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3. Comparison with the Dubins-Savage Formalization of the Gambler's Problem |
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. . . much of the mathematical essence of a theory of gambling consists of the discovery and demonstration of sharp inequalities for stochastic processes . . . this theory is closely akin to dynamic programming and Bayesian statistics. In the reviewer's opinion, [How to Gamble If You Must] is one of the most original books published since World War II.
M. Iosifescu. Math. Rev. 38, 5, Review 5276 (1969). |
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For those who have read, or can be induced to read, Dubins and Savage's influential book, this section (which requires special knowledge not essential for subsequent development) shows how to translate their formulation of the abstract gambler's problem to the present framework and vice versa. Briefly, their formulation is based on a progression of fortunes f0,f1,f2, .. . which the gambler attainsby a sequence of gambles. A gamble is naturally given as a probability distribution over the set of all possible fortunes F. The gambler's range of choice at any time t depends directly and only upon his current fortune ft so that, as Dubins and Savage remark, the word "state" might be more appropriate than "fortune." The gambler's range of choice for each fortune f is dictated by the gambling house G. The strategy sfor confronting the house is a function which at each time t selects a gamble in G on the basis of the sequence or partial history of fortunes to that time (f0,f1, . . ., ft). Finally the utility of a given fortune f to the gambler is specified by a utility function u. Thus an abstract gambler's problem is well posed when the objects (F, G, u)have been specified; the gambler's response to the problem is given by his strategy s. |
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The objects of the Dubins-Savage framework can be put in a one-to-one correspondence with formally equivalent objects in the present framework. With the help of this correspondence any theorem proved in one framework can automatically be translated to a statement which is a valid theorem in the other framework. The relation between the intended interpretations of corresponding objects is in itself enlightening, but the real advantage accrues from the ability to transfer results from one framework to the other with a guarantee of validity. |
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The following table presents the formal correspondence with an indication of the intended interpretation of each formal object. In this table the superscript "*" on a set will indicate the set of all finite sequences (or strings) which can be formed from that set; thus F* is the set of all partial histories. |
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