Distributional Robust Kelly Gambling

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Code In classic Kelly gambling, bets are chosen to maximize the expected log growth, under a known probability distribution. In this note we consider the distributional robust version of the Kelly gambling problem, in which the probability distribution is not known, but lies in a given set of possible distribitions. The bet is chosen to maximize the worst-case (smallest) log growth among the distributions in the given set. This distributional robust Kelly gambling problem is convex, but in general need not be tractable. We show that it can be tractably solved in the case of a finite number of outcomes, and some useful sets of distributions.