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Uniform Strategy with Card Counting
This article was originally published in the Blackjack Insider, a Blackjack e-mail newsletter edited and published by Henry Tamburin, author of "Blackjack: Take the Money and Run". The article was written by Dan Pronovost, President of DeepNet Technologies.
In the last issue of the "Blackjack Insider", we explored the effect of using a single uniform basic strategy in different blackjack games. The surprising conclusion was that players using only basic strategy need not worry about using the exact action tables that match the casino rules. For all numbers of decks and standard rule variations (DAS/noDAS and h17/s17), the differences in expectation were very small. Using the 'eight deck/DAS' basic strategy table delivered good results in all games.
Is the same conclusion true when you are using a blackjack count system to vary your bets according to the advantage, and/or indices to change your plays? This article explores this question with detailed analysis to determine statistically the cost of using uniform strategy when card counting in blackjack. We usedBlackjack Audit, an advanced blackjack simulator from DeepNet Technologies, to perform all the statistical calculations in this article.
Basics of Card Counting in Blackjack
Count systems in blackjack are surprisingly simple to understand. Values are assigned to each card rank, and the player accumulates a single running total as cards are exposed. Based on this count, players will make smaller or larger bets (bet range), and may also vary their basic strategy plays (indices). A count system will have a range of bet sizes based on the count, and may define indices for some of the plays which indicate the count value at which to use an alternate action. The net effect of a good count system is that in almost all blackjack games, a player will have a positive expectation (i.e. will make money on average playing blackjack).
This article uses Stanford Wong's popular and effective High-Low count system(1). Wong includes full tables and indices for a variety of casino rules and conditions, which makes it ideal for the analysis in this article.
In our last article, we saw that the ideal basic strategy depends on the casino rules, including the number of decks, DAS/noDAS, and s17/h17. Count systems are similar in that the play indices often vary with the rules as well.
Card counters usually use the count to vary their bets, and may also use play indices. Memorizing dozens of indices is a daunting task for serious card counters. Do the frequent differences in indices that accompany rule changes have a significant impact on player expectation?
The chart below shows the results ofBlackjack Audit simulations using different High-Low count system strategies with different rules (a minimum of 250 millions blackjack hands for each entry). We used a 1 to 10 bet range in the eight deck games, a 1 to 8 bet range in the six deck games, and the full published play indices for each column's strategy (Appendix A from "Professional Blackjack", High-Low 4-deck tables). 75% shoe penetration was assumed for all multi-deck simulation runs, re-splitting of aces was allowed, and surrender was not permitted. The expectation is the percentage of each bet you should expect to win or lose, on average, given a specific set of casino rules and play method. It is calculated by dividing the total profit for the simulation run, by the total sum of all wagers, expressed as a percentage. The bold entry in each row shows the best performing strategy column.
Table 1: Multi-deck expectations using the Full High-Low count system
To determine the negative impact of a uniform count strategy, the best metric is to compare hourly earning rates. For all games, the average total wagers per hand ranged from $10.33 to $11.32 (using a minimum bet of $5 and the designated bet range). We can compute the maximum loss per hour in dollars as follows:
(maximum disadvantage) x (maximum average wagers/hand) x 100 hands/hr
Hence, using the worst possible strategy matrix in the worst game situation cannot reduce your profit by more than 29 cents per hour. Since your hourly return rate for the above games ranges from $2.00 to $5.85, we once again have to conclude that playing a uniform count strategy does not overly penalize the card counter. If you are going to play with one multi-deck count strategy, use the 'eight deck, DAS, H17' table to get good performance in all games.
Many card counters use few play indices, or none at all. The only difference from a basic strategy player in this case is that they are using the count to establish their bet size. Table 2 shows the results using this playing method. We used a 1 to 10 bet range in the eight deck games, a 1 to 8 bet range in the six deck games, and a 1 to 4 bet range in the two deck games. 75% shoe penetration was assumed for all simulations in Table 2, re-splitting of aces was allowed, and surrender was not permitted. The bold entry in each row shows the best performing strategy column.
Table 2: Expectations using a partial High-Low count system: no indices
A careful look at these results shows an incredible result: the best expectation in the eight deck games is not achieved with the eight-deck strategy! This surprising result (which did not occur in Table 1, or in our prior basic strategy article) is not an error: it is a real side effect of not using play indices.
