How True is Your True Count?
From ET Fan:
How True is Your True Count? Implications for Advantage, Win Rate, Betting, and Table Hopping in Blackjack Card CountingBy Arnold Snyder (with Blackjack Simulations by John Gwynn)
(From Blackjack Forum Volume II #3, September 1982)
© Blackjack Forum 1982
Brace yourself, dear card counter, because this is another one of those all-the-blackjack-experts-have-been-wrong bombshells I've been having so much fun dropping on my faithful followers lately. John Gwynn dropped this one on true counts on me eight months ago, and it's taken me this long to put it all together with some coherence.
If you own a copy of the Zen Count, you will see, on page 4, that I list the various true counts at which I estimated a card counter would have a ˝% advantage in various blackjack games. When Gwynn completed his first 4-deck blackjack computer simulation runs of the Zen Count, he wrote to me that my advice was only partially true. He pointed out that the player advantage using the Zen Count was, as I advised, ˝% or better at a true count of +4 in the 4-deck strip game, but if 62.5% or more of the cards had been dealt out, the player would have an edge of ˝% or better at a true count of only +3.
This was a revelation to me. The whole purpose of adjusting running count to true count is to obtain an accurate estimate of your advantage at any deck level. Expert opinion in blackjack has always held that a running count of +6 with one deck dealt out of a 4-deck game indicates a player advantage equivalent to that of a running count of +4 with 2 decks dealt out or +1 with 3˝ decks dealt out. In all cases, the "true" count is +2 (on a count-per-deck basis.)
John Gwynn's 23+ million hand simulations of blackjack systems indicate otherwise. Since Gwynn's original comment to me regarding the Zen advantage, he discovered an error in his simulation program (Blackjack Forum Vol. II, #2). His corrected data show that his original observation, that a Zen player would average a ˝% edge at +3 in the 4-deck Strip game with 2˝ decks dealt, to be untrue (but close). However, his remark led me to examine closely the corrected data for any tendency of the true count to prove significantly "untrue". The results, to me, are startling.
Actual Advantage for any True Count Changes with the Depth of the Deal
For instance, let's look at the simulation results for Hi-Opt I (4 decks, no ace count, Vegas Strip rules). The depth of the deal is listed horizontally along the top of the table. The true count is listed vertically on the left. The table entries show the actual rate of win (or loss) for the player at each given true count. True counts are rounded, i.e., +2 = +1˝ to +2˝. The advantage shown is the cumulative advantage for all hands played up to that point:
Hi-Opt I Advantage (in %)
Note here that in 18 instances, the player advantage rose as depletion increased. In two instances, player advantage fell. In four instances, it remained the same.
Note how much each true point is worth with only 50% dealt out. Note how much each true point is worth with 87.5% dealt out.
Let me explain briefly how these simulations were done. Gwynn's computer is programmed to play through 23+ million hands (or 700,000 shoes), using any inserted system. It then tallies the data for the various circumstances. Separate runs are not necessary to obtain data for the various shuffle-points and true-count values.
To obtain the player advantage at a true count of +2 with only 50% dealt, the computer is simply instructed to tally all results obtained when the true count was between +1.5 and +2.5, and to disregard the results of all hands played after two decks have been depleted. Because of this methodology, the total number of hands played at the various levels of deck depletion differ.
At the 87.5% shuffle-point, 23.6 million total hands were played. With 75%: 20.3 million. With 62.5%: 17 million. And with 50%: 13.6 million hands. This also means that these advantages in the table are cumulative, i.e., the listings at 75% do not indicate the player edge only at 75% depletion, but the average advantage of all hands up to 75% depletion.
For instance, look at the entries for a +2 true count:
What this means is that up until 50% deck depletion, the player gained .53% on all his bets placed when the true count was +2. By the time 62.5% were dealt out, the player had gained .57% on all these bets at +2 true count. By the time 75% of the cards were dealt out, he was averaging a .62% advantage, etc. This is a very important point, because…
If the player was averaging .53% on all +2 true hands up to a 50% shuffle-point, then it would take an average advantage of .73% to raise this figure to .57% by the time a 62.5% shuffle-point was reached. And, to raise the .57% to .62% (at the 75% shuffle point), it would take an average edge of .87% on those hands played between these points.
