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St. Petersburg portfolio games. (English) Zbl 1262.91047
Gavaldà, Ricard (ed.) et al., Algorithmic learning theory. 20th international conference, ALT 2009, Porto, Portugal, October 3–5, 2009. Proceedings. Berlin: Springer (ISBN 978-3-642-04413-7/pbk). Lecture Notes in Computer Science 5809. Lecture Notes in Artificial Intelligence, 83-96 (2009).

Summary: We investigate the performance of the constantly rebalanced portfolios, when the random vectors of the market process $\left\{{𝐗}_{i}\right\}$ are independent, and each of them distributed as (${X}^{\left(1\right)},{X}^{\left(2\right)},\cdots ,{X}^{\left(d\right)},1\right)$, $d\ge 1$, where ${X}^{\left(1\right)},{X}^{\left(2\right)},\cdots ,{X}^{\left(d\right)}$ are nonnegative iid random variables.

Under general conditions we show that the optimal strategy is the uniform: $\left(1/d,\cdots ,1/d,0\right)$, at least for $d$ large enough. In case of St. Petersburg components we compute the average growth rate and the optimal strategy for $d=1$, 2. In order to make the problem non-trivial, a commission factor is introduced and tuned to result in zero growth rate on any individual St. Petersburg components. One of the interesting observations made is that a combination of two components of zero growth can result in a strictly positive growth. For $d\ge 3$ we prove that the uniform strategy is the best, and we obtain tight asymptotic results for the growth rate.

##### MSC:
 91A60 Probabilistic games; gambling 91G10 Portfolio theory