Hole Carding Kelly Bet Calculation.

Hi folks,

A few questions, all related:

The Wizard lists the advantage for seeing the Dealer’s card all the time at 10.1% in his BJ Appendix 16 - “Dealer Exposes Both Cards”. This strikes me as pretty high, so...

1 ) What is the advantage if you can only see the card 50% the time, 25% of the time, 1/10 of the time, etc. I don’t need exact numbers here, just want to know at what point it’s no longer an advantage. I figure it’s linear, since anytime you can’t see the card you are playing the normal HE. So a 10.1% advantage would probably be enough to overcome the house edge, even if you can only see the card 10% of the time. 5% or less would be a lot closer to call.

2 ) When late surrender is available, could this increase the advantage further? I am working on a spreadsheet to figure this out the individual decisions, since the Wizard’s strategy sheet in Appendix 16 doesn’t list them. However there are many obvious ones, such as to surrender 17, 18 or 19 against a dealer 20, etc. I’m wondering if anyone knows some of these hands.

3 ) How to calculate the Kelly bet – my main reason for posting, as this is the part that I really don’t understand.

Most Kelly calculators seem setup for simpler bets, I guess for sports, etc. I’m at a loss as to how to calculate this for Blackjack, based on the % advantage that could be gained by seeing the dealer’s card as outlined above. I also don’t understand how variance plays into the calculation, if at all, and the Wizard’s variance tables are all Greek to me.

I’m willing to be educated though, and am grateful for any help!

DJ

Quote: DJGenius

1 ) What is the advantage if you can only see the card 50% the time, 25% of the time, 1/10 of the time, etc. I don’t need exact numbers here, just want to know at what point it’s no longer an advantage. I figure it’s linear, since anytime you can’t see the card you are playing the normal HE. So a 10.1% advantage would probably be enough to overcome the house edge, even if you can only see the card 10% of the time. 5% or less would be a lot closer to call.

Just take the advantage for all the time and multiply it by how often you see the hole card. In other words, a pro-rata share.

Quote:

2 ) When late surrender is available, could this increase the advantage further? I am working on a spreadsheet to figure this out the individual decisions, since the Wizard’s strategy sheet in Appendix 16 doesn’t list them. However there are many obvious ones, such as to surrender 17, 18 or 19 against a dealer 20, etc. I’m wondering if anyone knows some of these hands.

That would increase the advantage significantly if surrender were available.

Quote:

3 ) How to calculate the Kelly bet – my main reason for posting, as this is the part that I really don’t understand.

Most Kelly calculators seem setup for simpler bets, I guess for sports, etc. I’m at a loss as to how to calculate this for Blackjack, based on the % advantage that could be gained by seeing the dealer’s card as outlined above. I also don’t understand how variance plays into the calculation, if at all, and the Wizard’s variance tables are all Greek to me.

Just use the approximate of advantage/variance, which is about 1.3 for blackjack. Most people find betting full Kelly to be too much for their stomach, including me, so adjust accordingly.

For practical or more real life purposes when you are calculating your initial advantage I’d add that I’d estimate -1% instead of the basic strategy house edge if you are counting no reads and misreads together. There will be errors, guaranteed, unless you’re lying to yourself. I’d also recommend chopping off about 35% of perfect play edge good read hands for discrete plays and because there will be hands that you count as good that your confidence level was sort of a guess at what you see.

I track every hand played rather than make assumptions for hands per hour played or read % and over thousands of hands and years with those adjustments the EV/actual are very close, wouldn’t be remotely close if I was assuming either 10+% or 100 hands/hr.

Not sure I’ve ever used any form of Kelly or fractional Kelly betting on one of those types of games, it’s typically just not practical in the real world unless you’re working on a small amount of capital. More relevant factors like these sort of dealers will sometimes freeze up at too much action and it ruins the game, or some places just won’t tolerate a win above a certain size and throw you out even if they don’t detect anything etc.

PLEASE DON’T HIT HARD 17+ OR SURRENDER 18+ EVER ON THESE. It’s just dumb. Negative future EV take the loss and move on.

As far as calculating edge, (as an example) say 10% when it’s good and -1% when bad.

50% read = 0.5 * 0.1 - 0.5 * 0.01 = 0.5 - 0.05 = +4.5%
25% read: 0.25 * 0.1 - 0.75 * 0.01 = 0.25 - 0.075 = +1.75%

So yeah, readability makes a huge difference.


I have no idea what the SD is for home carding, I’d imagine it’d be a bit more because you’ll have hands where you’ll be much more aggressive. But I really don’t know.

If you do know the variance, then a close approximation to optimal Kelly is to bet edge/variance * bankroll.


If variance is 2, edge is 9, bankroll is $100k, then you’d bet 0.09/2 * 100k = $4,500. But that’s way too much to be betting, like Mcallister said.

Thanks everyone for the helpful explanations and math!

