Anomaly?

London Colin,

I don't want to get too deep into simulations for BJ tournaments. As far as I'm concerned, there are too many "judgment" calls and "human calls" for deviations from Basic Strategy that make any simulation of questionable value. Add to that the fact that BJ tournament rules are far from uniform which make simulations very difficult. However, I will take a few moments to respond to your questions.

My observation of your tables is that the expected "flow" of probabilities is not what I would consider normal. What I mean is this: For a moment, lets disregard the ".5" probabilities. If you look at the probability changes from 1 unit to 2 units to 3 units etc., the numerically difference varies too widely. Sometimes this difference is maybe .1 and sometimes it is .9. If it is .9 then the next difference may drop down to maybe .5. All these numbers are for illustration purposes only. I'm not up to doing the actual calculations. I would have expected the differences in the probabilities to change as one goes from 1 unit to 2 units to 3 units etc. but to change in a more uniform manner. Hope I made my self clear but I'm not sure. Alzheimer's you know - or is it Mad Cow - can't remember.

My second thought on your simulation is the rules you used. I have no idea about how you programmed basic strategy, if you used the right Basic Strategy chart, if you handled Double Downs and Splits correctly, if you handled a BJ Push correctly, if you handled insurance correctly, is using a single deck correct or should a shoe be used, was the shuffle really random, etc. . I'm not implying you made any specific error just raising the thought that this is complicated and errors can and do creep in the logic of programming. These errors then become "embedded" in code and are difficult to find. But of course you already know this.

The other problem I have with BJ Tournament simulations is that they need to be customized to the individual's way of playing a tournament in order to reveal reliable information for that single individual. Let's take insurance for example. I almost always insure a BJ because adding to my chip count is more important in a tournament than the expected value of not taking insurance. BUT, if I need to get that extra "1/2" then it's no insurance for me. As for surrender, most tournaments don't offer it. I've come across a couple of places where they didn't even know what surrender was. The list goes on and on about the deviations possible in tournament play. How can one account for all that?

Anyway, good luck on your efforts.

 

Thanks Toolman

Firstly, I should clarify that this is in no way meant to be a tournament simulator. It simply represents a single player, flat-betting using a fixed strategy, against a given set of rules. That should be enough to test the fundamentals of the whole 'skewing' issue, and is what Ken asked for.

Regarding the expected flow of probabilities, the bottom-line measure of how accurate the simulation is must be how closely the overall calculated EV matches the known EV for the given set of rules and playing strategy. As I've mentioned I don't really know how many hands are needed to guarantee a particular level of accuracy. I'm hoping a maths wizard might be able to advise me. I'm not sure, but I suspect that the number of 'sequences' required to obtain the same level of accuracy in the distribution of units would be at least as large and probably larger, so if 250M hands are needed to calculate the EV, two billion or more hands would be needed to accurately get the distribution of probabilities for 8-hand sequences! Again, any advice from the mathematically inclined would be gratefully received.

The 'known EV' that is quoted by the software is produced using combinatorial analysis. The same software components that are used to arrive at this figure are also used in the simulation itself to decide what action to take when dealt a particular hand, and utlimately arrive at the 'calculated EV'. So with more and more hands simulated, the latter figure should home in on the former. It's presently set up for 'perfect' composition-dependent basic strategy (no charts are involved, it's computed from first principles). This is all based on Eric Farmer's work which I have mentioned previously; it would have taken me years to arrive at this point, starting from scratch. I could substitute other strategies or rules and the 'known EV' would change accordingly, as of course should the 'calculated EV'.

I recently recorded the results for 250M single hands. The EVs match pretty closely -

Code:

Known EV: -0.5274%
Hands Played: 250000000   Units Won: -1.31438e+06
Calculated EV = -0.5258%

[...]

