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| Dubins-Savage | Adaptive Systems | | F, fortunes |  , basic structures (see t below). | | g, a probability distribution over fortunes or a gamble. | P, a probability distribution over structures, i.e.,  . | | G, a function assigning a set of gambles to each f Î F, the house. | The (induced) function which assigns to each  the set of distributions ( ={w(A), wÎW}. | | s: F*® {g}, a strategy which assigns to each partial history p Î F* a gamble G(f), where f is the latest fortune in the sequence p. | t:  ®  , an adaptive plan; t uses only the retained history  in = X , but t has the same generality as s if = F and = F*. | | u: F ® Reals, utility. |  . |
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As implied by their terminology, Dubins and Savage treat situations wherein the expectation for any strategy s,given an initial fortune F,is less than F. That is, the strategies are operating in environments wherein continued operation makes degraded performance ever more likely. (This is similar to adaptation in an environment having only nonreplaceable resources, so that performance can only decline in the long run.) In contrast, the present work is primarily concerned with complex environments wherein performance can be permanently improved, if only the right information can be acquired and exploited. Despite the differences, or more likely because of them, theorems from one framework have interesting, and sometimes surprising, translations in the other framework. |
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