I understand that Kelly's theorem applied to BJ is "you should bet your edge as a % of your bankroll".
But surely this is an approximation, because standard deviation must have an effect somewhere ? Is there a "generalized Kelly theorem" taking into account the std deviation ?
(the standard deviation for a hand of BJ is usually close to 100%, so the effect should be small most of the time).
Bet = Bankroll * Edge / Variance
The variance is the square of the standard deviation. In BJ, the variance is about 1.3.
as far as I understand Kelly's paper, the real criterion is
Max EV [Log (Bankroll + Payoff)
with EV(Payoff) = Bet x Edge
and Var(Payoff) = Bet^2 x Var
Can you prove that this system is solved by Bet = Edge / Var ?
The Kelly Criterion was originally developed for communications theory. It can be extended to gambling, and used to determine the bet size that will maximize the LONG TERM growth rate (compound growth rate) of your bankroll. The original Kelly formula assumed that you either win or lose your bet. In this case:
Kelly Bet = Expectation x Bankroll
(note: Expectation is the expected $win per $bet)
Blackjack is more complex than a win-or-lose bet (due to doubling, splitting, insurance, naturals etc.). For games with complex expectations, and when the bet size is small relative to the bankroll, Kelly can be defined as follows:
Kelly Bet = Expectation x Bankroll / Variance
(note: Variance is Standard Deviation squared)
(note: Variance and Expectation are defined relative to initial bet for a hand)
If a player strictly follows Kelly, they will resize their bet each time their bankroll or expectation changes. If that player has exact information about their bankroll and expectation, and if they follow Kelly exactly, then they will never go broke and they will maximize their long term compound growth rate.
However, under casino conditions it is not possible to strictly follow Kelly. First, the expectation we think we have is only an estimate, and our actual expectation is probably lower than we believe, due to counting, playing and betting errors. In addition, most casinos do not allow perfectly sized bets (for example $36.41). And finally, we sometimes make deliberate errors in the interest of camouflage.
These are some of the reasons that the Kelly criterion should be only a starting point for determining your ideal bet size. In addition it is important to understand how your growth rate and its variability relate to your bet size. This can be seen be examining the effect of Kelly Fraction (KF), which is defined as the size of the actual bet as a fraction of the full Kelly Bet.
KF=0.0 You are betting nothing (therefore you don't win anything)
KF=0.5 Bankroll grows at 75% of optimal - Low Variability
KF=1.0 Optimal bankroll growth - Intermediate Variability
KF=1.5 Bankroll grows at 75% of optimal - High Variability
KF=2.0 Bankroll eventually goes to zero (you go broke)
KF<2.0 You go broke faster
Whenever the Kelly fraction is greater than one, two undesirable things happen: growth rate goes down and variability goes up. So there is absolutely no benefit in using a Kelly Fraction greater than one, and it makes a lot of sense to keep the Kelly fraction significantly lower than that. My understanding is that professional gamblers typically use 0.3 as their Kelly Fraction. At this betting level, their bankroll will grow at about 50% of the optimal rate, variability will be relatively low, and there will be an allowance for the inevitable errors that occur in casino play.
Editorial Comment: The post below corrects three typos that occurred in my original post. Two errors were inconsequential (spelling), but the third might have caused some confusion: The original post stated "KF<2.0 You go broke faster", while this corrected post states: "KF>2.0 You go broke faster" (> means "greater than"; < means "less than"). Sorry for any misunderstanding. I look forward to your comments, corrections or suggestions. Thanks...Freddie
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KELLY EXPLANATION
The Kelly Criterion was originally developed for communications theory. It can be extended to gambling, and used to determine the bet size that will maximize the LONG TERM growth rate (compound growth rate) of your bankroll. The original Kelly formula assumed that you either win or lose your bet. In this case:
Kelly Bet = Expectation x Bankroll
(note: Expectation is the expected $win per $bet)
Blackjack is more complex than a win-or-lose bet (due to doubling, splitting, insurance, naturals etc.). For games with complex expectations, and when the bet size is small relative to the bankroll, Kelly can be defined as follows:
Kelly Bet = Expectation x Bankroll / Variance
(note: Variance is Standard Deviation squared)
(note: Variance and Expectation are defined relative to initial bet for a hand)
If a player strictly follows Kelly, they will resize their bet each time their bankroll or expectation changes. If that player has exact information about their bankroll and expectation, and if they follow Kelly exactly, then they will never go broke and they will maximize their long term compound growth rate.
