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I keep reading that the expected time for your capital to reach any predefined number is minimized by the strategy that sizes bets according to the Kelly Criterion. But it seems trivially easy to come up with a counterexample.

Suppose we are playing a game (the details of which are unimportant) for which Kelly dictates that you bet 99% of your capital on each round (i.e. it's a highly favorable game for us). Say our goal is to increase our capital by a mere 1%. If we bet 99% on round one and lose, then it will take many more rounds for us to be able to reach our goal. On the other hand, if on each round we only bet as much as will allow us to exactly reach our target capital and no more when we win (1% on round one in this example), we retain the ability to be able to bet enough to reach our target on round two (as well as on a few more rounds after successive losses). So clearly the second strategy will on average reach the target capital in fewer time periods.

So either I am missing something, or the way that claim about the Kelly Criterion is stated is not completely accurate. Which is it? If the latter, how can the "overshooting" of Kelly in such scenarios be addressed rigorously?

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I think the way you characterize Kelly's criterion is not entirely correct. Kelly wanted to maximize the CAGR (Compound Annual Growth Rate) given a game where you have an unfair advantage. The gist of his is idea is that you should bet roughly in proportion of how unfair the game is in your direction. If you do that, then, in the long run you will achieve maximum CAGR. No other strategy will do that.

For example, say you have a biased coin with both sides being heads. How much of your capital should you bet on the next coin toss coming up as heads? Well, 100%, right? Go all in. Bet the farm. You can't lose and you double your money each round.

OK, so how about your example? Well, if Kelly's criterion says you should bet 99%, then you're extremely likely to win and you're going to double your money. (I assume the payoff is like a coin toss.) So your gains are going to be $100 + $99 = $199. That's 99% CAGR. It'll take roughly 70 years to replicate this with 1% gains.

In other words, when you win, you win so big, that it more than makes up for any 'lost time' (especially with such a highly skewed game). Also, the payoff here is important: if you're only specifying two percentages (win/lose), the assumption is 100% profit in case you win (the coin toss scenario).

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  • Yeah I understand all that, and I have no qualms about Kelly reaching targets the fastest in continuous cases such as stocks, but here I'm talking about discrete games and a specified target. Since my target is $101, reaching $101 and $199 is the same thing to me. I don't think your response addresses the time element that I was getting at. (But you were right in pointing out that I should have made explicit what the payoffs are; I was assuming 100% profit/loss when you win/lose.) – tobakudan May 2 '18 at 8:45
  • @tobakudan If I can paraphase Lajos' point, the Kelly Criterion is not about reaching a specific target regardless of the odds or payoff, but about maximizing return. If your goal is a 1% return, then you should choose a strategy where the expected return is 1% with the least amount of risk. IOW, either choose a different game (one with less risk) or structure your "bets" such that you don't lose 99% of your bankroll on the first go. – D Stanley May 2 '18 at 14:04
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It would be nice to know where you saw this characterization of the Kelly Criterion. Wikipedia says:

In probability theory and intertemporal portfolio choice, the Kelly criterion, Kelly strategy, Kelly formula, or Kelly bet is a formula used to determine the optimal size of a series of bets in order to maximise the logarithm of wealth. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run (that is, over a span of time in which the observed fraction of bets that are successful equals the probability that any given bet will be successful). It was described by J. L. Kelly, Jr, a researcher at Bell Labs, in 1956.[1] The practical use of the formula has been demonstrated.

[emphasis added]

Wikipedia does seem to fail to specify what metric is used to define "better". I believe that they are evaluating strategies by their median values; the mean value is maximized by betting everything. Minimizing time to a specified value would probably also be accomplished by the Kelly Criterion, but as you note, this can fail on small scale.

So either I am missing something, or the way that claim about the Kelly Criterion is stated is not completely accurate. Which is it?

The way you apparently have found the Kelly Criterion stated in not completely accurate. However, this is not the standard statement. The main feature of the Kelly Criterion is that it maximizes the expected value of the logarithm of the ending capital. This property then leads to others, such as maximizing the median.

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