Why not just do a simulation to prove or disprove?Originally Posted by peterlee
It's a theoretical finding. To me, that trumps simulation. If you ask me what the probability is of getting heads on a coin flip, and I tell you 50%, if you then run a sim and find a different answer, do you go with the sim result?
Don
Well, it won't be 50% with a real coin as coins are heavier on one side than the other.Blackjack rules are also not as simple. Sims are used to design nuclear reactors, dams, airplanes, large bridges, pipelines as the calculations are too complex for pure math. They start with the theory and fine tune with simulations. Hurricanes and other weather phenom are still beyond both. But, they're slowly getting closer with atmospheric simulation.
"I don't think outside the box; I think of what I can do with the box." - Henri Matisse
Usually theories base on logic.
A round end up with a high card more often…If we agree with this, my logic is simple, after the first round, more high cards are used on average, then before the second round, RC drops.
So where is wrong with this simple logic?
At one point I ran a series of sims relative to CCE.
1. Single deck, 1 player versus dealer, 10,000,000 rounds.
2. CD basic strategy was used. TD basic could have been used instead.
3. Cut card postion was varied to follow fourth through fifteenth cards from the top of a full single deck. I stopped at fifteenth card because this was the first position that the cut card was never once encountered on round 1 of 10,000,000 rounds. For the other positions the first round was also the last round a variable number of times because cut card was encountered on round 1.
4. Results for 2 rounds were recorded:
Precision of results was of little importance, only exposure of basic tendency of CCE.Code:round before cut card is encountered round when cut card is encountered, causing a reshuffle
what this showed is somewhat common sense.
Player can eliminate CCE by wonging out after the first round where cut card encounter prob > 0. A small mistake in estimating this point results in a smaller reduction in basic strategy EV whereas a larger mistake results in a much larger reduction.Code:When the cut card is sufficiently deep that there is zero probability of it being encountered: Overall EV using basic strategy = full shoe EV using basic strategy on that round. When cut card encounter prob > 0 on a round: 1. If cut card is actually encountered there is no CCE because reshuffle occurs 2. If cut card is not encountered CCE is manifested on following round
These sim results are on my (disorganized) website: http://www.bjstrat.net/simResults.html
I'll leave any analysis of CSM to someone else.
k_c
One of the major misses people have had in evaluating the One2Six is considering the lag in the buffers, but not the bult in lag in the slot picking mechanism. Realize that it is important for a CSM to avoid recycling cards too quickly as that could make them beatable in a different manner. So, they build in a lag not existing in a casino shuffle. Unlike ASMs, CMSs are less random.
"I don't think outside the box; I think of what I can do with the box." - Henri Matisse
Drops from WHAT? What was it before the ten was dealt? Why is everyone fixated on this alleged fact that the first round ends with a ten more than 4/13 of the time? IF that is true (and I'm not at all sure that it is), then maybe before the ten is dealt, on average, more than 9/13 of the cards are non-tens. You simply cannot say that, by playing blackjack, you can force the RC to be negative after a round. I don't believe that. If it were true, then BS edge wouldn't be the same as the first round ... and it is, for sure. I'd be willing to bet on that.
Don
Drops from before the first round, a new shoe, that is RC=0.
I got this idea from the followings:
1. People say a hand end up with a high card more often than end up with a low card.
My thinking, from the top of a shoe, I choice to a point that is a T, and count from the top to that T, likely more often to be negative RC. So I accept the "end up with a high card more often" idea.
2. A friend of mine showed me the page
https://wizardofodds.com/games/black...t-card-effect/
and ask me how do I think about the figures in the page.
I found it strange, CSM should not dealt out high card more often. After I read the introduction, I think it use one round per shoe to simulate as a CSM machine. So I think one round per shoe will produce those figures: more high cards are dealt.
3. Then I ask a friend to do simulation of one round per shoe, the result shows that after the first round, RC drop to negative on average. The RC is compared with the RC0 before the first round.
Of course, the difference is very slightly. But enough for me to doubt about the WOO csm simulation.
There are some unproven concepts and unverified conclusions floating around this thread and also this one:
https://www.blackjacktheforum.com/sh...CSM-simulation
I've run a few sims in an effort to confirm the facts about what is being discussed.
1 - Last Card of a Round Bias?