However, under casino conditions it is not possible to strictly follow Kelly. First, the expectation we think we have is only an estimate, and our actual expectation is probably lower than we believe, due to counting, playing and betting errors. In addition, most casinos do not allow perfectly sized bets (for example $36.41). And finally, we sometimes make deliberate errors in the interest of camouflage.
These are some of the reasons that the Kelly criterion should be only a starting point for determining your ideal bet size. In addition it is important to understand how your growth rate and its variability relate to your bet size. This can be seen by examining the effect of Kelly Fraction (KF), which is defined as the size of the actual bet as a fraction of the full Kelly Bet.
KF=0.0 You are betting nothing (therefore you don't win anything)
KF=0.5 Bankroll grows at 75% of optimal - Low Variability
KF=1.0 Optimal bankroll growth - Intermediate Variability
KF=1.5 Bankroll grows at 75% of optimal - High Variability
KF=2.0 Bankroll eventually goes to zero (you go broke)
KF>2.0 You go broke faster
Whenever the Kelly fraction is greater than one, two undesirable things happen: growth rate goes down and variability goes up. So there is absolutely no benefit in using a Kelly Fraction greater than one, and it makes a lot of sense to keep the Kelly fraction significantly lower than that. My understanding is that professional gamblers typically use 0.3 as their Kelly Fraction. At this betting level, their bankroll will grow at about 50% of the optimal rate, variability will be relatively low, and there will be an allowance for the inevitable errors that occur in casino play.
If a player strictly follows Kelly, they will resize their bet each time their bankroll or expectation changes. If that player has exact information about their bankroll and expectation, and if they follow Kelly exactly, then they will never go broke and they will maximize their long term compound growth rate.
This isn't a special property of Kelly, though you could be more than forgiven for thinking so given the way the subject is covered in the literature. Any betting system where you never bet 100% of your bank, even betting 99.99% of your bank continuously, would theoretically never completely wipe out your bank.
KF=2.0 Bankroll eventually goes to zero (you go broke)
It is important to understand that ROR when you bet 2xkelly isn't zero, though it is very close to it. If it was, then you would be able to beat negative expectation games with a large enough bank. One in a thousand 2xKelly bettors would win a fortune compensating for all the other bankruptcies and restoring the overall positive expectation.
Which BS play would change if variance is taken into account ?
I would think that any split with a negative expectation (such as splitting 3-3 against dealer 7) go against Kelly's theorem because you put more money on the table (=more variance) with a negative expectation.
Maybe you could increase your Sharpe ratio (return/std deviation) if you modify certain plays: insure good hands, split less often, double less often.
There are risk averse strategies that take variance into account. You could alter your play based on those strategies, but they work best in conjunction with the count and the amount of money you have on the table.
The short answer is no, it doesn't affect basic strategy, since basic strategy tells you the best play to make based on your hand and the dealer's upcard. If that is the only information you have, then the BS play is the one which will get you the most money in the long run. If you have more information, such as what you learn from counting the cards, you may alter basic strategy based on the count and, if you wish to use risk averse strategies, you can alter based on the count and the amount of money you have on the table.
Thanks, actually I meant Playing Decisions, not Basic Strategy, because I understand that the count must be included in the decision.
I have run a few simulations and they seem to show that risk-averse strategies are hurting the edge too much to be useful.
Do you know where I can find a discussion of risk-averse strategies ? In you book ?
I have run a few simulations and they seem to show that risk-averse strategies are hurting the edge too much to be useful.
They can be useful, increasing your overall win rate by 3% or so without any significant extra effort. The problem may be that you cannot "guess" at risk-averse strategy, you have to use complicated formulas to determine what strike numbers to use, as the figures you get will often seem counter-intuitive.
Detailed files on this subject exist in the archives at bjmath.com.
The second edition of Blackjack Attack covers risk averse strategies pretty well. I didn't include them in my book because I thought Don did cover it well and I don't think it adds all that much to a person's game.
>> The short answer is no, it doesn't affect basic strategy, since basic strategy tells you the best play to make based on your hand and the dealer's upcard.
This a very popular answer among blackjack experts, but it is technically incorrect. The "best play" isn't absolute, it is relative to some criteria (whether expressed or implied) for measuring two plays in order to determine which is "better." The implicit criteria for basic strategy is EV[bankroll], Kelly betting is founded on maximizing EV[log(bankroll)], and the same criteria should be used to optimize playing strategy. This complicates matters somewhat, because the log optimal playing strategy varies as a function of bet fraction. So, although the strategy is more complicated, it is mathematically possible to determine the log optimal playing strategy for blackjack.
