When and Why Kelly "Breaks Down".

Posted by MathProf on 21 Sep 1997, at 6:50 a.m.

A question has come up as to whether or not you should follow the Kelly Criterion when it calls for a big bet, such as 52% of your bankroll. Someone wrote to me and asked about the mathematics of the situation and asked me top post the comments here. While much of this material is already known to many correspondents here, there are still many who do not understand it. For that reason, I have taken the liberty to start a new thread with a rather lengthy exposition.

The key point to realize is that in BlackJack, the Kelly formula is just an approximation. Kelly originally considered the situation of games with simple payoffs. You either lose your bet or win your bet. For such games, the "optimal bet size" (for maximizing the logarithm of your expectations) is exactly your expectation times your bankroll. If you have a 70% advantage in that situation, you would bet 70% of your bankroll.

However, BJ is not a simple game. It has a more complex set of payoffs. You can win/lose double your original bet, or, with splitting, resplitting, and DAS, up to 8 times your original bet. Indeed you can theoretically lose 8.5 times your bet if you take insurance. Now we may approximate BJ by an equivalent simple game. If we do this, then the optimal fraction is Expectation divided by Sum of Squares ( E(x)/E(x^2) ). This is approximately the same exp/Variance.

Now this is an approximation that is valid for small betting fractions. It breaks down with large fractions. The technical mathematic reason is that the technique depends on approximing terms log(1+xf) by xf +( xf)^2, its second-order Taylor Polynomial. Here x is a payoff and f is a betting fraction. This is an excellent approximation as long as xf is small. For practical situations, where you fraction is a small percentage of your bankroll, then the approximation is excellent.

However when you obtain a big advantage in BJ, leading to big values of f then the approximation breaks down. Even if you never studied Calculus, common sense should still indicate this. Obviously if you bet more 50% of your bankroll, then you cannot double down and so you are no longer playing the same game. If you bet more than 25% of your bankroll, you could not resplit completely, etc.

Now, for a game with a fixed set of payoffs, one could determine the optimum fraction, in a specific case, by going back to the original equation. This equation involves these awful logarithmic terms and requires a trial and error solution, but one that could be easily done with a computer. Then you would have the correct fraction, in this specific case, that would maximize your logarithm.

Strategy Changes

However this approach will not work with BlackJack. In BJ, the payoffs are determined by Players strategy and a Kelly player will not follow the conventional strategy when they have a large portion of their capital at risk. Standard BlackJack strategies are based upon maximizing expectation whereas Kelly strategies would maximize the expectation of logarithms. When only a small of capital is wagered, then the difference between these two strategies is negligible. However this breaks down when larger amounts are bet. For example, a Kelly player with a big bet out should insure "good hands" even if the expectation on the Insurance Bet were negative. (There is an excellent discussion of this point in the 1986 edition of Griffin's "Theory of BJ"). Similarly, Kelly strategy would dictate passing up many double down bets when the gain from doubling did not justify the increased risk. (Or, double down for less if allowed.)

For a specific question, one could determine the optimal fraction and corresponding Kelly strategy. For each play (double, split, insurance) you could determine a corresponding Kelly fraction. Then through a trial and error process determine the optimal betting fraction. For example, in the case of what to do if you "knew" your first card was an Ace, it might turn out (I don't know; I haven't done the calculations) that the optimal play might be to bet very big, give up on double downs and splits and insure Black Jacks for even-money.

However, I don't believe the results of this exercise would have much practical significance. For example, suppose you "knew" that your next card was going to be an Ace. If you made an enormous jump in your betting, several times your typical maximum, you very well would be exceeding the Table Limit and would also be attracting attention to yourself. They might decide to burn the next card, or to bar you from play. I would only bet my ordinary maximum in this situation.