The idea from which most of the rest of discussion stemmed was that the running count during the first round of a shoe tends toward the negative. We'll deal with that soon, but this idea was supported by the suggestion that a round of basic strategy blackjack tends to end with a high card and, in particular with a 10-valued card.
Norm correctly pointed out that there are more 10s than any other rank (4/13 vs 1/13), so this makes some sense. But if we're using this notion to support the idea that the running count tends toward the negative during the first round, what we really want to know is whether the negatively tagged ranks tend to end a round more than one would expect by random chance. The cards are dealt randomly (or at least unpredictably) and so the final card, which is the only one subject to a stopping decision, can be the only source of such a bias.
Because it's the most commonly used counting system and because it's probably the one being assumed here, let's use Hi-Lo. The negatively tagged ranks are Ace and the 10-valued cards. All sims were run using 6D S17 DOA DAS SPL3 SA1 (one card on split aces) and were run for 2 billion rounds using my own software.
I first wanted to know whether this bias exists in general, so I ran a sim with 7 players heads up vs the dealer and 4.5/6 decks in play. I used 7 players because I wanted to isolate the dealer's role in this issue. With 7 players at the table, the dealer almost always plays out his hand. Here is the resulting distribution of the final card dealt during the simulation. For reference 1/13 = 7.6923% and 4/13 = 30.7692%:
We can see that there is a general trend for which high value cards end the round more frequently and the low value cards less so. The Ace is a special case because it is both a high and low card. In particular though, we see that 10-valued cards end the round much more often than randomness suggests (38.7389% vs 30.7692%). Also, while the ace finishes a round less often than if it were random (it is used as a low card more of than than as a high card), the 10s and aces together finish a round 44.6624% of the time which is more than the sum of the probabilities for 2 through 6 which is 29.8644%.Code:A 2 3 4 5 6 7 8 9 10 ------------------------------------------------------------------------------------------------------------ 5.9235% 3.3775% 4.8635% 6.2381% 7.5187% 7.8666% 8.1897% 8.4965% 8.7871% 38.7389%
So the final card of a round does tend toward the negative significantly in general while the remaining cards dealt during the round should come randomly. This suggests a potential bias toward the negative in general. Here are the final card distributions for a few more scenarios:
4.5/6 decks, one player. The trend persists even when the dealer finishes his hand less often:
And since we're going to move on and talk about the initial round of the shoe next, here are the distributions for 7 and players and 1 player respectively for the initial round only:Code:A 2 3 4 5 6 7 8 9 10 ------------------------------------------------------------------------------------------------------------ 6.3118% 2.9461% 4.0992% 5.1693% 6.1641% 6.8743% 8.0664% 8.7357% 9.3868% 42.2462%
We can see that the frequencies for 1 round are very close to those for 4.5/6Code:7 Players: A 2 3 4 5 6 7 8 9 10 ------------------------------------------------------------------------------------------------------------ 5.9236% 3.3716% 4.8576% 6.2340% 7.5149% 7.8641% 8.1863% 8.4963% 8.7878% 38.7638% 1 Player: A 2 3 4 5 6 7 8 9 10 ------------------------------------------------------------------------------------------------------------ 6.3141% 2.9400% 4.0921% 5.1624% 6.1585% 6.8682% 8.0632% 8.7352% 9.3904% 42.2760%
Running Count Bias for the First Round of the Shoe?
