In Search of... Clumping (was Re: Reply to Ken Fuchs)
by Abdul Jalib
From r.g.b.m
"RUSSELL J. HALL" writes:
> We believe that the more players in the game, the more the
> cards clump up. Perhaps not for that number of players but for a fewer
> number of players. When you go from 7 to 5 players the game should get
> worse for the players. On the other hand 1 or 2 players vs the dealer have
> a tendency to unclump the cards.
Nrrr. The opposite is true, as I will show.
I arm myself with my realistic blackjack simulator and go In Search Of...
CLUMPING [insert discordant creepy music]
(Actually, I did a similar study around 1990, and I posted the results,
but I don't have a copy, so I have to do it from scratch here. Brace
yourself, this is going to be a long one. You can cheat by skipping
down to the line that says "*SUMMARY*".)
The game being simulated is 6 decks, S17, DOA, DAS. My program includes
very detailed implementations of real Atlantic City casino shuffles, but
I thought the clumpers would go bonkers trying to make all sorts of
modifications to the shuffle, so I opted for a simple shuffle: unplayed
stacked on top of played, cut in two, half deck picks, single riff each
pair of picks, stack until done, random cut. The interlace frequency
was modelled after an actual shitty dealer's drops: 66% chance 1 card,
26% chance 2 cards, 5% chance 3 cards, 2% chance 4 cards, and 1% 5 cards.
The grabs are subject to up to a +-20% error, evenly distributed. To avoid
any disputes about "card boxing" I stuck to a single riff. My program
puts the cards in the discard tray just like Atlantic City dealers
do. The hands are played using proper basic strategy (sometimes without
splitting as noted.) If clumping does not occur with this, then why
would it occur with more thorough casino shuffles?
The simulation is of 100,000 shoes, and in each shoe the first 169
cards are compared with their successors. (There's a method to this
madness of 169, but mostly it was unnecessary.) The theoretical
distribution of the 16,900,000 trials can be computed mathematically.
I show this theoretical frequency distribution of previous card (column)
to next card (row) below:
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
theoretical expected frequencies from math not simulation
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
96141 100322 100322 100322 100322 100322 100322 100322 100322 401286
100322 96141 100322 100322 100322 100322 100322 100322 100322 401286
100322 100322 96141 100322 100322 100322 100322 100322 100322 401286
100322 100322 100322 96141 100322 100322 100322 100322 100322 401286
100322 100322 100322 100322 96141 100322 100322 100322 100322 401286
100322 100322 100322 100322 100322 96141 100322 100322 100322 401286
100322 100322 100322 100322 100322 100322 96141 100322 100322 401286
100322 100322 100322 100322 100322 100322 100322 96141 100322 401286
100322 100322 100322 100322 100322 100322 100322 100322 96141 401286
401286 401286 401286 401286 401286 401286 401286 401286 401286 1588424
For the sims, the 3 standard deviation confidence intervals on the figures are:
For 96141: [95214, 97068]
For 100322: [99375,101269]
For 401286: [399408,403164]
For 1588424:[1584808,1592040]
For example, for the bottom right hand figure of the table, the 1588424
refers to the relative frequency of a 10 following a 10, and in our
simulation of 16,900,000 trials, it's not likely to occur less than
1584808 times or more than 1592040 times. We'll use these confidence
bounds to judge whether the simulations are exhibiting anything out of
the norm.
Next I'll present the control case, 7 hands, no splitting, with
the control being the random shuffle, since we're using it as a
crosscheck on the theoretical values...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
7 hands, no splitting, random shuffle (control)
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
96488 100400 100316 100294 100600 100617 100601 100044 100573 401561
100103 95999 100660 99633 100922 100333 100139 100021 100764 401423
100421 100777 95575 100067 99526 100491 100680 100504 100312 401374
100099 100391 100236 96338 100244 100827 100561 100124 99827 400342
100716 99963 100102 100582 96325 100725 99828 100446 99939 401405
100528 100342 100270 100160 100816 95720 99984 100364 100231 401853
100154 100007 101524> 100226 99714 100731 95913 100164 100273 401679
100225 100590 100221 100022 99652 100193 100236 96167 100370 401483
100481 99855 100419 100089 100303 100166 100902 100229 95677 401666
402240 401665 400569 401764 401816 400378 401646 401059 401803 1587223
Interpretation: This is just a "control" simulation - I used a random
shuffle, even though I already know what the results should be from theory.
One result falls outside the 3 standard deviation confidence interval.
I've indicated it with a greater than sign (">"), meaning it's above the
upper bound. When you see the less than signs ("<") later, you can
guess what they mean. This is not totally unexpected, as there are 100
entries in the table, and there is about a 1 in 500 chance of any one
entry being outside the confidence bounds strictly by random fluctuation.