The concept can be extended to an infinite variety of utility functions other than log(). Each different utility function has the potential to lead to a different playing strategy that is optimal for that utility function. One interesting consequence of non-linear utility functions is that "double for less" is sometimes the best play, while DFL is correct under basic strategy only with extremely unusual house rules (doubling allowed after splitting aces, but hitting not allowed.) There was some recent discussion of this topic on the bjmath site.
I have run a number of simulations and (although my study is not exhaustive at all), it tends to show that :
- if your utility is U=E[log (bankroll)] and your betting strategy is optimal, then U is very highly correlated with your Edge. I would therefore postulate that whatever hurts your edge, hurts your utility.
- most usual playing decisions (i.e. the ones that are optimal for your edge) are still optimal for your utility.
- in particular, doubling for less is always sub-optimal (results in lower utility).
I know that my research is not very scientific, but based on those few observations, I cannot think of any playing decision which would optimize your edge while being sub-optimal for your utility.
I have read the discussion on bjmath and it is not very specific.
> I have run a number of simulations and (although my study is not exhaustive at all), it tends to show that :
> - if your utility is U=E[log (bankroll)] and your betting strategy is optimal, then U is very highly correlated with your Edge. I would therefore postulate that whatever hurts your edge, hurts your utility.
If "whatever" happens to leave variance unchanged, then increased EV correlates perfectly with increased U. Also, if EV stays fixed while variance _decreases_ then U will increase.
> - most usual playing decisions (i.e. the ones that are optimal for your edge) are still optimal for your utility.
Yes, the vast majority of playing decisions remain the same. This is partly due to your stated constraint that bet size is optimal. As bet size shrinks, log optimal strategy approaches basic strategy. With a small enough bet fraction, the two strategies become identical. With large bet fractions, the double-downs and pair splits start to disappear from the log optimal strategy.
> - in particular, doubling for less is always sub-optimal (results in lower utility).
> I know that my research is not very scientific, but based on those few observations, I cannot think of any playing decision which would optimize your edge while being sub-optimal for your utility.
You have to look for "close" decisions involving doubling or splitting. For example, in single deck S17 games, doubling has a tiny bit more EV than standing. Standing becomes log optimal when the bet fraction is greater than 0.0192% of bankroll.
A,6 vs. 2 is another example. Here basic strategy is to double, but with a bet fraction larger than 0.365% of bankroll it is better to double for less, and hitting becomes the best play with a bet fraction larger than 0.729% of bankroll.
What was the first example? You omitted the cards. I guess either A7 v. 2 or A8 v. 6. Whatever it is, this is interesting as a piece of trivia.
I guess SBA would generate RA indices, and we are essentially looking for those that go from negative to positive.
Lefty
L> What was the first example? You omitted the cards. I guess either A7 v. 2 or A8 v. 6. Whatever it is, this is interesting as a piece of trivia.
Oops. It was A,8 v. 6.
L> I guess SBA would generate RA indices, and we are essentially looking for those that go from negative to positive.
That might be one way. I'm using my CA program. I ought to modify it so that it automatically finds the crossover point. A,8 v. 6 is the "first" crossover from basic strategy to log-optimal strategy. Some more crossover points where doubling becomes incorrect:
A,8 v. 6 Stand if f > 0.0192%
A,2 v. 4 Hit if f > 0.34%
A,6 v. 2 DFL if f > 0.365%, hit if f > 0.729%
The A,2 v. 4 case is interesting because hitting becomes best before DFL becomes best. The log utility for doubling in this case "peaks" at a bet fraction of 3.02%, and doubling becomes negative EV at about 6.02%. The crossover point comes at a lower bet fraction than the peak, so there is no DFL region. In order to have a DFL region, the crossover has to occur at a bet fraction that is higher than the point where the utility for doubling reaches the peak.
More cases:
4,4 v. 5 Hit if f > 0.591%
5,3 v. 5 Hit if f > 0.981%
4,4 v. 6 Hit if f > 1.27%
A,3 v. 4 Hit if f > 1.27%
A,7 v. 3 Hit if f > 1.57%
A,4 v. 4 Hit if f > 1.63%
A,5 v. 4 DFL if f > 1.63 %, hit if f > 1.71%
5,3 v. 6 Hit if f > 1.86%
One of the last double-downs to disappear would be 11 vs. 6.
9,2 v. 6 DFL if f > 18.5%, hit if f > 24.41%
8,3 v. 6 DFL if f > 19.1%, hit if f > 25.11%
7,4 v. 6 DFL if f > 19.6%, hit if f > 25.75%
6,5 v. 6 DFL if f > 20.3%, hit if f > 26.65%