The two-deck strategy table differs from the eight deck table mostly by adding some pivotal doubles and splits: 9/2:D, 11/A:D, A8/6:D, 66/7:P, etc. The play indices for these actions are close to zero in the High-Low count system, so they will occur frequently. When the count is high, the bets will be much larger (especially with a 1 to 10 bet range). Hence, the additional earnings made by the more aggressive (and correct) plays when there are larger bets outweighs the disadvantage from the overly aggressive (and incorrect) plays when the count is low but the bet is small. Of course, using play indices eliminates the trade off completely, delivering further gains.
As an example, consider player hard nine versus dealer two. Normally, you hit this hand. In High-the Low count system, you double instead at a true count of one or greater (D8, DAS, H17). What the index play means is that you will make more money doubling 9/2 once the true count is one or greater. This will occur on 31% of the 9/2 hands. But at a true count of one you also increase your bet to three units, then to five units at a true count of two, then to ten units at a true count of three or higher. Since 69% of the hands occur when the true count is less than one, you might think you're better off always hitting. But you have much larger bets out when the count is one or greater. The added profit by doubling these fewer high-bet hands outweighs the slight loss from the more frequent low-bet hands played with the incorrect double action.
So, if you count to set your bets but don't use play indices, your will earn more money by using the 'two-deck, DAS' strategy table for all games, especially for six or eight deck games. But you should note that the maximum loss is about 50 cents per hour, using the same method from the analysis after Table 1. Although larger than the loss when playing with indices, you should still feel comfortable playing whatever strategy you are used to in multi-deck games.
Astute observers may compare Table 1 and 2 to determine the impact on expectation of not using play indices. For example, Table 1 shows that the best expectation in the "eight deck, DAS, s17" game is 0.517% with indices. Without play indices, we can get at most 0.404% using the "two deck, DAS" strategy from Table 2. This is a 25% difference, which will have approximately the same percentage impact on your hourly earnings. In the next article in our continuing series, we will statistically analyze the advantage and disadvantage of other 'fine tuning' blackjack methods, such as Wonging and hand spreading.
Single-Deck Basic Strategy Analysis
The options used in our single deck analysis are: single deck, 1 to 3 bet range, and the full published play indices for each column's strategy (Appendix A from "Professional Blackjack", High-Low single-deck tables), 50% shoe penetration, re-splitting of aces allowed, no surrender.
Table 3: Single-deck expectations using Full High-Low: 1-3 bet range, all indices
Using the same earnings per hour analysis from before, the average total wagers per round varies from $11.06 to $11.16 ($5 minimum bet was used through out. The tighter range is due to the smaller 1 to 3 bet range). The maximum loss per hour in dollars is:
(maximum disadvantage) x (maximum average wagers/hand) x 100 hands/hr
As with the multi-deck games, playing a uniform count strategy has little impact on potential earnings. The worst hourly earning rate above is $6.57, turning the 29 cents an hour drop into a maximum loss in earnings of 4.4%.
Notice also that the eight deck strategy performs quite well, particularly the 'D8/DAS/H17' strategy we've already selected as the favored system for multi-deck games.
Once again, we have to conclude that uniform strategy has minimal negative impact on hourly earnings in blackjack, particularly when using bet spreads and play indices. If you are going to use uniform count strategy, choose the High-Low, "eight deck, DAS, H17" tables. If you are not yet a practitioner of card counting and this article has made you decide that it's time to start, we highly recommend Stanford Wong's superb book "Professional Blackjack" that includes all of the High-Low count system details used in this analysis. Use the Four Deck tables in Appendix A (which apply equally to eight deck games). Additionally,DeepNet Technologies sells Blackjack Counter and a High-Low supplementary database that provides all the training software you need to teach yourself this count system (Windows or Palm OS): learn more at www.deepnettech.com.
If you count only for the purposes of establishing your bet and do not use play indices, then you are better off using the two deck/DAS basic strategy tables, especially in six and eight deck games. This counter-intuitive result may well change the way some card counters play the game of blackjack!
The observations in this analysis will generally apply to most other count systems, especially balanced systems like High-Low (i.e. OmegaII or HiOpt). Unbalanced systems such as Knock-Out are not the same, since the number of decks has a much larger impact on the system parameters (such as the initial run count and pivots).
1) "Professional Blackjack", Stanford Wong, Pi Yee Press, La Jolla, California. 1994.