Anyway, the random shuffle appears pretty random, which is all I wanted
to verify.
Now let's see what happens if we use a realistic shuffle...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
7 hands, no splitting, realistic shuffle
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
97869> 100280 100185 99945 99900 99757 100521 99980 100371 400399
99667 97643> 100847 101419> 101306> 101113 101236 100652 100303 398742<
100299 100680 96271 100690 101248 100596 100598 100610 100203 399898
99848 100745 100932 97056 100424 100224 100883 100514 100311 400604
99578 100396 100969 101247 96496 100457 100700 101181 100648 400132
99529 100371 100238 100202 100317 95867 100330 100485 100335 400506
100580 100551 100683 100591 100570 99969 96702 100555 100165 401565
100279 100893 100671 100311 100329 100407 100196 96083 100124 401948
99965 101144 100595 100157 100078 100372 100569 100289 95658 400698
401603 400405 399824 399689 401176 399613 400116 400841 401333 1587950
Interpretation: With 7 players, no splitting, and a realistic shuffle,
small cards have a slight tendency to clump with each other. Aces tend
to follow aces, and deuces tend to follow deuces, fours, and fives.
Now let's see what happens if we allow splitting...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
7 hands, realistic shuffle, splitting allowed
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
88899< 101521> 101363> 100931 100398 100417 100990 102048> 100837 402606
100736 91932< 102310> 101434> 101258 101218 101959> 101838> 100863 398538<
101111 101793> 91535< 101136 101119 101368> 101387> 101420> 100931 400498
100231 101730> 101152 94828< 100515 100429 101459> 101016 100548 399079<
99960 101043 101245 100035 96142 100191 101023 100805 100286 399613
100273 100592 100997 100308 100787 92974< 100627 101903> 100677 400945
101266 101291> 101190 101175 100972 101113 91008< 101468> 101479> 401813
102018> 101058 101185 101190 101456> 101369> 101612> 86059< 101785> 402378
101128 100490 100289 101040 100252 100920 101119 101428> 90735< 402118
404498> 400587 401115 399084< 397442< 400217 401354 402250 401374 1583868<
Interpretation: If we allow splitting, then the results change fairly
dramatically. Now aces *hate* aces, and deuces *hate* deuces, and
similarly for all the other split cards. Note that 5's get along just
fine with each other. The sudden profusion of values over their max
bounds must be attributed to being a side effect of the split cards'
distaste for each other. For example, if an 8 tends not to follow an
8, then every other card is more likely to follow an 8 than it would be
with a random shuffle. The anticlumping of 10's is probably also a side
effect, since sequences like 88TT tend to be less likely than sequences
8T8T - runs of tens get broken up in the haste of split cards to get
far away from each other. In case it's not obvious, the split cards
don't really hate each other - it's just that the shuffle is too weak
to bring back together cards that have been split apart and had a few
cards inserted in between.
Now let's see what happens when we drop from seven hands to one hand...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
1 hand, splitting allowed, realistic shuffle
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
90751< 101179> 100944 99986 100937 101031 101019 101883> 100474 402763
102033> 95692 103085> 102691> 101271> 102070> 100564 100163 99654 393693<
101527> 103890> 94247< 101760> 101464> 101149 100342 99589 99399 396220<
101218 102470> 102079> 96545 101349> 101497> 100198 100342 99623 395246<
100945 102176> 102079> 101784> 96266 100478 100363 99957 99666 395885<
100567 102148> 101794> 101079 100603 93924< 100539 100145 100400 397574<
101247 100103 99690 99919 99686 100242 92658< 101676> 101131 403862>
100620 99230< 100032 99491 100369 100096 101290> 90260< 101274> 407644>
100807 99190< 99041< 100411 99591 99783 100855 101735> 93129< 405038>
401266 394761< 396652< 396605< 398235< 398799< 402399 404454> 404920>1601400>
Interpretation: when dropping to heads up, still with splitting and realistic
pick-up and shuffling procedures, split cards still anti-clump and small
nonidentical cards clump more. The big difference is that now big (8,9,T)
cards clump (except that 8's anticlump and 9's anticlump within themselves).
The largest effect is the 8 and ace anticlumping, which is about equivalent
to the effect of an ace/8 or two being removed from a six deck shoe when
the previous card was an ace/8. The ten clumping effect is the equivalent
of an extra ten being in the shoe when the previous card was a ten. This
could be used as a tie breaker for close strategy decisions, but no extreme
strategy or betting changes would be warranted.
