Bet Sizing in Theoretical Game

Assume a theoretical blackjack game where we can calculate our exact advantage before each hand, and can play perfect composition-dependent strategy. We can also play or not play at any given time. We cannot play more than one hand at a time.

Given this scenario, what would be an appropriate betting strategy?

Table min/max is 1 to 50
Our Bankroll is 500 units
We can bet size in any whole number amount from 1 to 50.

What would our plan be for an ROR of 5%?
How about for an ROR of 20%?

I was experimenting with various Kelly betting schemes, but didn't like the results because the advantage is not directly related to our percent chance of winning.

I think it is a safe assumption that we will never play if our edge is <=0, but do we start playing if we have any positive edge, no matter how tiny?

Any help in tackling this problem would be appreciated. Thanks in advance.

given this is all theoretical, i think it would come down to a balance between time (i.e. expected units/hr) and ror.

if time is not a factor then just bet when there is nothing left in the deck but aces and tens

also, besides no specified time constraint there is also no target goal for the end bankroll..

Good points. Thanks for your posts.

As for a time constraint: Let's assume you must play 1000 (6 deck) shoes with another player. You can play as few or as many hands as you like. If you play them all, you will see something like an average of 30 hands/shoe (240 cards). But you can play fewer, or none. (It is quite rare with composition-dependent strategy to have zero hands in a shoe where you have an advantage, if pen is at all decent.)

The goal is to maximize profit while staying within the two RORs I provided (5 and 20%.)

Does this help?

ran a sim and got 21% ror for a 1-16 bet spread, wonging at -1 using vegas 6d rules with 75% pen, counting with wong halves and full indexes.

if you want something more exact, i don't think cvdata will let me set up a bet spread based on the exact player advantage.. and i probably could have used a better system than wong halves

Thanks. I have CVData, and have used it for this, but like you said it won't do a betting strategy based on an exact advantage. I guess one way I could do it would be to use the best/most complex count, and use that. But a Composition Dependent determined edge would still be far better, as would the decisions that are made.

What do you want to maximize? Your EV? Logarithmic growth? You said that you didn't like the kelly betting because advantage is not directly related to the chance of winning. Somewhat true in general, but for bj, the variance is (relatively) low, which gives a high correlation between advantage and probability of winning.

The optimal betting scheme for bankroll growth would be a (fractional) kelly bet depending on the advantage that the deck gives for the next hand calculated using the composition strategy (assuming by composition you mean deck composition, not hand composition, which is somewhat negligible).

I have fairly recently been working with iCountNTrack on a project to make the data output from my cdca (composition dependent combinatorial analyzer) program available to a scientific analysis program he uses.

There turned out to be a few problems to overcome first, though.

First thing I did was compile a .dll. Although the data output from the .dll could be read and used by programs written in languages that could access a Microsoft C++ .dll the scientific program required an extra interface in order to access the .dll. The .dll could be read out of the box, though, in Excel using VBA.

Finally with iCountNTrack's help in trying to learn enough of his scientific program to create an external procedure to it in C++ we were able to get the necessary interface working.

Next I added the ability to sim either optimal composition dependent perfect play strategy or composition dependent full shoe basic strategy to the .dll. However, we found that my original programming was faulty because after about 125,000 rounds of perfect play simulation, the program crashed because it had a memory leak.

I found the source of the memory problem and I rewrote my code in all of my programs that had the faulty code.

The simulation interface to the .dll requires the user of the it to supply his/her own shuffle. iCountNTrack's program has what seemed to be a very fast shuffle when the .dll was employed from within his program as compared to my shuffle algorithm using the C++ random number generator when the .dll was emplyed from within a C++ program. In any case, nobody can accuse the .dll simulation of being biased due to a faulty shuffle because the shuffle is external to the .dll.

Simulation of basic strategy is pretty fast. iCountNtrack's program is faster than my C++ test program I think mainly because the shuffle is faster.

A basic CD strategy sim to a pen of 4 cards should have shown an EV virtually equal to the computed full shoe CD EV and that proved to be the case so the shuffles employed seemed to be working adequately.

Optimal play simulation takes more time because each play needs to be computed. It could probably be faster because my program computes all up cards on all calculations even though there is only one up card for each round of a sim.

I think we were able to show EV for flat betting (single deck, 3:2 bj, s17, DOA, NDAS, split 2-10 once, split aces once, one card to split aces, full peek, no surrender, penetration of 35 cards) is something like +.7%. In this scenario every play made is as good as it can be using everything known when the play is made. This includes all split hands.

I was mainly working to get the process working but I think I completed about 100,000 rounds of above a couple of times. iCountNTrack may have done more once the bugs were worked out, I don't know.

I have never been one to worry too much about optimal betting, choosing instead to be satisfied with ensuring positive EV and using common sense. However, one could add the step of computing pre-deal EV and applying a system of bet spreading to the above. This would require even more computer time (to get pre-deal EV) but since everything is computed an overwhelming amount of data wouldn't be required.

However, remember at the end of the day this is only theoretical. It's easy to forget that.

A future thing to do might be to record all of the intermediate results of an optimal sim. That type of data might prove to be a bit more practical to use. I'm not doing much programming at this time, though.