Average Point Value and Deck Depletion
Another way to analyze this Hi-Opt I data is to estimate the value of an average true point between various levels of depletion. If, as Peter Griffin tells us, the starting advantage in this game is -.48%, we can estimate the average point's value up to the 50% shuffle-point by taking the difference between the player edge at true counts of +5 and 0, and dividing by the 5 true points. We can then calculate the value of an average point between two levels of deck depletion, according to the point value necessary to cause such a change in "average" point value.
This raises serious questions about the "truth" of the true count. Gwynn's data for the Hi-Opt I system are generally consistent in showing notable increases in player advantage at any true count as deck depletion increases. The value of a true point appears to depend on a number of factors, one of which is the level of deck depletion. Another factor seems to be the system itself. Here is a similar table for Uston's APC:
Uston APC (with Ace)
Here, we'll note that in 22 instances, player advantage rose as deck depletion increased. It fell in five instances. In three instances, it remained the same. The average point values for Uston's APC, at various levels of deck depletion:
This data indicates that the radical change seen in the value of a true point for the Hi-Opt I system may not necessarily be expected for any system. But the difference is still significant.
Look at a similar table of player advantages for the Zen count (with 25 indices):
Zen Count Advantage (in %)
Here, in 27 instances, player advantage rose as deck depletion increased. In nine instances, player advantage fell. The average point values for the Zen Count, at various deck levels:
Here again, this table as a whole shows the same tendency as Uston's APC; the results are less consistent and more erratic.
Looking at the results for all three of these systems, it appears we cannot say with any degree of certainty what a true point is worth for any one of them. Occasionally, I get a letter asking me to clarify the precise value of a point for some system. Assigning such a value appears to be, at best, an oversimplification.
Gwynn's Hi-Opt I results are the most consistent and dramatic. Based on this simulation, it appears that a single true point is worth twice as much (or more) deep in the shoe as shallow.
The results I used in this analysis were for player advantages up to about 2˝%. Gwynn provided no data for true counts below -2. I used these results because they are the most frequently occurring true counts, thus the most significant. The data distorts radically at progressively higher true counts due to chance fluctuation.
But look at this table, which shows the Hi-Opt I advantage for all true counts from 0 through +20, with 87.5% of the cards dealt out:
The figure in parentheses, the "point value", shows how much each individual true count raised the player advantage over the previous true count. For instance at +6, the win rate was 2.80%. This particular true point raised the player advantage by .58% over the player win rate at a true count of +5, which was 2.22%. Thus the "point value" of this particular true point was .58%.
The data in this table run contrary to one currently held theory that the "strategy gain" from card counting increases dramatically at higher true counts. Gwynn was employing all 201 strategy indices in this 23.6 million hand simulation.
According to The World's Greatest Blackjack Book (Humble and Cooper), a single Hi-Opt I point is worth .515% (pp. 265-66). The formula they provide for estimating advantage at any true count is to multiply .515% times the true count, then to add both the "strategy gain" and the Starting Advantage of the game. Applying this formula to a +12 true count, we get:
(.515% x 12) + 3% - .48% = 8.7%
The -.48% is our starting advantage in this 4-deck Strip game. The 3% is the amount of "strategy gain" that Humble and Cooper explain is added at a true count of +12. (They also explain the strategy gain would be 2% at 8, and 1% at 6). Humble's predicted 8.7% edge, at a true count of +12, is more than 3% higher than the computer simulation result.
Gwynn's +12 advantage of only 5.64% suggests that Humble's "strategy gain" is either virtually non-existent in the 4-deck game, at least as any highly significant factor, or that the average value of a Hi-Opt I point is quite a bit lower than .515% in the 4-deck game. This 3+% discrepancy between the Humble/Cooper estimate and Gwynn's simulation results cannot be attributed to normal fluctuation. Gwynn's computer played more than 52 thousand hands at this +12 true count, so one standard deviation is only .48%.
