Evolutionary Computation : A Unified Approach

6.3. SELECTION-ONLY MODELS 125
So, truncation selection leads to extremely rapid convergence. When k = 1, it takes only
one generation and when k = 2, it takes only two generations.
For finite-population models, the top k individuals can be used to produce offspring
either deterministically or stochastically. If breeding occurs deterministically, then offspring
are produced in multiples of k. For non-overlapping-generation models with m = n, this
restricts the choice of k to k = m/b where b is the “brood size”. Since only the offspring
survive, everything is deterministic, resulting in convergence to the same fixed points and
the same takeover times as the corresponding infinite population model in which the top
k/m fraction are selected.
However, if the offspring population is created from the k parents using stochasticuniform
selection, then we have the additional effects of drift to account for. Although
difficult to analyze mathematically, we can experimentally study this effect using the same
experimental setup that we used to study drift by initializing the population with m unique
genotypes and estimate the time, fixed(m), it takes for a population of size m to converge
to a fixed point. In addition, in keeping with the model assumptions, each genotype is
assigned a unique fitness (arbitrarily chosen from the interval [0,100]).
Figure 6.3 presents these results for the case of m = 50 and contrasts them with the
predicted values of the infinite population model. Notice how the effects of drift, namely
faster convergence times, increase with k, which makes sense since increasing k in the absence
of drift increases the number of generations before convergence and, as we saw in the earlier
section, drift effects increase linearly with the number of generations.
Perhaps more important to note is that, in addition to faster convergence times, the
finite population models are also converging to different (non-optimal) homogeneous fixed
points than their infinite population counterparts. Figure 6.4 illustrates this for m = 50.
Rank-Proportional Selection
Another frequently used selection scheme that is based on sorting the members of a selection
pool by fitness is one in which the probability of selecting a genotype is based linearly on
its rank in the fitness-sorted selection pool. So, rather than have all of the probability mass
assigned to the top k individuals, each genotype is assigned a selection probability, ranging
from (2/m) − ɛ for the best fit genotype, to a probability of ɛ for the worst fit. By doing
so, the total probability mass sums to 1.0 andɛ controls the slope of the linear probability
distribution by ranging from 0.0 to1/m. To be more precise, the probability of selecting a
genotype of rank i is given by:
prob(i) = ( 2
m − ɛ) − ( 2
m − 2ɛ) ∗ i−1
m−1
when the best genotype is assigned a rank of 1 and the worst a rank of m.
Clearly, when ɛ =1/m, the slope is zero and the distribution is uniform over the entire
selection pool. The more interesting case is when ɛ approaches zero producing a maximal
slope, in the sense that each genotype has a nonzero probability of being selected, but with
maximum selection differential based on rank. To understand the selection pressure induced
in this case, we again focus on p t , the proportion of the population occupied by the best
genotype at generation t. This corresponds to the set of genotypes with rank i ≤ p t ∗ m,
and so we have:
p t+1 = ∑ p t∗m
i=1
prob(i)