Win  Seqs: 105685786    (42.27%)        +128508158.50 units.    Value: +121.59%
Lose Seqs: 123482143    (49.39%)        -129822540.50 units.    Value: -105.13%
Push Seqs: 20832071     (8.33%)

So, I can imagine that there might be a lack of accuracy in some of the probabilities due to a lack of sufficient recorded hands, but I can't really see how a bug could cause the overall 'shape' of the distribution to be wrong without causing the EV itself to also be wrong. (which is not to say that I rule out that possibility entirely.)

I might paste the results into a spreadsheet and see if I can persuade it to plot a graph for me. Perhaps a beautiful bell curve will emerge. (Or possibly a horrible mess.)

The trouble with all this is that once you reach the point where the results are not totally and obviously wrong, it's hard to come up with tests to determine if they are in fact totally right.

Thanks for your good wishes. I foresee a long and difficult road ahead, but I shall endeavour to keep going.

 

Nice post and explanation London Colin. Understood everything you said, it was good.

You are one persistent SOB. Working on this type of simulation is indeed a long road. Maybe you can package and sell the final software. Again, GOOD LUCK on your endeavors.

PS: When I run simulations, I use 250 million hands. So I would suspect that 2 billion (American billion that is) hands would be needed since you need 7 or 8 hands to make a "sequence". That's just my opinion.

 

A picture is worth a thousand words

This is the 20M x 8 data viewed as a graph. Make of it what you will ...

 

Don't think I haven't thought about it! Probably the only way I could ever make money from blackjack.

That was my thinking, but I wonderedf if it needs to be even higher than that, because the rare events for a sequence are so very rare indeed. I think a mathematician's answer might feature phrases like 'standard deviation' and other things that will make me wish I had payed more attention in school.

 

London Colin,

I wouldn't think that those very rare events are any real concern. If they don't show up much in 2 billion hands then they are a real "anomaly" indeed. A mathematician may have other thoughts but that's mine.

I like the graph. Puts the data in prospective.

One question. Why do you think the probabilities for .5 units gain/loss (i.e. 2.5) always drop off substantially from it's whole number gain/loss? Example: The probability to win 2 units is much greater than the probability to win 2.5 units. The probability to win 3 units is less than 2 units (as expected) but greater than 2.5 units. I'll put this in numbers copied from your work:
2 unit win = probability of 5.6607
2.5 unit win = probability of 3.8373
3 unit win = probability of 4.4738
I assume the .5 units come from a BJ or surrender.

 

BJs are rare

I think you've answered your own question. For 0.5 of a unit to appear, there has to have been a BJ or a surrendered hand in the 8-hand sequence. These are rare events. Most sequences will contain neither and so must yield an integer gain/loss.

 

Of course. Makes sense. Now I think it's about 2:30am in your part of the world. Don't you sleep? (Last question for tonight, I promise)

 

Last answer for tonight

Yes I do. And I plan to right now.

 

spinning out!!!

I can't read all these posts currently so I'll review them in more detail tomorrow. One comment I have is that the number of push sequences is lower than I would have thought. If I understand push sequences to mean sequences of hands where you break even (i.e wins=losses) then I would have expected them to be higher than the figures your simulation shows.

I'm finding it difficult to articulate what I want to say at the momnet so I'll stop right now and try again later.

Cheers

Reachy

 

pushing

By a push sequence, I mean any that yields zero change in your balance. For short sequences with an even number of hands, I expect that would often be when won hands = lost hands. There will also be various combinations of surrender and BJ that can cancel each other out, or one DD/split balanced by two single bets.

Clearly, for an odd-numbered sequence, won hands cannot equal lost hands without at least one push.

 

8

Can you confirm what your win/push/lose results are for 8 hand sequences.

Cheers

Reachy

 

No Problem

It's listed in an earlier post, but here it is again ...

Code:

Win  Seqs: 9117027      (45.59%)        +25486460.00 units.     Value: +34.94%
Lose Seqs: 9459499      (47.30%)        -26327213.00 units.     Value: -34.79%
Push Seqs: 1423474      (7.12%)

I've got my new version just about working. I should be able to post a fuller set of results soon.