Again, these results question the validity of many currently held beliefs about true count. Gwynn's simulation data indicates that the value of a true point for any system varies with both deck depletion and, as we shall see, with the number of decks in play.
The Hi-Opt I data indicates that a true point may be worth only .3% to .4% early in a 4-deck shoe and .7% to .9% deep in the shoe, with an average value of about .5%. From the data Gwynn has provided, it is impossible to tell how much lower these values would be without the playing strategy indices which the computer employed throughout. It appears that the total gain for the Hi-Opt I player, including the "strategy gain", with each increase in true count, averages to about .5% by the time three decks have been depleted.
Guidelines for Card Counters
My recommendations: Since the Hi-Opt I data suggest such a radical departure from long-standing card counting theory and because the Zen and Uston APC data are more erratic, though still supportive of the "untrue" true count notion, with greater point values at deep shuffle points, I'll be cautious in my recommendations. It may be that an entirely new method of adjusting running count to true count is needed.
The concept of true count goes back to E.O. Thorp. In his original ten-count (Beat the Dealer, 1962), Thorp described his method of estimating advantage according to the ratio of tens to non-tens. In essence, this simple ratio provided the first true count. John Gwynn has produced a body of data which leads me to question the validity of Thorp's assumption and methodology.
Gwynn's data does suggest certain guidelines for players. First of all, it appears to be a waste of time arguing about the actual value of a true point. This value depends not only on deck level, but also on the precise point in question. The point between +4 and +5 may be significantly more or less valuable than the point between +3 and +4.
If you are a table hopper, attempting to bet in proportion to your advantage, I would advise more conservative estimates of advantage, especially if you are in the habit of adding a "strategy gain." None of the three systems for which Gwynn provided data indicates that true points are consistently worth more as true counts become higher. It would be interesting to see a Hi-Opt I run using no indices, but playing basic strategy. Such a run would by comparison show the actual strategy gains at the various true counts and deck levels in this 4-deck game.
I will suggest being more conservative in sizing bets early in a shoe, and somewhat more aggressive later. Table hoppers will tend to play far more hands at lower levels of deck penetration than players who keep their seats through the negative counts. Thus, the high true counts they see will more often be indicative of less of an edge than has generally been assumed by blackjack experts.
This is not an argument against table-hopping, which is still your best multi-deck count strategy. This is simply a caution to be more conservative in estimating your advantage. Essentially, Gwynn's data show that any oversimplified methods of estimating advantage must be viewed as rough approximation techniques only.
Alas, true count, like true love, is rarely true.
Let's look at some one-deck data for these three systems. These simulations were done in the same way, though results were tabulated at only three deck levels: 25%, 50% and 75%. A total of 20.1 million hands were played by the 75% level. Again, these results are for Vegas Strip rules, and assume that no ace side counts are being used.
Again, note that in all three systems the win rate at any given true count increases as deck depletion increases. There are a few variations from this tendency, but the overall effect of deck depletion on true point value is consistent with our 4-deck findings. In fact, in these single-deck runs, this tendency appears even stronger than in 4-deck games. A single-deck true point generally appears to be worth more than a 4-deck true point. The effect of deck depletion in single-deck games is even more radical than in 4-deck games.
For all three systems, look at the win rates at true counts of +2. Compare the win rates at 25% depletion and 75% depletion.
How can you use this knowledge at the tables? First of all, I don't believe anyone is going to come up with a highly accurate method of adjusting running count to true count. Nor can anyone define the value of a point as specifically as most experts and writers, including myself, have been doing for years.
Nor does it appear feasible to develop a practical betting scheme that allows you to bet in proportion to your advantage with any high degree of accuracy. I'm not suggesting that you throw out your attempt to bet in proportion to your advantage, but that you realize how rough any estimate of your advantage is.
