MORE EDGE IN AGGRESSIVE CARD COUNTING: A BACKUP ANALYSIS by Dan Pronovost Introduction
When Card counting is not so great Last month, BJI author Alan Krigman provided an excellent study of risk of ruin when counting cards in blackjack (see www.bjinsider.com/newsletter_97_agg.shtml). His surprising and counterintuitive conclusion was that using more aggressive strategies as a card counter to get your edge higher might actually mean less chance of winning, depending on your bankroll (i.e., you may have more losing sessions than winning sessions). This just doesn't seem right on the surface... surely if your method of play means a larger mathematical edge over the casino, then you'll have a better chance of winning (by increasing your bet spread, for example)? As it turns out, bankroll is everything: to make more money in blackjack as a card counter, you generally have to bet more when you have the edge, and that means more volatility in your bankroll. If you don't increase your cash to play, then you may well go broke before seeing the extra theoretical advantage over the casino. In the long-term, given a sufficiently large bankroll and time, a positive edge over the casino through card counting is your ticket to play for profit. But the reality is that we all play with limited funds for a very small amount of time in relation to the volatility that large card counting bet spreads inevitably bring on. Understanding this and using that knowledge to back your play with healthy session bankrolls, is the key to success as a card counter. Alan's study was very interesting to me, since I wrote a similar article a few years ago ("The Unbeatable Card-Counter Myth", http://www.bjinsider.com/newsletter_34_bankroll.shtml). I took a different line of attack on the problem, but Alan and I effectively leveraged the same principle: your bankroll as a card-counter must be matched to your betting strategy and risk of ruin. Many card counters make the mistake of thinking that a 0.5 percent to 1 percent edge over the house means they are unbeatable, no matter what money they start with in their pocket. Proper bankroll and risk of ruin is the most important lesson a new card counter can learn, and it is probably the most common mistake I've seen in years of teaching novice players (www.goldentouchblack.com). So, I decided one simply can't say enough on this subject, and asked Alan's permission to expand on his article and data. Alan kindly obliged, and in fact includes his own further analysis on this subject in this issue as well (www.bjinsider.com/newsletter_98_agg2.shtml). The hypothetical game In Alan's prior analysis, he made these assumptions: - "Conservative card counters might get a 0.5 percent edge by pressing their bets cautiously on rising counts. A reasonable 1-4 betting spread..."
- "More aggressive card counters might achieve a 1 percent advantage with the following 1-8 betting spread..."
Rather than use a blackjack simulator, Alan started with these assumptions and then applied standard statistical equations to determine the risk of ruin for different bankrolls and goals. He also backed this up by running a statistical simulation in software (again, using only the assumed edge and bankroll conditions, not actually using a blackjack simulator). This is a valid approach, but it does mean the conclusions depend on the validity of the starting conditions. As the developer of a line blackjack training products and our own advanced blackjack simulator (www.HandheldBlackjack.com), I decided to try and replicate these conditions in a realistic and common game. Assuming this was possible, I could then use my Blackjack Audit risk of ruin simulator and calculators to further back up Alan's analysis. Then, there would be no doubt that the conclusions were sound and fair. I started with a common Vegas two-deck game that I guessed would come close to modeling Alan's assumptions: - 2 decks, DAS, H17, 66.7% penetration, head up against the dealer.
- High-Low count system, as published by Stanford Wong in, "Professional Blackjack" (PiYee Press, www.bji21.com). Using only the 'Fab18' most useful indices, and insurance.
- 1 to 4 bet spread as follows: True Count < 0: 1 bet unit, TC >=0: 2 units, TC=1: 3 bet units, TC>=3: 4 units. While not an optimal bet spread, this is a realistic bet spread that includes a bit of cover camouflage since you're betting 2 units "off the top" after the shuffle. This spread was also chosen to try and come close to Alan's assumptions about the percentage of hands bet at each level (see below).
And this is the data we get for this game:
This is about as close as we can get to the initial assumptions, given a realistic blackjack game. Now, let's compare corresponding bankroll and risk of ruin values:
This data both shows that Alan's assumptions about the blackjack game are acceptable, and his conclusions follow the actual risk of ruin characteristics closely. And the same general conclusion applies to both methods: if you play with a low bankroll, you stand a good chance of going home broke (only a 30 to 40% chance of going home a winner with a starting bankroll of $500, and a goal of $1000 in profit). Readers may notice that the values in the last two rows for the ROR blackjack simulator vary a fair bit from the calculator method (about 10%). This is a consequence of statistical assumptions in the underlying 'double barrier' risk of ruin equations, and the very small bankroll in relation to the goals... the equations can be off by 5 to 10%. My theory is that this is due to blackjack card counting not being an To approximate Alan's 1% edge 'aggressive' game, I used these blackjack parameters: - Exact same game as before, but a different bet spread.
- 1 to 8 bet spread as follows: True Count < 1: 1 bet unit, TC >= 1: 2 units, 2: 4, 3: 5, 4: 6, 5: 8. This time, we are only moving to two bet units at a true count of 1 (not zero). This definitely yields a better edge, at the loss of some playing camouflage.
And this is the data we get for this game:
We can see once again that this is a good matching game to Alan's assumptions. Let's look at the corresponding bankroll and risk of ruin values:
Once again, we have good corroboration and supporting evidence for Alan's study. In particular, we can see the counter intuitive result that we stand a greater chance of surviving 400 rounds at the 0.5% edge game compared to the higher risk 1% game (i.e. both Alan's and my data indicate the same result). Our chances of earning $1000 before busting is slightly better at the 1% game, but the more important observation is that you still have 51% chance, The ROR simulator in my Blackjack Audit program provides additional information to help explain the counter-intuitive results of this study. Specifically, how you can have less than 50% chance of having a winning session, yet have a positive edge? Consider the worst case scenario: playing to earn $1000 profit with a meager $500 bankroll. In both the 0.5% and 1% games, you only have about a 1 in three chance of going home a winner (i.e. 2 out of 3 times you'll go bust). But when you are a winner, you win big (twice your original bankroll). So some rough math yields 2/3 * $500 + 1/3 * $1,000 = $0. The ROR simulation shows that the actual average profit per session in the 0.5% game case is about $130, with an average of 1200 rounds per session. Using the exact average bet size and edge provided in table 1, we can double check this result: 1200 * $22.21 * .005 = $133. Conclusions Being a good card counter is not enough to win: you need to play with a bankroll sufficiently large in comparison to your bet spread and game. How much is enough? Generally, to last a 12 hour session of blackjack, 10 times your average bet size is a good guide (or 20 times your unit bet size). So, for $10 players, you shouldn't walk into the casino with less than $2,000. And that still leaves you with a 5% chance of losing that session bankroll... you're going to walk out with empty pockets 1 in 20 times! A good blackjack simulator and calculator are essential tools once you have honed your skills with good training software. All of our blackjack, craps and poker training products are free to download and try out, and we have software for Windows, Palm OS, and Pocket PC. Visit our web site for details: www.HandheldBlackjack.com.
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