I hadn’t considered the cost of errors made while trying to read, which as you guys point out could be significant (seems like the consensus is around -1%). Of course I guess that will depend on my actual error rate, but that is not too difficult to track.

The example I gave about 17s, 18s, etc. was only to illustrate my question about when surrendering would give you a statistical advantage. The point is well taken though, about not playing hands in a way that make it obvious you are hole carding, so thanks for the reminder.

So let’s assume the 50% rate which results in an advantage of 4.5% according to RS’s formulation.

So with a bankroll of 1000 units that would be .045 / 1.3 * 1000 = 34.6. So say a bet of 35 units for a full Kelly, 17 or so for a half? And I guess the idea is to do the Kelly calculation with every single bet, so it’ll increase slightly every time you win a hand, and decrease slightly every time you lose?

How does this compare with flat betting? Slower rate of return, but lower risk of ruin?

Quote: Wizard

Just use the approximate of advantage/variance, which is about 1.3 for blackjack. Most people find betting full Kelly to be too much for their stomach, including me, so adjust accordingly.

How do you get to that approximate ?
The paper by the Czech guys is sloppy. They give an approximate of EV / E[V^2]. Is that the source?

So we have an approximate by Kelly that is approximated by the Czechs which is further approximated by advantage/variance. And then partial Kelly which is not justified by any reasoning ( except deciding that your capital is a portion of what it is — but isn’t that the notion of bankroll?)

Quote: DJGenius

Thanks everyone for the helpful explanations and math!

I hadn’t considered the cost of errors made while trying to read, which as you guys point out could be significant (seems like the consensus is around -1%). Of course I guess that will depend on my actual error rate, but that is not too difficult to track.

The example I gave about 17s, 18s, etc. was only to illustrate my question about when surrendering would give you a statistical advantage. The point is well taken though, about not playing hands in a way that make it obvious you are hole carding, so thanks for the reminder.

So let’s assume the 50% rate which results in an advantage of 4.5% according to RS’s formulation.

So with a bankroll of 1000 units that would be .045 / 1.3 * 1000 = 34.6. So say a bet of 35 units for a full Kelly, 17 or so for a half? And I guess the idea is to do the Kelly calculation with every single bet, so it’ll increase slightly every time you win a hand, and decrease slightly every time you lose?

How does this compare with flat betting? Slower rate of return, but lower risk of ruin?

AFAICT, that seems about right.

Of course, I was using an example of a 10% edge and -1% (dis)advantage. The reality is probably much different.


I haven't done a bunch of holecarding, but it's likely harder to track misreads than you think. Likely you're going to see a range. Also, a mistake can be huuuuugely costly. You're "sure" you saw an ace, but it's really a deuce. Or you're "sure" the HC is a 6 but it's really a 7, upcard is a T.....that's not going to end well for ya. Or there's a 9 up and you think a 3 down but really it's a 2 down.


Yeah, as you win you'll be betting more and as you lose you'll be betting less. Although IMO it's not something you should actively think about while playing. Okay, so you start off with $1,000 and you're making $17 bets. You play for a while and you're up 10 bets, now you're going to be betting like $19?



Don't quote me on this 100%, but from what I remember doing some math & sims on flat betting vs kelly or fractional kelly..... Over some set period of time, say 100 hours, a flat bettor's results are going to be more equally distributed and more likely to come out ahead. The flat bettor's losses are going to be greater and his wins are going to be smaller -- BUT, he is more likely to come out ahead. OTOH, the kelly bettor is less likely to come out ahead than the flat-bettor, but his losses are going to be smaller and his wins are going to be greater.

What most APs do, as far as I can tell, is they just use whatever fraction of kelly they're comfortable with, say 1/4 kelly. Then if their bankroll goes up by 25% or down 25% (or whatever number they choose), then they resize their bets. IOW: They don't resize during a session.


Generally, you're going to not be limited by your bankroll but what kind of action the casino is willing to take.

Thanks RS for following up.

In terms of it being easy to track, I only meant that when the dealer does flip over the hole card, I’ll be able to know whether my read was accurate or not. So the longer I play the better picture I’ll get of my own accuracy as a percentage. It may differ from session to session, or from day to day, and will of course be affected by a number of factors, but I could use my worst case scenario to determine whether I am actually playing with an edge or not, and also to calculate the Kelly bet.

Sounds like I may need to do some sims of my own for the different approaches. I had just assumed that you would recalculate the amount every single bet... but now I see what you mean. So I’ll take some time to program a bit and see if there is much of a difference. It would be a lot less work to only calculate it for each session.

I am a pretty casual player and my bankroll is not very big. I’m just interested in this as a hobby and for some reason I enjoy the challenge of figuring out some way to get an edge somewhere. Believe it or not, I actually enjoy the math, programming and spreadsheets etc.

If I can actually turn this into a profitable situation for myself, I’ll be pleased. I doubt the casino will notice. Hopefully I’m not wrong!