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2-armed bandit problem). The reader can learn a great deal more about convergence properties of reproductive plans from N. Martin's excellent 1973 study. |
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Since convergence is not a useful guide, we must turn to the stronger "minimal expected losses" criterion introduced in chapter 5. Results there (Theorems 5.1 and 5.3) indicate that the number of trials allocated to the observed best option should be an exponential function of the trials allocated to all other options. It is at once clear that enumerative plans do not fare well under this criterion. Enumeration, by definition, allocates trials in a uniform fashion, with no increase in the number of trials allocated to the observed best at any state prior to completion; accordingly, as the number of observations increases, expected losses climb precipitously in comparison to the criterion. On the other hand, plans of type  do award an exponentially increasing number of trials to the observed best, as we shall see in a moment. More importantly, plans of this type actually treat schemata from X as options, rather than structures from  . In doing this the plans exhibit intrinsic parallelism, effectively modifying the rank of large numbers of schemata each time a structure  is tried. The effect is pronounced, even in an example as simple as that of Figure 13, which illustrates 2 generations of a small population (M = 8, l = 9) undergoing reproduction and crossover. Specifically, under plans of type the number of instances of a schema increases (or decreases) at a rate closely related to its observed performance  at each instant. That is, the portion Mx(t) of each schema x represented in population  changes simultaneously according to an equation much like that suggested at the end of chapter 5: |
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The foregoing statements can be established with the help of |
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LEMMA 7.2: Under a plan of type , given Mx(t0) instances of x in the population  at time t0, the expected number of instances of x at time t, Mx(t), is bounded below by |
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a time-invariant constant generally close to zero, depending only upon the parameters of the plan, the length l(x) of x, and the number l0(x) of defining positions for x. |
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