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The Kelly criterion gives a simple formula to calculate the fraction of one's current wealth/bankroll.

On the page above it says

Assuming that the expected returns are known, the Kelly criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity).

Down below in article also, there is a heuristic argument given, saying that it maximizes some expected value. The heuristic argument of course remains vague.

To me the argument remains unconvincing, since obviously the set of all strategies is uncountable (i.e. at each step choose a real number between 0 and current wealth) and a natural measure on that set, does not appear to be obvious.

Now the question:

  1. Is there a simple reason or heuristic argument that explains why there should even be a unique single strategy out of the uncountably many that is optimal? A nice compactness argument would make me happy.
  2. Assuming 1), is there a simple reason or heuristic argument that explains that Kelly's criterion takes such a simple form — namely a constant factor of the wealth rather than some other more complicated function of wealth?

Update

Let me address some of the criticism in the comments. First of all I did, I tried reading the original paper but could not get far due to the different framework of information theory. Also I read both the proof for fair bet case below, which I can follow, but even the sentence:

So in the long run, final wealth is maximized by setting $\Delta$ to zero, which means following the Kelly strategy seems handwavy to me.

Another thing I realized is, that the argument given for my question 2 below I found completely convincing, with one caveat: We implicitly assume that the random variables are non-discrete, as the scaling invariance would break down if there were discreteness involved (which actually would be the case for real-world money where we can only can go down to 1 cent).

This and also "So no matter what heuristic argument is proposed, one can fault it for being not totally convincing." made me thinking, that my real question is more about having the framework for stating the Kelly criterion and proving the existence of a solution as in question 1. So let me try to make this more precise.

Question 1': Do you know of any general formulation/framework for betting strategies (e.g. covering both continous and discrete random variables) which covers the situation of the Kelly criterion together with results that either imply a unique solution or maybe show that an optimal strategy might not need to be unique?

P.S.: I am still hoping that compactness, will pop up somehow.

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  • $\begingroup$ I do not know why this is downvoted and voted to close, it is a good question... $\endgroup$
    – Nate River
    Commented Jun 20 at 9:47
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    $\begingroup$ That Wikipedia article cites various serious-looking sources (published articles in maths and economics) for the arguments/claims you’re asking about. Have you followed up those citations? They may answer your doubts, or (if they turn out to be equally unsatisfying) you can improve the question by pinpointing your issues more precisely. @NateRiver: I guess that’s the reason this question is downvoted: it complains that arguments in a Wikipedia article are vague/imprecise, but shows no evidence of having followed back to the citations, which is the obvious first thing to try. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Jun 20 at 11:59
  • $\begingroup$ @PeterLeFanuLumsdaine Ah, fair enough. $\endgroup$
    – Nate River
    Commented Jun 20 at 12:09
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    $\begingroup$ For #2, it makes sense the criterion would be scale linearly: it shouldn't matter whether you count your money in dollars or yen. For #1, one way to think of the Kelly strategy is maximizing time diversification—and anything other than a single fixed percentage would be weighting some bets more than others. However, intuition only takes you so far. An actual rigorous proof is kind of messy! $\endgroup$
    – Martin M. W.
    Commented Jun 20 at 12:10
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    $\begingroup$ I agree with Martin M. W. Asking for a convincing heuristic argument is a bit delicate. If the argument were totally convincing, then it wouldn't be a heuristic; it would be a rigorous proof. So no matter what heuristic argument is proposed, one can fault it for being not totally convincing. But certainly, if it's optimal to bet 200 dollars when you have 1000 dollars, then it is intuitive that it's optimal to bet 2 dollars when you have 10 dollars. After all, if it's optimal to bet 200 dollars with 1000 dollars then it's surely optimal to bet 200 cents with 1000 cents. $\endgroup$
    – Timothy Chow
    Commented Jun 21 at 3:31

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