 

Wasn't sure

I wanted to make sure I was reading the right data so thanks for humouring me

As I said before I would of expected more push sequences than your data set suggests. Your outcomes for the 8 hand sequences seem to mirror the outcomes for single hands and that does suprise me.

Cheers

Reachy

 

Sneak Preview

I think the figure is just coincidentally close to the single hand figure. I don't want to clutter the forum up with lots of random data; I plan to generate one further set of large-scale results. But in the mean time, you may be interested in the % figures for push sequences produced by viewing a single set of 10M hands as sequences of various lengths, from 1 to 1000 -

Code:

[b]10	9 	8	7	6	5	4	3	2	1[/B]
6.04	6.54	7.14	7.82	9	9.64	13.2	11.41	26.23	8.33

[B]100	90	80	70	60	50	40	30	20	10[/B]
1.76	1.78	1.97	2.12	2.24	2.45	2.82	3.22	3.87	6.04		

[B]1000	900	800	700	600	500	400	300	200	100[/B]
0.41	0.57	0.64	0.58	0.75	0.77	0.86	1.02	1.34	1.76

 

More results

I've now got the results for one billion hands. One detail that stands out is that (far from short sequences being favoured) the probability of finishing ahead rises to a maximim at a sequence length of around sixty before starting to decline.

Code:

Known EV: -0.5274%
Hands Played: 1000000000   Units Won: -5.26654e+06
Calculated EV = -0.5267%

Sequence size: 1000
Sequences: 1000000
Win  Seqs: 438744       (43.87%)        +11931720.50 units.     Value: +2.72%
Lose Seqs: 555820       (55.58%)        -17198224.00 units.     Value: -3.09%
Push Seqs: 5436 (0.54%)

******************************

Sequence size: 900
Sequences: 1111111
Win  Seqs: 490603       (44.15%)        +12709004.50 units.     Value: +2.88%
Lose Seqs: 614233       (55.28%)        -17975533.50 units.     Value: -3.25%
Push Seqs: 6275 (0.56%)

******************************

Sequence size: 800
Sequences: 1250000
Win  Seqs: 555762       (44.46%)        +13625830.00 units.     Value: +3.06%
Lose Seqs: 686638       (54.93%)        -18892336.00 units.     Value: -3.44%
Push Seqs: 7600 (0.61%)

******************************

Sequence size: 700
Sequences: 1428571
Win  Seqs: 639344       (44.75%)        +14721324.50 units.     Value: +3.29%
Lose Seqs: 779839       (54.59%)        -19987826.50 units.     Value: -3.66%
Push Seqs: 9388 (0.66%)

******************************

Sequence size: 600
Sequences: 1666666
Win  Seqs: 751889       (45.11%)        +16093942.50 units.     Value: +3.57%
Lose Seqs: 902954       (54.18%)        -21360451.50 units.     Value: -3.94%
Push Seqs: 11823        (0.71%)

******************************

Sequence size: 500
Sequences: 2000000
Win  Seqs: 909332       (45.47%)        +17859647.50 units.     Value: +3.93%
Lose Seqs: 1075410      (53.77%)        -23126151.50 units.     Value: -4.30%
Push Seqs: 15258        (0.76%)

******************************

Sequence size: 400
Sequences: 2500000
Win  Seqs: 1145101      (45.80%)        +20268190.00 units.     Value: +4.42%
Lose Seqs: 1333155      (53.33%)        -25534699.00 units.     Value: -4.79%
Push Seqs: 21744        (0.87%)

******************************

Sequence size: 300
Sequences: 3333333
Win  Seqs: 1542045      (46.26%)        +23790041.00 units.     Value: +5.14%
Lose Seqs: 1758121      (52.74%)        -29056570.00 units.     Value: -5.51%
Push Seqs: 33167        (1.00%)

******************************

Sequence size: 200
Sequences: 5000000
Win  Seqs: 2334432      (46.69%)        +29694554.00 units.     Value: +6.36%
Lose Seqs: 2603911      (52.08%)        -34961069.50 units.     Value: -6.71%
Push Seqs: 61657        (1.23%)