Now let's see what happens if we disallow splitting for the one player...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
1 hand, no splitting, realistic shuffle
================================PREVIOUS CARD==================================
96041 100423 100099 100191 100733 99929 100077 100999 100509 400881
101402> 99533> 102966> 102205> 101747> 101345> 99983 100104 99300< 392658<
100564 103723> 98063> 101878> 101700> 101387> 99683 99116< 98702 395634<
100922 102454> 101778> 97342> 101023 101115 100843 99893 99483 395733<
100819 102755> 100867 101434> 97044 100931 100064 100052 99736 396415<
100728 101463 101321> 101339> 100291 96395 100423 99697 99836 397927<
100450 100275 100474 99648 100295 100092 95817 99554 100810 402956
99819 98798< 99435 99588 99767 100125 100945 96871 100629 404870>
99826 98467< 98821< 99320< 99250< 100265 100937 100851 96595 405255>
399102< 393338< 396890< 397695< 398298< 397917< 401709 403712> 404096>1604740>
Interpretation: Clumping ahoy! Small cards (A-6) tend to clump, as to
big cards (8,9,T). This seems to be the most pronounced clumping case.
Even so, the clumping effect of a ten preceding a ten only just neutralizes
the effect of removal of that first ten, so there's not much exploitability
here. Also, it costs 1% to never split, so you should not consider
refraining from splitting.
Now just for completeness, let's see the control case for 1 hand...
FREQUENCY DISTRIBUTION OF PREVIOUS CARD TO NEXT
(Rows are the next card, A,2,3...T, from top to bottom)
1 hand, no splitting, random shuffle (control)
================================PREVIOUS CARD==================================
A 2 3 4 5 6 7 8 9 T
------- ------- ------- ------- ------- ------- ------- ------- ------- -------
95545 100294 100775 100116 99664 99950 100721 100262 100143 400991
100501 96080 100458 100598 100191 101010 100328 100630 100526 401427
100509 100375 95680 100050 100912 99809 100739 100193 100029 401498
100548 100330 100035 96252 100353 100281 100533 100277 100275 400985
99751 100577 100061 100417 95902 100517 100404 100349 100311 401254
100320 100050 100696 100236 100175 96113 100578 99612 100172 401382
100202 100247 100480 100520 100482 100424 95939 100176 100274 401806
100262 100734 99776 101132 100695 100024 100230 95941 100501 400540
99743 100296 100520 100170 100834 100381 100078 100386 96038 401881
401012 402847 401293 400400 400132 400935 400838 402024 402103 1588954
Interpretation: The previous control had one value outside the bounds; this
one does not, which may allay concerns about the random number generator
and whatnot.
If you've gotten this far, you're probably overwhelmed by numbers, so
I'll try to summarize briefly.
*SUMMARY*
% CLUMPING OF TENS
SPLITTING
Yes No
H ------ ------
A 1 | +0.817 +1.03
N
D 7 | -0.287 -0.036
S
The above table shows you how far off the tens-follow-tens results are
for each of the four experiments. You can infer that the fewer the hands,
the less the anti-clumping of tens (ultimately becoming clumping). Also,
if splitting is not allowed, then the anti-clumping is reduced (or the
clumping is increased), though this is a much less important factor
than the number of hands.
Before clumpers go insane with glee, note that you've probably got
things exactly backwards, as anti-clumping is probably the rule for
multiplayer games, and in any case, the effects are very small,
only about as large as removing a single card from a six deck shoe,
which any card counter knows is worth very little.
I'm sure the clumpers will cry for simulations showing the
correlations of the previous N cards to the next M cards. I did
such a study back around 1990, and it's a similar story - the
clumping effect can be observed, but it's miniscule in size, much
smaller than the effects I revealed above. In any case, it looks
like someone is about to post such a study.
The simulations I've done here, and back in 1990, should have been
done by the clumpers long ago, long before they ever dreamed of trying
to exploit clumps in the casinos, long before they ever dreamed to
declare that clumping was real and not a perceptual artifact. By not
doing their homework, they got things pretty much exactly backwards,
and vastly overestimated the strengths of the effects and the impact
of casino shuffles on normal players and counters.
The fact that the card distributions are not random does not
invalidate card counting, because the card distributions are *almost*
random. Simulations show that if there is any impact on player/counter
expectation of a casino shuffle compared to a random shuffle, it's below
.02% advantage, because most very long simulations show no statistically
significant difference.
--
Abdul Jalib wearing the hat of | May you never be tapped on the shoulder
Professional Degenerate Gambler| in the New Year.
AbdulJ_DELETE_@PosEV.com | (Delete _DELETE_ to reply via email.)