Based on the bias confirmed above, it was suggested that the running count during the first round of a shoe should tend toward the negative. I once again rans sims using 7 players and one player. Here is the distribution of the Hi-Lo running count after one round has been played off the top of the shoe with 7 players:
The running count does indeed end up negative more often than positive. However, since RC=0 is part of the TC=0 bin, the true count remains non-negative more often than it goes negative.Code:-19.0: 4 = 0.00% -18.0: 54 = 0.00% -17.0: 390 = 0.00% -16.0: 2,787 = 0.00% -15.0: 16,178 = 0.00% -14.0: 79,986 = 0.00% -13.0: 315,037 = 0.02% -12.0: 1,014,843 = 0.05% -11.0: 2,761,620 = 0.14% -10.0: 6,500,571 = 0.33% -9.0: 13,532,477 = 0.68% -8.0: 25,288,297 = 1.26% -7.0: 42,962,343 = 2.15% -6.0: 66,837,622 = 3.34% -5.0: 95,921,440 = 4.80% -4.0: 127,625,071 = 6.38% -3.0: 158,002,569 = 7.90% -2.0: 182,605,477 = 9.13% -1.0: 197,475,935 = 9.87% 0.0: 200,250,564 = 10.01% 1.0: 190,742,782 = 9.54% 2.0: 171,053,267 = 8.55% 3.0: 144,616,397 = 7.23% 4.0: 115,489,675 = 5.77% 5.0: 87,247,180 = 4.36% 6.0: 62,435,664 = 3.12% 7.0: 42,386,408 = 2.12% 8.0: 27,326,377 = 1.37% 9.0: 16,739,413 = 0.84% 10.0: 9,760,852 = 0.49% 11.0: 5,422,912 = 0.27% 12.0: 2,868,135 = 0.14% 13.0: 1,450,538 = 0.07% 14.0: 699,716 = 0.03% 15.0: 322,818 = 0.02% 16.0: 143,470 = 0.01% 17.0: 60,947 = 0.00% 18.0: 24,541 = 0.00% 19.0: 9,769 = 0.00% 20.0: 3,772 = 0.00% 21.0: 1,359 = 0.00% 22.0: 493 = 0.00% 23.0: 161 = 0.00% 24.0: 60 = 0.00% 25.0: 21 = 0.00% 26.0: 4 = 0.00% 27.0: 4 = 0.00% Positive = 878,806,735 = 43.94% Zero = 200,250,564 = 10.01% Negative = 920,942,701 = 46.05%
A similar trend is shown with a single player in action:
This is where Don referred us to Thorp's theorum stating that EV is invariant for all basic strategy rounds throughout the shoe and asked: How can this be if the second round of the shoe is biased toward beginning with a negative running count? I can't say whether these results violate Thorp's theorum, mainly because I don't have access to the proof in order to evaluate the situation mathematically and I don't want to speculate.Code:-10.0: 24 = 0.00% -9.0: 320 = 0.00% -8.0: 3,335 = 0.00% -7.0: 26,437 = 0.00% -6.0: 373,720 = 0.02% -5.0: 1,581,197 = 0.08% -4.0: 49,153,892 = 2.46% -3.0: 155,829,745 = 7.79% -2.0: 292,373,869 = 14.62% -1.0: 344,666,114 = 17.23% 0.0: 397,224,920 = 19.86% 1.0: 311,213,292 = 15.56% 2.0: 219,965,408 = 11.00% 3.0: 124,904,784 = 6.25% 4.0: 63,080,489 = 3.15% 5.0: 26,309,655 = 1.32% 6.0: 9,326,538 = 0.47% 7.0: 2,910,546 = 0.15% 8.0: 799,753 = 0.04% 9.0: 190,590 = 0.01% 10.0: 45,864 = 0.00% 11.0: 13,323 = 0.00% 12.0: 4,292 = 0.00% 13.0: 1,359 = 0.00% 14.0: 379 = 0.00% 15.0: 109 = 0.00% 16.0: 35 = 0.00% 17.0: 9 = 0.00% 18.0: 2 = 0.00% Positive = 758,766,427 = 37.94% Zero = 397,224,920 = 19.86% Negative = 844,008,653 = 42.20%
Very nice work, Dave. But everyone needs to understand something: When the TC is floored, there is no symmetry around zero for positive and negative counts. For over 40 years, anyone could look at TC distributions from Wong's work and then mine and Norm's in chapter 10 to see the asymmetry. When you floor, the TC is "pulled" to the left; the bin to the right of exactly zero, from 0 to 1, gets lumped in with TC = 0, while the corresponding bin to the left of exactly zero, from -1 to 0 becomes TC = -1. As a result, there will always be a much larger frequency of negative counts than positive ones. However, this fact alone has absolutely nothing to do with the edge that the BS player has from one round to the next. If it did, that player would constantly be experiencing a bias to the BS edge that is less than that of the off-the-top edge. Again, this is simply NOT the case.
Despite Dave's very lovely work, above, it does not in any way disprove Thorp's fundamental theorems that state that BS edge is INVARIANT from one round to the next. And while that may not seem logical or intuitive to some of you, it certainly does to me.
Don
Last edited by DSchles; 01-15-2023 at 08:25 AM.
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