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THE RE-OPENING OF DUBINS AND SAVAGE CASINO IN THE ERA OF DIVERSIFICATION

Published online by Cambridge University Press:  19 November 2013

Isaac Meilijson
Isaac Meilijson*
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: [email protected]
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Abstract

In Dynamic Programing, mixed strategies consist of randomizing the choice of actions. In some problems, such as portfolio management, it makes sense to diversify actions rather than choosing among them purely or randomly. Optimal betting in casinos and roulette by a gambler with fixed goal was studied by Dubins and Savage [9] and their school without the element of diversification (betting simultaneously on different holes of the roulette), once it was proved (Smith's theorem - Smith [16], Dubins [8] and Gilat and Weiss [10]) that diversification does not increase the probability of reaching the goal. We question the scope of this finding, that was based on the assumption that the holes on which gamblers can bet are disjoint, such as 1 and BLACK in regular roulette. A counter example is provided in which holes are nested, such as 1 and RED. Thus, it may be rational for gamblers with a fixed goal to place chips on more than one hole at the table.

This note is related to a joint work with Michèle Cohen on the preference for safety in the Choquet Expected Utility model.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 1 , January 2014 , pp. 55 - 65
Copyright
Copyright © Cambridge University Press 2013 

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References

1.Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l’école américaine. Econometrica 21: 503–546.CrossRefGoogle Scholar
2.Allais, M. (1988). The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function. In Risk, Decision and Rationality. Munier, B.R., (ed.), Reidel: Dordrecht, pp. 233–289.Google Scholar
3.Bellman, R.E. (2003). Dynamic Programming. Princeton University Press: Princeton, NJ (1957). Dover paperback edition, ISBN 0-486-42809-5.Google Scholar
4.Chateauneuf, A., Cohen, M. & Meilijson, I. (2004). Four notions of mean-preserving increase in risk, risk attitudes and applications to the Rank-dependent Expected Utility model. Journal of Mathematical Economics 40: 547–571.Google Scholar
5.Chateauneuf, A., Cohen, M. & Meilijson, I. (2005). More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model. Economic Theory 25(3).Google Scholar
6.Cohen, M. & Meilijson, I. (2013). Preference for safety under the Choquet model: in search of a characterization. Economic Theory.Google Scholar
7.Dubins, L.E. (1967). A simpler proof of Smith's Roulette Theorem. Annals Mathematics and Statistics 39: 390–393.CrossRefGoogle Scholar
8.Dubins, L.E. (1972). On roulette when the holes are of various sizes. Israel Journal of Mathematics 11: 153–158.CrossRefGoogle Scholar
9.Dubins, L.E. & Savage, L.J. (1965). How to gamble if you must. New York: McGraw-Hill. Re-published as Inequalities for stochastic processes, New York: Dover Publications, 1976.Google Scholar
10.Gilat, D. & Weiss, E.A. (1976). On roulette which allows stakes on infinitely many holes. Israel Journal of Mathematics 24: 282–285.CrossRefGoogle Scholar
11.Kahneman, D. & Tversky, A. (1979). Prospect theory: an analysis of Decision under Risk. Econometrica 47: 263–292.Google Scholar
12.Pratt, J.W. (1964). Risk Aversion in the small and large, Econometrica 32: 122–136.Google Scholar
13.Schmeidler, D. (1986). Integral representation without additivity. Proceedings of the American Mathematical Society 97: 255–261.Google Scholar
14.Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57: 517–587. First version: Subjective expected utility without additivity, Forder Institute Working Paper (1982).CrossRefGoogle Scholar
15.Shapley, L. (1953). Stochastic games. Proceedings of the National Academy of Sciences of the United States of America, 39: 1095–1100.Google Scholar
16.Smith, G. (1967). Optimal strategy at roulette. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 8: 91–100.Google Scholar