******************************

Sequence size: 100
Sequences: 10000000
Win  Seqs: 4718225      (47.18%)        +43048533.50 units.     Value: +9.12%
Lose Seqs: 5108271      (51.08%)        -48315062.50 units.     Value: -9.46%
Push Seqs: 173504       (1.74%)

******************************

Sequence size: 90
Sequences: 11111111
Win  Seqs: 5246430      (47.22%)        +45523875.00 units.     Value: +9.64%
Lose Seqs: 5661249      (50.95%)        -50790416.50 units.     Value: -9.97%
Push Seqs: 203432       (1.83%)

******************************

Sequence size: 80
Sequences: 12500000
Win  Seqs: 5905802      (47.25%)        +48451517.50 units.     Value: +10.26%
Lose Seqs: 6351350      (50.81%)        -53718047.50 units.     Value: -10.57%
Push Seqs: 242848       (1.94%)

******************************

Sequence size: 70
Sequences: 14285714
Win  Seqs: 6752742      (47.27%)        +51980742.50 units.     Value: +11.00%
Lose Seqs: 7235872      (50.65%)        -57247280.50 units.     Value: -11.30%
Push Seqs: 297100       (2.08%)

******************************

Sequence size: 60
Sequences: 16666666
Win  Seqs: 7884668      (47.31%)        +56354418.50 units.     Value: +11.91%
Lose Seqs: 8408243      (50.45%)        -61620950.50 units.     Value: -12.21%
Push Seqs: 373755       (2.24%)

******************************

Sequence size: 50
Sequences: 20000000
Win  Seqs: 9457219      (47.29%)        +61979853.50 units.     Value: +13.11%
Lose Seqs: 10050611     (50.25%)        -67246386.00 units.     Value: -13.38%
Push Seqs: 492170       (2.46%)

******************************

Sequence size: 40
Sequences: 25000000
Win  Seqs: 11818758     (47.28%)        +69621772.00 units.     Value: +14.73%
Lose Seqs: 12495326     (49.98%)        -74888304.00 units.     Value: -14.98%
Push Seqs: 685916       (2.74%)

******************************

Sequence size: 30
Sequences: 33333333
Win  Seqs: 15731508     (47.19%)        +80820772.50 units.     Value: +17.13%
Lose Seqs: 16545260     (49.64%)        -86087314.00 units.     Value: -17.34%
Push Seqs: 1056565      (3.17%)

******************************

Sequence size: 20
Sequences: 50000000
Win  Seqs: 23479092     (46.96%)        +99625368.50 units.     Value: +21.22%
Lose Seqs: 24561292     (49.12%)        -104891906.50 units.    Value: -21.35%
Push Seqs: 1959616      (3.92%)

******************************

Sequence size: 10
Sequences: 100000000
Win  Seqs: 46046119     (46.05%)        +142158152.50 units.    Value: +30.87%
Lose Seqs: 47910050     (47.91%)        -147424694.00 units.    Value: -30.77%
Push Seqs: 6043831      (6.04%)

******************************

Sequence size: 9
Sequences: 111111111
Win  Seqs: 50861599     (45.78%)        +150011893.50 units.    Value: +32.77%
Lose Seqs: 52982958     (47.68%)        -155278433.00 units.    Value: -32.56%
Push Seqs: 7266554      (6.54%)

******************************

Sequence size: 8
Sequences: 125000000
Win  Seqs: 56970733     (45.58%)        +159323604.50 units.    Value: +34.96%
Lose Seqs: 59123096     (47.30%)        -164590145.00 units.    Value: -34.80%
Push Seqs: 8906171      (7.12%)

******************************

Sequence size: 7
Sequences: 142857142
Win  Seqs: 64384206     (45.07%)        +170518260.00 units.    Value: +37.83%
Lose Seqs: 67285633     (47.10%)        -175784801.50 units.    Value: -37.32%
Push Seqs: 11187303     (7.83%)

