COMPARING THE PERFORMANCE OF BALANCED vs. UNBALANCED COUNTING SYSTEMS
POSTED BY PERMISSION OF HENRY TAMBURIN,
PUBLISHER OF THE BLACKJACK INSIDER NEWSLETTER
Blackjack Insider Newsletter, April 2002, #29
http://www.bjinsider.com/newsletter_29.shtml
COMPARING THE PERFORMANCE OF BALANCED vs. UNBALANCED COUNTING SYSTEMS
BY FRED RENZEY
When "unbalanced" card counting arrived on the scene about fifteen years ago, it opened up the door for a multitude of losing blackjack players to become winners at the game. These new "contenders" for the casinos' money were players who until then just couldn't quite master the full rigors of traditional card counting.
It wasn't that adding and subtracting points for high and low cards were so tricky. And it wasn't that memorizing how much to bet at various counts was too much to handle. No! What knocked 90% of the would-be successful card counters out of the box was converting the "running count" to the "true count". For most, this was just too much mental gymnastics! First, you had to keep a running count of high vs. low cards. Then you needed to divide that running count by the number (or fraction) of decks that remained before playing your hand and sizing your next bet. And finally, you had to revert back to the running count and update it once the dealer began dealing the next hand. That was the problem!
Then in the early 80's, Arnold Snyder of Blackbelt in Blackjack fame introduced his Red 7 Count. For its first dozen years, it never received the recognition it deserved. It just seemed like a " squirrelly" concept. I mean, why would you want to count only the red 7's and not the black ones? Snyder had an excellent reason.
By unbalancing the traditional Hi/Lo Count to include the red 7's as low cards along with all the 2's through 6's, the player had himself a built in, reasonably accurate true count calibrator. Using the unbalanced Red 7 system, the "running count" was in fact a close approximation of the "true count"! That meant no more converting to play your hands or to bet your chips. All was done strictly by the running count!
It wasn't long before other unbalanced count variations followed. But until the advent of home blackjack analysis software in the 1990's, it wasn't clearly known that unbalanced counts were a legitimate item and could actually cut the mustard. That's when unbalanced card counts became a hot commodity. Today, another unbalanced system, the KO Count challenges the old tried and true Hi/Lo Count for most widely used count system in play.
SPLITTING THE HAIRS OF PERFORMANCE
So what about it? If unbalanced card counting can win at blackjack without the tedium of true count converting, just how well does it perform next to its balanced counterpart? As it turns out, the answer lies in how many decks you're using.
You see, unbalanced card counts merely estimate your true count. The fewer cards that are involved, the more inaccurate (proportionally) they become. An unbalanced counting system's Achilles' heel, if it has one is the single deck game. Being only approximately accurate there is a tangible flaw -- but not a fatal one.
To learn just how these two types of card counts fare against each other, I ran 2.8 billion simulated blackjack hands with Stanford Wong's Blackjack Count Analyzer. 200 million hands each were run with three different single level unbalanced counts along side the standard balanced Hi/Lo Count in single deck play. This exercise was then repeated for double deck, and finally another 300 million hands each were run with the six-deck shoe. The three unbalanced systems were Snyder's Red 7 Count from his Blackbelt in Blackjack, Fuchs and Vancura's KO Count from their popular Knockout Blackjack and the StageII Black Ace Count from my own Blackjack Bluebook. All four of the systems tested are described below.
COUNT COUNT C A R D T A G S RANKS BET PLAY
NAME TYPE 2 thru 6 r7 b7 8 9 10 bA rA COUNTED CORR. EFF.
Hi/Lo balanced +1 0 0 0 0 -1 -1 -1 10 97% 51%
Blk Ace unbal. +1 0 0 0 0 -1 -1 0 9.5 93.5% 55.5%
Red 7 unbal. +1 +1 0 0 0 -1 -1 -1 10.5 97% 53%
KO unbal. +1 +1 +1 0 0 -1 -1 -1 11 97.5% 55%
The Hi/Lo Count tracks five low ranks of cards (2's thru 6's) and five high ranks (10's thru Aces), thus it is a balanced card count. The Black Ace Count mimics the Hi/Lo, but drops the red Aces right out of its count, thereby unbalancing it for "auto-calibrating" purposes. The Red 7 Count goes the opposite direction by shadowing the Hi/Lo, then throwing in the two red 7's as low cards to accomplish its unbalancing objective. The KO Count takes it one small step further by tracking all 2's thru 7's against the 10's and Aces. In doing this, it unbalances its count by one full rank of cards.
