Leavable gambling problems with unbounded utilities
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- by A. Maitra, R. Purves and W. Sudderth PDF
- Trans. Amer. Math. Soc. 320 (1990), 543-567 Request permission
Abstract:
The optimal return function $U$ of a Borel measurable gambling problem with a positive utility function is known to be universally measurable. With a negative utility function, however, $U$ may not be so measurable. As shown here, the measurability of $U$ for all Borel gambling problems with negative utility functions is equivalent to the measurability of all PCA sets, a property of such sets known to be independent of the usual axioms of set theory. If the utility function is further required to satisfy certain uniform integrability conditions, or if the gambling problem corresponds to an optimal stopping problem, the optimal return function is measurable. Another return function $W$ is introduced as an alternative to $U$. It is shown that $W$ is always measurable and coincides with $U$ when the utility function is positive.References
- Howard Becker, Pointwise limits of subsequences and $\Sigma ^1_2$ sets, Fund. Math. 128 (1987), no. 3, 159–170. MR 922568, DOI 10.4064/fm-128-3-159-170
- D. Blackwell, D. Freedman, and M. Orkin, The optimal reward operator in dynamic programming, Ann. Probability 2 (1974), 926–941. MR 359818, DOI 10.1214/aop/1176996558
- D. Blackwell and S. Ramakrishnan, Stationary plans need not be uniformly adequate for leavable, Borel gambling problems, Proc. Amer. Math. Soc. 102 (1988), no. 4, 1024–1027. MR 934886, DOI 10.1090/S0002-9939-1988-0934886-3 —, Transformations analytiques: théorèmes de capacitabilité, de séparation et d’itération transfinie, Séminaire Initiation à l’Analyse 20, Publ. Math. Univ. Pierre et Marie Curie 46 (1980-81), 16-01-16-27.
- C. Dellacherie, Quelques résultats sur les maisons de jeux analytiques, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 222–229 (French). MR 889480, DOI 10.1007/BFb0075851 —, Les sous-noyaux élémentaires, Théorie du Potentiel Proceedings (Orsay 1983), Lecture Notes in Math., vol. 1096, Springer-Verlag, Berlin and New York, 1983, pp. 183-222.
- C. Dellacherie and P. A. Meyer, Ensembles analytiques et temps d’arrêt, Séminaire de Probabilités, IX (Seconde Partie, Univ. Strasbourg, Strasbourg, années universitaires 1973/1974 et 1974/1975), Lecture Notes in Math.,., Vol. 465, Springer, Berlin, 1975, pp. 373–389 (French). MR 0426138
- Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV, Hermann, Paris, 1975 (French). Chapitres I à IV; Édition entièrement refondue. MR 0488194
- Lester Dubins and David Freedman, Measurable sets of measures, Pacific J. Math. 14 (1964), 1211–1222. MR 174687
- Lester E. Dubins and Leonard J. Savage, Inequalities for stochastic processes (how to gamble if you must), Dover Publications, Inc., New York, 1976. Corrected republication of the 1965 edition. MR 0410875
- Lester E. Dubins and William D. Sudderth, Countably additive gambling and optimal stopping, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 1, 59–72. MR 471074, DOI 10.1007/BF00535014
- Lester E. Dubins and William D. Sudderth, On stationary strategies for absolutely continuous houses, Ann. Probab. 7 (1979), no. 3, 461–476. MR 528324
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751 A. Louveau, Capacitabilité et sélections Boreliennes, Séminaire Initiation à l’Analyse 21, Publ. Math. Univ. Pierre et Marie Curie 54 (1981-82), 19-01-19-21.
- Ashok Maitra, Victor Pestien, and S. Ramakrishnan, Domination by Borel stopping times and some separation properties, Fund. Math. 135 (1990), no. 3, 189–201. MR 1077510, DOI 10.4064/fm-135-3-189-201
- D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), no. 2, 143–178. MR 270904, DOI 10.1016/0003-4843(70)90009-4
- P. A. Meyer, Réduites et jeux de hasard, Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972), Lecture Notes in Math., Vol. 321, Springer, Berlin, 1973, pp. 155–171 (French). Exposé du travail de Mohammed Traki avec un appendice qui est une rédaction de commentaires de G. Mokobodzki. MR 0370753
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
- S. Ramakrishnan and W. D. Sudderth, The expected value of an everywhere stopped martingale, Ann. Probab. 14 (1986), no. 3, 1075–1079. MR 841607
- Ralph E. Strauch, Measurable gambling houses, Trans. Amer. Math. Soc. 126 (1967), 64–72. MR 205352, DOI 10.1090/S0002-9947-1967-0205352-9
- William D. Sudderth, On the existence of good stationary strategies, Trans. Amer. Math. Soc. 135 (1969), 399–414. MR 233595, DOI 10.1090/S0002-9947-1969-0233595-9
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 320 (1990), 543-567
- MSC: Primary 60G40; Secondary 03E15, 03E35, 62L15, 90D60, 93E20
- DOI: https://doi.org/10.1090/S0002-9947-1990-0989581-5
- MathSciNet review: 989581