******************************

Sequence size: 6
Sequences: 166666666
Win  Seqs: 74686902     (44.81%)        +184375808.00 units.    Value: +41.14%
Lose Seqs: 76975995     (46.19%)        -189642348.50 units.    Value: -41.06%
Push Seqs: 15003769     (9.00%)

******************************

Sequence size: 5
Sequences: 200000000
Win  Seqs: 87765142     (43.88%)        +202563072.00 units.    Value: +46.16%
Lose Seqs: 92941261     (46.47%)        -207829613.50 units.    Value: -44.72%
Push Seqs: 19293597     (9.65%)

******************************

Sequence size: 4
Sequences: 250000000
Win  Seqs: 108051734    (43.22%)        +225760505.50 units.    Value: +52.23%
Lose Seqs: 108972884    (43.59%)        -231027046.00 units.    Value: -53.00%
Push Seqs: 32975382     (13.19%)

******************************

Sequence size: 3
Sequences: 333333333
Win  Seqs: 140582183    (42.17%)        +264881001.00 units.    Value: +62.81%
Lose Seqs: 154667204    (46.40%)        -270147540.50 units.    Value: -58.22%
Push Seqs: 38083946     (11.43%)

******************************

Sequence size: 2
Sequences: 500000000
Win  Seqs: 188920785    (37.78%)        +311466638.50 units.    Value: +82.43%
Lose Seqs: 179956234    (35.99%)        -316733179.00 units.    Value: -88.00%
Push Seqs: 131122981    (26.22%)

******************************

Sequence size: 1
Sequences: 1000000000
Win  Seqs: 422765392    (42.28%)        +514035188.50 units.    Value: +121.59%
Lose Seqs: 493934660    (49.39%)        -519301728.00 units.    Value: -105.14%
Push Seqs: 83299948     (8.33%)

******************************

 

Graph of distribution for sequences of length 1 to 10

Not sure it conveys much meaning, but it's quite pretty.

 

First off, if I were rich, I would use your second graph as a model for a super casino in Las Vegas! What do you think?

As for the results of 1 billion hands, I'm a little surprised that for say the 1000 hand sequences one would end up a net winner 44.83% of the time. I would have thought it would be a lot lower since the negative expectations of the game would have seemed to have a greater effect than your simulation shows. As for pushes, the more hands you play the greater chance for the probabilities to work as expected and therefore a lower number of times one would end up a net push after x number of hands. But now we have a conflict.

The pushes work as expected to the point that they are almost eliminated. But the win rate does not go down by any substantial percentage. The 44.83% win rate for 1000 hands is about what is expected for 6 hands. So the negative probability of the game is not reducing one's probability of winning if one plays 1000 hands per session as opposed to 6 hands per session. Am I missing something.

 

The Toolman Tower

If you build it, they will come.

Actually those two graphs are the same edifice, viewed from different angles. Rather than produce ten separate graphs, I combined them into one by going 3D, but produced two views to try and show all the detail. The equivalent for the 10 to 100 sequences looks even prettier - swooping curves, without the jaggedness.

Where do you get 44.83% from? The figure for 1000-hand sequences is 43.87%. I don't have an explanation for why, but it seems to rise between the lengths of 4 and 60, and then start to fall away.

 

Binary

This is really interesting data Colin. These results are essentially binary as far as I can see since they give no information on the size of win or loss sequences (which may answer some of Toolmans queries, maybe not). The only thing we have exact data on are the push sequences since they = zero. Interestingly the loss sequences go over 50% somewhere between 40-50 hands. Currently I'm not sure what to do with this information since I have "man flu" and am spaced out on co-codamol and must go to bed right now!

Cheers

Reachy

PS. Co-codamol (a mixture of codiene and paracetomol) isn't something you will get in the US since Paracetomol is banned over there due to it's liver toxicity I believe. About 50% of liver transplants in the UK are due to overdoses of Paracetomol and paracetomol related drugs and there are calls to get it banned over here too.