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Theorem 5.3 rewritten in the terms of the genetic plan's allocation of trials, Nx* and n', noting that r'q(Nx*, n') is an upper bound on q(n1, . . ., nr).) It is critical to what follows that r' ·n' need not be equal to nx*. As  is transformed into  by the genetic plan, each schema x having instances in can be expected to have  instances in . Thus, over the course of several time-steps, the number of schemata r' receiving n' trials will be much, much greater than the number of trials allocated to individuals  , even when n' approaches or exceeds nx*. This observation, that generally r'·n' >> nx*, is an explicit consequence of the genetic plan's intrinsic parallelism. |
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With these observations for guidance, we can establish that the losses of genetic plans are decreased by a factor 1/(r' - 1) in comparison to the losses under optimal allocation. Specifically, we have |
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THEOREM 7.4: If r' is the number of schemata for which |
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i.e., if r' is the number of schemata for which the number of trials n' increases at least proportionally to nx*,0, then for any performance function  , |
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as N ® ¥, where the parameters are defined as in Lemma 7.3. |
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Proof: Substituting the expression for Nx* (from Lemma 7.3) and the expression for q(Nx*, n') (from the proof of Theorem 5.1) in , and noting that (1 - r'q(Nx*, n')) nx* < nx*, gives |
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If  , it is clear that the first term (the exponential term) decreases as nx*,0 increases, but the second term, nx*,0, increases. In other words, if  i.e., if n' increases at least proportionally with nx*,0, the expected loss per trial will soon depend almost entirely on the second term. We have already seen (in the proof of Corollary 5.2) that the same holds for the second term of the expression for expected loss under an optimal allocation of N trials. Thus, for r' and n' as specified, the ratio of upper bound on the reproductive plan's losses to the lower bound on the optimal allocation's losses approaches |
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