Player Advantage is not Linear
Advantage is not in any sense linear in the game of blackjack, nor does it appear to follow any pattern without variation. The value of a true point depends on the deck level. However, even at a true count of 0, the player is significantly better off in single-deck games deeper in the deck.
Although I reproduced here only the Vegas Strip results, Gwynn's data shows the same tendency for all three count systems vs. Northern Nevada rules, to about the same degree. Although Gwynn's printout does not show win rates at true counts below -2, it appears that you are also better off at the same negative true counts later in the deck, i.e., a true count of -2 is not so bad with 75% dealt out as with only 25% dealt out.
Strange quirks exist throughout all of the systems. The Zen Count, for instance, in the single-deck Vegas Strip game, actually indicates a player edge at a true count of -2, but a house edge at -1. This is not due to an isolated weird run of hands.
Even with Northern Nevada rules, you are better off at a true count of -2 than -1 if you are using the Zen Count. Likewise, you have a greater advantage at a true count of +10 than you do at a true count of +11, and also greater at +2 than +3, regardless of rules; and you usually have a greater advantage at +4 than at +5, depending on the penetration of the deck.
Most of these same anomalies exist to a lesser extent in 4-deck games with the Zen Count. Similar type occurrences are apparent in the other count systems, but occurring at different true counts. A precise explanation of such phenomena does not present itself.
There are undoubtedly precision system fanatics who would attempt to modify their play according to such data as Gwynn has developed. They would raise their bets at a true count of -2, lower them at -1, and raise them again at 0 (providing more than 50% of the deck was depleted). In sizing their bets, they would realize that they were always better off at a true count of +2 than +3, and better off at +4 than +5 (if 75% of the deck had been depleted), etc. However, such a betting scheme would drive most players nuts.
Precision Doesn't Pay
Gwynn's data indicate to me that some players are wasting a lot of time and effort with precision techniques. Gwynn programmed his computer to adjust to true count by counting the exact number of cards, "accurate to the gnat's ass," as he put it, and also to adjust to true count by estimating to the nearest quarter-deck (13 cards). There was no significant difference in win rates between these two methods.
Since true count isn't true anyway, this is understandable. Why use a toothbrush to scrub the side of a barn? Such precision appears to be a waste of time.
I'm not advocating sloppy play. Most players use a quarter-deck true count approximation and should stick with it. In single-deck games, using a half-deck approximation is too sloppy. Gwynn tried this with Uston's APC and found that it was no better than Hi-Opt I, when Hi-Opt I used a quarter-deck true count approximation.
In the next issue of Blackjack Forum I'll continue my examination of John Gwynn's massive body of simulation data. In particular, I'll be comparing the single-deck win rates of various systems in single-deck games with both Vegas and Reno rules with various shuffle-points. Gwynn has already provided me with this data for Hi-Opt I, Uston's APC, and the Zen Count. He hopes to have his single-deck runs completed for Hi-Opt II and the Hi-Lo count in time for the next issue.
This computer simulation data that John Gwynn has been making available to me for publication in these pages since Blackjack Forum Vol. I #4 is, to my knowledge, the most extensive and important computer research being done on blackjack today. Analysis of Gwynn's extensive data answers many questions, but raises many others.
I am grateful to Dr. Gwynn for allowing me to publish his extremely valuable data with my initial analyses and commentary. I realize that this "rush to press" methodology leaves many questions unanswered, but I feel the information contained in this data is of such importance to blackjack players that they should get this information as soon as possible. Hopefully, further research on these findings, and input from other experts, will ultimately lead all of us to a better understanding of the game. ♠
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||The Impossibility of Accuracy in Blackjack Card Counting
Beginning blackjack players tend agonize over the choice of card counting system, and even some professional-level blackjack players never realize that the pursuit of perfection in card counting is not where the money is. Arnold Snyder shows why true counts can never be true, and provides practical answers to getting the most money out of blackjack card counting.