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Optimal strategies for repeated games

Published online by Cambridge University Press:  01 July 2016

Mark Finkelstein  and
Robert Whitley
Mark Finkelstein*
Affiliation:
University of California, Irvine
Robert Whitley*
Affiliation:
University of California, Irvine
*
∗Postal address: Department of Mathematics, University of California, Irvine, CA 92717, U.S.A.
∗Postal address: Department of Mathematics, University of California, Irvine, CA 92717, U.S.A.
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Abstract

We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let Fn∗ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that Fn/Fn∗ is a supermartingale with E(Fn/Fn∗) ≤ 1 and, consequently, E(lim Fn/Fn∗) ≤ 1. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that E(Fn/Fn∗) = 1 for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that Fn∗/Fn is a submartingale with E(Fn∗/Fn) ≤ 1 and E(lim Fn∗/Fn) ≤ 1; and E(Fn∗/Fn) = 1 if and only if at each time period j, 1 ≤ j ≤ n, the strategies leading to Fn and Fn∗ are ‘the same’.

Keywords

KELLY CRITERIONOPTIMAL STRATEGYFAVORABLE GAMEOPTIMAL GAMBLING SYSTEMPORTFOLIO SELECTIONCAPITAL GROWTH MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 415 - 428
Copyright
Copyright © Applied Probability Trust 1981 

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References

[1] Abramowitz, M. and Stegun, A. (1965) Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. Google Scholar
[2] Aucamp, D. (1975) Comment on “the random nature of stock market prices” authored by Barrett and Wright. Operat. Res. 23, 587–591.Google Scholar
[3] Bicksler, J. and Thorp, E. (1973) The capital growth model: an empirical investigation. J. Finance and Quant. Anal. VIII, 273–287.CrossRefGoogle Scholar
[4] Breiman, L. (1961) Optimal gambling systems for favorable games. Proc. 4th Berkeley Symp. Math. Statist. Prob. 1, 65–78.Google Scholar
[5] Breiman, L. (1968) Probability. Addison-Wesley, New York.Google Scholar
[6] Durham, S. (1975) An optimal branching migration process. J. Appl. Prob. 12, 569–573.CrossRefGoogle Scholar
[7] Ferguson, T. (1965) Betting systems which minimize the probability of ruin. J. SIAM 13, 795–818.Google Scholar
[8] Hewitt, E. and Stromberg, K. (1965) Real and Abstract Analysis. Springer-Verlag, New York.Google Scholar
[9] Hakansson, N. (1971) Capital growth and the mean-variance approach to portfolio selection. J. Finance and Quant. Anal. VI, 517–557.Google Scholar
[10] Kelly, J. (1956) A new interpretation of information rate. Bell System Tech. J. 35, 917–926.CrossRefGoogle Scholar
[11] Latané, H. (1959) Criteria for choice among risky ventures. J. Political Econ. 67, 144–155.Google Scholar
[12] Latané, H., Tuttle, D., and James, C. (1975) Security Analysis and Portfolio Management. Wiley, New York.Google Scholar
[13] Olver, F. (1974) Introduction to Asymptotics and Special Functions. Academic Press, New York.Google Scholar
[14] Roberts, A. and Varberg, D. (1973) Convex Functions. Academic Press, New York.Google Scholar
[15] Thorp, E. (1971) Portfolio choice and the Kelly criterion. Proc. Amer. Statist. Assoc, Business, Econ. and Stat. Section, 215–224.Google Scholar
[16] Thorp, E. (1969) Optimal gambling systems for favorable games. Rev. Internat. Statist. Inst. 37, 273–293.Google Scholar
[17] Thorp, E. and Whitley, R. (1972) Concave utilities are distinguished by their optimal strategies. Coll. Math. Soc. Janos Bolya, European Meeting of Statisticians, Budapest (Hungary), 813–830.Google Scholar