing new lineages. Genetic lineages can, however, be lost at different rates. In the above model of selection, if the fitness of the root parent is passed to its child. Since every initial individual represents a unique lineage, the next generation will contain at most $\frac{\mu}{T}$ distinct lineages. If the individuals all have unique fitness values that are passed unchanged to children, the ordering by fitness of children will be the same as the parents ordering in the previous generation. Thus, at generation $g$, the maximum number of distinct lineages will be $\frac{\mu}{T^g}$, with a minimum of $1$.

If the fitness of children is not transfered from the root-parent, it is possible to think about upper and lower bounds on the loss of lineages. To consider the upper and lower bounds of genetic lineage loss, remember that the $\frac{\mu}{T}$ members of partition $p^T$ are sorted according to their unique fitness values. The lower bound (or the least) number of lineages preserved in the next generation will occur when the children of individuals from $p^T$ have a fitness ordering consistent with their root-parent ordering. Interestingly, this also holds true when the fitness ordering is the complete opposite of the root-parent ordering. In the former case, offspring produced by an application of the operator to their root-parent undergo small fitness value change from the parent. The upper bound (or the most) number of genetic lineages preserved in the next generation from $p^T$ will be when the children are ordered randomly compared to the root-parent ordering in the previous generation.

Admittedly, this caricature of tournament selection is elitist toward the better individuals, which are found in partition $p^T$. This behaviour is emphasised to illustrate the loss of lineages, and that in a set of runs, those runs with more genetic lineages are probably performing a more parallel type of search.