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Proof: This is immediately established by noting that, when p1 and p2 are constant, the expected lifespan of A is 1/p2 and the expected number of offspring is simply the number of offspring expected during the expected lifespan, i.e., p1/p2. In more detail, the probability of A surviving for exactly T time-steps is p(T)= (1 - p2)T-1 ·p2, and the expected number of ''offspring" during that interval is  . Thus, the expected number of "offspring" during A's lifespan is |
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But  converges to (1/p2)2 (as may be easily verified by taking the derivative of both sides of the identity (1/1 - x) = 1 + x + x2 + . . . ). Therefore |
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For plans in  the interpretation of this lemma is direct: The probability of Ah being selected to produce an offspring A' during time t is  where  , while the probability of Ah being deleted at the end of that time-step is 1/M. Hence, if  changes negligibly over Ah's lifespan, the expected number of offspring is |
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 can be looked upon as a "normalized" payoff, the "usefulness" of Ahbeing measured relative to the average performance of the other members in the population. With this arrangement the expected number of offspring of Ah is greater than 1 just in case Ah's performance is above average. Since  is not stationary for plans  , the probability p1 does not in fact remain constant (though, over the expected lifespan of a structure, it will not often change greatly). If increases (as it will generally with a good plan), then Ah will receive fewer offspring than predicted by the calculation of p1 at the time Ah originated. That is, the performance of Ah looks less promising relative to the current average, so trials of Ah are curtailed. If decreases, the opposite effect occurs. Still, the expected number of offspring varies in direct relation to Ah's relative performance, so that plans in satisfy the (informal) characterization of reproductive plans. |
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A slight change in the form of the algorithms in yields a class of algorithms Rd wherein a time-step is a "generation" during which each individual  is replaced, deterministically instead of as an expectation, by  offspring. Thus, for Rd,  consists of the set of all offspring of the individuals in  . (To keep the population level at M individuals a special kind of rounding |
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