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On the other hand, under reproductive plans of type  , operator equilibrium is persistently destroyed by reproduction. In effect, useful linkages are preserved and nonlinearities (epistases) are exploited. Indeed, it would seem that the term "coadapted" is only reasonably used when alleles are peculiarly suited to each other, giving a performance when combined which is not simply the sum of their individual performances. Following Lemma 7.2, each coadapted set of alleles (schema) changes its proportion at a rate determined by the particular average (observed) fitness of its instances, not by the sum of the fitnesses of its component alleles. |
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(Because of the stochastic nature of the operators in genetic plans, each chromosome  has a probability of appearing in the next generation  , a probability which is conditional on the elements appearing in  . If there are enough instances of x in , the central limit theorem assures that  , where µx is the expected fitness of the coadapted set x under the given probability distribution over  . Thus the observed rate of increase of a coadapted set of alleles x will closely approximate the theoretical expectation once x gains a foothold in the population.) |
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Returning to the example just above, but now for genetic ( ) plans, we see (from Table 3) that  has a rate of change given by |
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while  changes as |
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Consequently, the coadapted set of alleles with the higher average fitness quickly predominates. Thus, when epistasis is important, plans of type (and the corresponding theorems involving schemata) provide a better hypothesis than the hypothesis of independent selection (and least mean squares estimates of the fitness of sets of alleles). |
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We see from Lemmas 7.2 and 7.3 that, under a genetic plan, a schema x which persists in the population for more than a generation or two will be ranked according to its observed performance. This is accomplished in a way which satisfies the desiderata put forth at the end of chapter 5. Specifically, the proportion of x's instances in the population will grow at a rate proportional to the |
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