Whether a single level unbalanced count should be "offset" by a half rank or a full rank of cards can be argued, but I believe a half rank offset takes better advantage of the reason you unbalance a count to begin with. Here's the reason for my stance. All unbalanced counts are perfectly accurate at having their running count indicate the true count at one particular point -- and at that point only. As their running counts stray away from that particular point, inaccuracy begins to set in and increases the further away you get. With unbalanced counts that have a half rank offset, the running count will indicate to perfection whenever you have a true count of +2. Unbalanced counts with a full rank offset are perfect at telling you when your true count is +4.
In short, when using the Red 7 Count (half rank offset) in a two deck game, whenever your running count has risen four points above your starting count, you'll have a true count of exactly +2 -- regardless of how many cards have been played out. But a running count eight points above your starting count won't necessarily be +4 true. It may be +5 true or even +7 true depending upon your current depth into the deck.
Conversely with the KO Count (full rank offset), a running count eight points above your starting count will always be exactly +4 true. But a running count 4 points above your starting count rather than being +2 true, could be +1 true and maybe even -1 true, again depending upon where you are in the deck.
Since +2 true is near the threshold of where your player edge begins and since it occurs much more often than +4 true, that's the point I'd want to be more sure of. Also, a greater number of basic strategy departures kick in around +2 rather than at +4 true. For these reasons, I believe a half rank offset makes better use of what unbalanced counts do best.
TEST METHODOLOGY
To be sure of comparing apples to apples in my computer runs, a major criterion was to keep the hourly standard deviation at the same level for all four counting systems. That is, I wanted to ensure that we were risking the same amount of bankroll dollars in all cases. This necessitated using a slower betting ramp in systems with a higher betting correlation due to their more frequent recognition of advantageous deck compositions. The net effect was that each system reached its maximum bet the same percentage of times and at about the same percentage of net advantage -- though not at the same true count. (This makes sense when you think about it since a +3 true count with the Black Ace system yields a greater edge than a +3 true count with the KO system, although the KO system will produce a +3 true count more often. This is true because more expansive count systems track more cards, therefore finding more advantageous situations. However, its extra tracked cards are generally the less important ones so that a shortage of six 2's thru 7's matters less than a shortage of six 2's thru 6's. Still, the overall betting spreads remained identical with all systems, as did the average bet size.)
Table I compares the performances for single deck play.
TABLE I
SINGLE DECK PERFORMANCES
COUNT SYSTEM HI/LO BLK. ACE RED 7 KO
BETTING SPREAD $25-$75 $25-$75 $25-$75 $25-$75
AVERAGE BET $49 $49 $49 $49
STANDARD DEV./HR. $461 $461 $461 $461
NET GAIN % 0.95% 0.80% 0.82% 0.83%
HOURLY WIN (100 hands) $47 $40 $41 $41
40,000 HAND WIN PROB. 98% 96% 96% 96%
Rules for the test were; S17, DOA, NO DAS, NO SR. The shuffle point came when 25 or fewer cards remained. The standard error for the percentage gain of each system was less than 0.01%. All systems were played at a three-handed table and the counter bet $50 off the top of each deck, then went to $25 or $75 depending upon the count. The Hi/Lo Count employed strategy departures at all true counts from -4 to +8 (using 61 index numbers). All three unbalanced counts used 22 hand calculated indices -- the most negative of which was 12 against a 6 and the most positive being 16 against a 9 -- and all were activated strictly by the running count. The bottom column in the table, "40,000 Hand Win Probability" represents the players chance to be ahead of the game after roughly 400 hours of play.
