Abstract
In their book,How To Gamble, If You Must, Lester E. Dubins and Leonard J. Savage defined the utility μ(σ) of a strategy σ with respect to a utility functionu by μ(σ)=lim sup t→∞μ(σ,t)=lim sup t→∞ E(σ,u t) where “lim sup” is taken over the directed set of all stop rules onH, u t is the real-valued function onH defined byu t(h)=u(ft(h)) ifh=(f 1,f2,…ft(h)…) andE(σ,u t) is the Eudoxus integral ofu t with respect to the strategy σ. In this note, we will show that μ(σ) is nothing but the σ-integral of the real-valued functionu * defined onH byu *(h)=lim sup n→∞ u(f n) ifh=(f 1,f2,…) inH. Finally, an application of this result is stated.
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References
L. E. Dubins and L. J. Savage,How To Gamble, If You Must, Inequalities for Stochastic Processes, McGraw-Hill, New York, 1965.
Nelson Dunford and Jacob T. Schwartz,Linear Operators, Part I, Wiley-Interscience, New York, 1958.
R.A. Purves and W.D. Sudderth,A finitely additive strong law, (1973), (preprint).
W. D. Sudderth,On measurable gambling problems, Ann. Math. Statist.,42 (1971), 260–269.
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Chen, R. A note on the Dubins-Savage utility of a strategy. Israel J. Math. 21, 1–6 (1975). https://doi.org/10.1007/BF02757128
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DOI: https://doi.org/10.1007/BF02757128