Prophet inequalities and order selection in optimal stopping problems

Author:
T. P. Hill

Journal:
Proc. Amer. Math. Soc. **88** (1983), 131-137

MSC:
Primary 60G40

MathSciNet review:
691293

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Abstract: A complete determination is made of the possible values for and for independent uniformly bounded random variables; this yields results of Krengel, Sucheston, and Garling, and of Hill and Kertz as easy corollaries.

In optimal stopping problems with independent random variables where the player is free to choose the order of observation of these variables it is shown that the player may do just as well with a prespecified fixed ordering as he can with order selections which depend sequentially on past outcomes.

A player's optimal expected gain if he is free to choose the order of observation is compared to that if he is not; for example, if the random variables are nonnegative and independent, he may never do better than double his optimal expected gain by rearranging the order of observation of a given sequence.

**[1]**Antoine Brunel and Ulrich Krengel,*Parier avec un prophète dans le cas d’un processus sous-additif*, C. R. Acad. Sci. Paris Sér. A-B**288**(1979), no. 1, A57–A60 (French, with English summary). MR**522020****[2]**Y. S. Chow, Herbert Robbins, and David Siegmund,*Great expectations: the theory of optimal stopping*, Houghton Mifflin Co., Boston, Mass., 1971. MR**0331675****[3]**Theodore P. Hill and Robert P. Kertz,*Ratio comparisons of supremum and stop rule expectations*, Z. Wahrsch. Verw. Gebiete**56**(1981), no. 2, 283–285. MR**618276**, 10.1007/BF00535745**[4]**T. P. Hill and Robert P. Kertz,*Additive comparisons of stop rule and supremum expectations of uniformly bounded independent random variables*, Proc. Amer. Math. Soc.**83**(1981), no. 3, 582–585. MR**627697**, 10.1090/S0002-9939-1981-0627697-3**[5]**T. P. Hill and Robert P. Kertz,*Comparisons of stop rule and supremum expectations of i.i.d. random variables*, Ann. Probab.**10**(1982), no. 2, 336–345. MR**647508****[6]**Theodore P. Hill and Victor C. Pestien,*The advantage of using nonmeasurable stop rules*, Ann. Probab.**11**(1983), no. 2, 442–450. MR**690141****[7]**Ulrich Krengel and Louis Sucheston,*Semiamarts and finite values*, Bull. Amer. Math. Soc.**83**(1977), no. 4, 745–747. MR**0436314**, 10.1090/S0002-9904-1977-14378-4**[8]**Ulrich Krengel and Louis Sucheston,*On semiamarts, amarts, and processes with finite value*, Probability on Banach spaces, Adv. Probab. Related Topics, vol. 4, Dekker, New York, 1978, pp. 197–266. MR**515432**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1983-0691293-4

Keywords:
Optimal stopping theory,
prophet inequalities

Article copyright:
© Copyright 1983
American Mathematical Society