There's one last significant point I need to bring up. In analyzing half rank offset unbalanced counts (such as the Red 7), Wong's software forces you to count all the 7's as +0.5 rather than half the 7's as +1. In essence then, it's actually testing a system with 98.5% and 54% efficiency ratings rather than the 97% and 53% ratings that the Red 7 actually carries. Similarly, the Black Ace Count actually brought 95% and 56.5% ratings into its simulation test rather than its true 93.5% and 55.5% strengths. Realizing this, I reviewed the relative performances of several different half rank unbalanced counts and correlated their performances with their efficiency ratings. Doing this revealed that for every percentage point you increase betting correlation and playing efficiency among the same species of system, its performance tends to improve by about 0.02%. Consequently, the net percentage gains reported in Table I for the Black Ace Count and the Red 7 Count have both already been adjusted downward by 0.03% to account for their probable decreased performance had they been tested in their literal forms. Their reported net dollar earnings and 40,000 hand win probabilities have also been decreased using similar extrapolations.
SINGLE DECK SUMMARY: Although the balanced Hi/Lo Count clearly outperformed all of its unbalanced counterparts, it didn't do it by a remarkable margin. The thing to keep in mind is that single deck play is the most demanding blackjack of all. Realistically, no human can play it as accurately as a computer since you're always estimating how many fractions of a deck are left and your true count division efforts are always rounded off. Whereas the computer plays each unbalanced system strictly by the running count, the same as any human would. One has to wonder after all the remaining deck estimations and division-rounding errors are factored in, how much difference would actually be left in the real world. One fact must be true: Only if the particular human using the balanced count did a more accurate job at true count converting than does an unbalanced structure, could there been anything left over at all! This I must admit even though I myself have been using a multi-level balanced count for the past 25 years.
Table II states the performances of each of the four systems for double deck play.
TABLE II
DOUBLE DECK PERFORMANCES
COUNT SYSTEM HI/LO BLK. ACE RED 7 KO
BETTING SPREAD $15-$90 $15-$90 $15-$90 $15-$90
AVERAGE BET $33 $33 $33 $33
STANDARD DEV./HR. $468 $468 $468 $468
NET GAIN % 0.74% 0.69% 0.74% 0.74%
HOURLY WIN (100 hands) $24 $22.50 $24 $24
40,000 HAND WIN PROB. 85% 84% 85% 84%
Rules were the same as for the single deck test. The shuffle came with 44 or fewer cards remaining. The player always came off the top with a $25 bet, and then moved upward or downward according to the count. No next bet was ever increased or decreased by more than a factor of three. Again, the Black Ace and Red 7 yields shown above have been lowered from 0.72% and 0.77% respectively in an effort to represent their actual performances most accurately (hourly wins were also adjusted upon the same basis).
DOUBLE DECK SUMMARY: Note how the performances of balanced vs. unbalanced counts here have bunched together. That's presumably because errors in unbalanced estimates of the true count are proportionally smaller and matter less when you're drawing from a larger supply
Through all of this, the fact remains that the vast majority of blackjack games are dealt from a shoe -- hence, Table III. The rules for the multi-deck shoe test were modified to allow doubling after splits -- all else remained unchanged. The shoe was shuffled with 94 or fewer cards remaining. The bet off the top of each shoe was $25 (6% of all hands). Also, as is advisable for shoe games, at any true count lower than -3 each player's wager was $0. This occurred 7% of the time. Because of the wider betting spread, 300 million hands with each system were needed to get the standard net gain error down to 0.01%. Finally, remember that 0.03% has already been deducted from the stated net gain percentages of the Black Ace and Red 7 Counts.
TABLE III
SIX DECK PERFORMANCES
COUNT SYSTEM HI/LO BLK. ACE RED 7 KO
BETTING SPREAD $10-$100 $10-$100 $10-$100 $10--$100
AVERAGE BET $26 $26 $26 $26
STANDARD DEV./HR. $464 $464 $464 $464
NET GAIN % 0.68% 0.63% 0.67% 0.65%
HOURLY WIN (100 hands) $18 $17 $18 $17.50
40,000 HAND WIN PROB. 79% 78% 79% 78%
All in all, I guess I'd have to say that unbalanced counts, if they're done right cut about 40% of the work out of card counting while retaining about 95% of the benefit.
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Editor's Note: Fred Renzey is an "advantage" blackjack player and author of the critically acclaimed "Blackjack Bluebook", a clearly detailed 188 page strategy manual for casino "21".
