The generalized Kelly Criterion

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In our previous introduction to the Kelly Criterion, we considered the importance of sizing your bets correctly via a discussion of the concept of expected growth and a presentation of the basics of the theory of bet sizing/portfolio optimization. The main question was whether there for any plausible betting proposition was possible to find a ‘perfect’ bet fraction, one that is optimal in the sense of always outperforming any other fraction. If the bettor’s objective is optimization of long-term growth, both existence and uniqueness (under some mild conditions) of such a fraction was proven.

Now, the realisation of the fact that a solution to the above problem of finding the ‘perfect’ fraction when offered a *single* bet exists, naturally gives rise to another engaging question. Could the same idea hold true in a more general, multidimensional case, where instead of a unique opportunity a full *set* of betting propositions are taken under consideration?

The theme of this post is to answer this question by discussing whether or not a construction of such an extension is achievable.