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The main point of this theorem quickly becomes apparent if we rearrange the results to give the number of trials N*(1)allocated to x(1) as a function of the number of trials n* allocated to x(2): |
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Thus the loss rate will be optimally reduced if the number of trials allocated x(1) grows slightly faster than an exponential function of the number of trials allocated x(2). This is true regardless of the form of the distributions defining x1 and x2. Later we will see that the random variables defined by schemata are similarly treated by reproductive plans. |
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It should be emphasized that the above approximation for n* will be misleading for small N when |
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(i)µ1 - µ2is small enough that, for small N,the standard deviation of  is large relative to µ1 - µ2 and, as a consequence, the approximation for the tail 1- F(x0) fails, |
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or |
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(ii) s2 is large relative to s1 so that, for small N,the approximation for x0 is inadequate. |
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Neither of these cases is important for our objectives here. The first is unimportant because the cumulative losses will be small until N is large since the cost of trying x2 is just µ1 - µ2.The second is unimportant because the uncertainty and therefore the expected loss depends primarily on x1 until N - n* is large; hence the expected loss rate will be reduced near optimally as long as N - n @ N (i.e., most trials go to x(1)), as will be the case if n is at least as small as the value given by the approximation for n*. |
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Finally, to get some idea of n* when s1 is not known, note that for bounded payoff, , the maximum variance occurs when all payoff is concentrated at the extremes, i.e., p(r0) = p(r1) = ½. Then |
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2. Realization of Minimal Losses |
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This section points out, and resolves, a difficulty in using L*(N)as a performance criterion. The difficulty occurs because, in a strict sense, the minimal expected loss rate just calculated cannot be obtained by any feasible plan for allocating trials in terms of observations. As such L*(N)constitutes an unattainable lower bound and, |
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