Answering your questions/points:

I want to maximize the total money made (EV).
Maybe I need to work with Kelly betting.
Composition is unplayed card composition.

Thanks. I appreciate your thoughts.

Interesting stuff, though I admittedly don't quite understand everything you are talking about.

some people here would beg to differ, including myself and i dont use halves :whip:

what about revere rapc? 1.0 BC

or perhaps uston apc with a side of aces

In a nutshell it's a simulation of a heads up game where player has the benefit of knowing his mathematically best play for every decision he must make.

I think you asked about the advantage using an optimal bet spread in such a game. The only sims I did used flat betting without worrying about pre-round EV but all I'm saying is a spread strategy could have been used instead based upon the actual calculated EV before each round.

The other stuff is just basically background.

Sorry if I'm confusing. :grin:

Ah! OK, got it. Thanks.

I haven't read it in awhile, but I believe Griffin's Theory of Blackjack shows a player advantage of something just under 1% if perfect composition dependent play is assumed.

Playing with CDCA, I see that it happens on occasion that the player advantage exceeds 5%, and even gets to 10% sometimes. How much to bet in such a situation? I guess it also depends on the likelihoods of various results such as a blackjack, a double, a split, a surrender, etc. And each of these would change based on not only the advantage, but more specifically the specific composition.

from the limited amount of information i've read on this, you want to bet x% of your total bankroll when you have y% advantage, where x is some portion (k) of y.

i'm not sure what the optimal k should be exactly, and i don't know if there exists any conclusive research on it, but i think most would agree that it is between .5 and .8?

i don't know how to calculate y given a certain deck composition, but it can be done (as k_c points out).

so for example let's say the deck is stacked and you have 5% adv at the beginning of a hand. you would want to bet something like 3.5% of your bankroll. and then you would play out your hand based on the deck composition. however, this constant bet sizing would be for a 0% ror. if you want something between 5-20%, it could be calculated but not with cvdata afaik..

i'm don't know what practical information can be had from this theoretical situation

Yes, if you can bet $0 at any <0% EV situations, which, from the "theoretical" perspective of the thread, it seems to be the case.

I'm curious too if there was a practical reason for the thread, or if it was just a nice little exercise in math

Try this experiment in cdca, Input 6 decks and remove all cards except the 7s, 8s, and 9s. Make sure the compute mode option is set to 'Use best CD strategy'. EV in that case is over +50% when only 1 split is allowed and over +60% when 3 splits are allowed.

Now change the compute mode to 'Use basic full shoe CD strategy'. What happens to EV?

Although the likelihood of ever encountering this composition is virtually nil, it illustrates why this is mainly a theoretical exercise. If you were a super human player that knew not only every exact composition but also how to play any situation, it wouldn't be theoretical. However, the rest of us that probably rely on counting to bet and play would play at an EV far removed from the theoretical EV.

Right. How can we determine what k is, or even what it should approximately be? How did one come up with the .5 and .8 figures, for example?

CDCA does this.

In this instance ror is not zero, right? Since there is a minimum bet, ror must be > 0?

This sort of gets back to the initial question: how can we calculate ROR based on a given k?

I have some ideas that I am working through. I'd love to share them when appropriate. Thanks so much for the input thus far.

ah yes well those are some interesting questions. from the little i've read on the topic, the reason x=ky instead of x=y is to account for the increased variance due to 3:2 payouts, doubling, splitting, etc.

you are correct that the practical ror isn't 0, but in order to determine that you would need to define some cutoff point (p) where you don't lower your bet spread and play with a "real" ror. but again, lowering k also lowers your chance of reaching that point.

but in order to sim all this (and find that magical k value) you would need to be able to specify a betting strategy based on your ever-changing bankroll (b). and when b <= p, you'd need to define a "minimum" betting spread (e.g. $10-$100). or you could just declare yourself "ruined" when b = p.

let's ask qfit to implement this in cvdata

Isn't The Answer Simple?

Theoretical or real world the answer is the same.

Bet Kelly!:joker::whip:

However!:joker::whip:

The Kelly criterion (constant resizing) assumes a simple plus minus payoff. In blackjack we do not have that and considering the variance of the game is not enough.

Becccccause:
Even if you had a 15% advantage if you were playing a DAS game you would probably not bet 15% of bank because you could not split fully. If you can split and double to make 4 hands and if you add a potential insurance bet one bet could turn into 8.5 bets.

Also, in the real world we are not certain of our advantage, it's only an approximation.

So!:joker::whip:
Given the above annnnnnd
The growth rate of betting .5 kelly and 1.5 kelly is the same, except the latter having a higher variace. Also, the growth rate of .5 kelly is 75% that of kelly. Finally, the chace of losing 90% of bank with kelly is 10% and with
.5 kelly is .1%. Even if one felt they had a good grasp of their real advantage there are many compelling reasons to not bet fully kelly.

So kelly is always the answer!:joker::whip:
errrrr
.5 kelly
perhaps even .75 kelly because "greed is good"
or a lower fraction then .5 kelly if one wishes to keep variance down