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CALCULATING THE HOUSE EDGE FOR ANY NUMBER OF DECKS AND BLACKJACK RULE SET
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CALCULATING THE HOUSE EDGE FOR ANY NUMBER OF DECKS AND ANY BLACKJACK RULE SET
By Arnold Snyder
(From Blackbelt in Blackjack, 3rd Edition, Cardoza Publishing 2005)
© 2005 Arnold Snyder
Quick Link to Charts for Effects of Rules on House Edge
Counting is Not Enough
Many card counters believe that as long as a game is called “blackjack,” and is being offered by a legitimate casino, they can win by applying their counting systems. But the fact is that while some games can be beaten by card counting strategies many can’t, and table conditions make the difference.
This article will give you simple guidelines you can follow that will help to keep you from throwing your money away in unbeatable games.
First, let’s define table conditions. There are four distinct conditions of any blackjack game that directly affect the profit potential for card counters:
1. The number of decks in play. In U.S. casinos, this may currently range from one to eight.
2. Rules. There are about two dozen common rule variations, and dozens more uncommon variations, in U.S. casinos.
3. Crowd conditions. You may be the only player at the table, or one of as many as seven. Crowded tables mean fewer hands per hour and lower earnings for card counters.
4. Depth of deal, or deck penetration, between shuffles. Anywhere from 2% to 90% of the cards may be dealt out.
The Depth of the Deal (Penetration)
Of all of these table conditions, penetration is by far the most important. When I published my first book, The Blackjack Formula, in 1980, many players were skeptical of the weight I gave to the effect of deck penetration. No other authors had mentioned penetration as an important factor up to that time, and I received numerous letters from players who simply could not believe that there was any great difference in profitability between a single-deck Reno game with 55% penetration and one with 65% penetration. “10% is only five cards!” one player wrote to me. "Yet your formula shows the advantage almost doubling with the same 1 to 4 spread. That’s impossible!” Other card counters, who were playing 4-deck downtown Vegas games with 70% penetration and 1 to 4 spreads, were incredulous of my claim that such a small spread, with such poor penetration, left them with barely a tenth of a percent advantage over the house.
These days, any decent book on card counting will tell you that penetration is the name of the game, but before my book in 1980 no one knew! None of the books on card counting had ever mentioned the importance of deck penetration before.
The general rule is this: The shallower the penetration, the larger the betting spread you must use to beat the game. With a bad set of rules and poor penetration, you may not be able to beat the game with any spread.
In most single deck games, you can’t win big unless more than 50% of the cards are dealt out between shuffles—with Reno rules (double 10/11 only and dealer hits soft 17), make that more than 60%. There are two main reasons for this: One, most single-deck games have poor rule sets; two, you generally can’t get away with a very big spread in single-deck games. With 2-deck games, you’ll want at least 65% dealt out. (But don’t even bother with a 2-decker when playing Reno rules.) With 4 or more decks, a bare minimum of 70% of the cards should be dealt out. Regardless of the number of decks in play, a 10% difference in penetration will make a huge difference in your profit potential: A 6-deck game with 85% penetration (about 5 decks dealt) is vastly superior to a 6-deck game with only 75% (about 4 ˝ decks dealt).
For more information on penetration, and a formula for quickly and easily calculating the profitability of any blackjack game, see the Snyder Profit Index in Chapter 11 of Blackbelt in Blackjack.
This rest of this article will deal with the number of decks in play and the effects of rules on the profitability of blackjack games. Before you can profit from any card counting system, you must overcome the house edge—that is, the cost in percent of playing the game. Below you will find all the information you need to quickly calculate the basic strategy house edge for any number of decks and any set of blackjack rules.
The Number of Decks in Play
Now let's consider the effect of the number of decks shuffled together. All other conditions being equal, single-deck games would be the most profitable for card counters. The more decks being used, the less profitable the game becomes, not only for card counters, but for basic strategy players as well. A single-deck Vegas Strip game (blackjack pays 3:2, double down on any two cards, and dealer stands on soft 17), is pretty close to being a break even proposition for a basic strategy player. With four or more decks in play, and the same set of rules, the house has about a ˝ percent edge. Use this chart to estimate your basic strategy (dis)advantage due to the number of decks in play:
The second condition you must consider is the set of rules used on the game. Some rules, notably those that offer the player more options, are favorable to the player, assuming the player applies the correct strategy. Such rules would be surrender, doubling after splitting allowed, resplitting aces allowed, etc. Those rules that limit the player’s options, such as doubling down on 10-11 only, or no resplits, are disadvantageous to the player.
Some rules neither limit nor offer options to the player, but alter the dealer’s procedure. An example of one such rule would be “dealer hits soft seventeen.” This is disadvantageous to the player. An advantageous dealer rule, used occasionally in short-term special promotions, would be “blackjack pays 2-to-1.”
A different type of advantageous rules for the player are the “bonus” rules, such as “dealer pays $XXX bonus to player hand of 6, 7, 8 same suit.” Most bonuses, due to the rarity of the bonus hand(s) occurring, have very small $ value to the player.
Now let's look at the approximate effect of each rule on your basic strategy expectation. By adding the effect of the number of decks in play to the effects of the rule variations, you will know the house advantage against basic strategy players. Card counters call this the starting advantage, or the advantage off the top.
Most rules, to be sure, affect card counters differently than they affect basic strategy players. The house edge off the top, however, is always an important consideration, as this is what your skillful play must overcome. For instance, insurance has no value to a basic strategy player, since correct basic strategy is to never take insurance. If a casino disallows insurance, however, this hurts card counters, since counters profit from their selective insurance bets. Likewise, the surrender option has little value to basic strategy players--less than one-tenth of 1 percent increase in expectation. For a card counter, however, surrender is, like insurance, very valuable.
In order to figure out our starting advantage, we need to begin by defining a benchmark game, i.e., a set of standard rules to which we can add or subtract the effects of the rule variations. Most authors define this benchmark game as Vegas Strip rules:
1. Dealer stands on soft 17.
2. You may double down on any 2 original cards.
3. You may not double down after splitting a pair.
4. You may split any pair.
5. You may resplit any pair except aces.
6. Split aces receive only one card each.
7. No surrender.
8. Dealer either receives a hole card, or the player’s original bet only is lost if the player doubles down or splits a pair and the dealer gets a blackjack.
9. Insurance is allowed up to one-half the player’s bet, and pays 2 to 1.
10. Player blackjack is paid 3 to 2.
Now the effect of any other rules must be accounted for in determining your starting advantage. These are the rule effects:
Most of these rule effects have been calculated by using data from Peter Griffin’s Theory of Blackjack. Note that the last five rules show effects of 00.00 percent for basic strategy players. When it comes to the “bonus” rules, such as 6,7,8 suited or 7,7,7 pays 2:1, the general rule is to never change your basic strategy to attempt to get a bonus payout.
In some cases, where a specific dollar amount is awarded for the bonus hand, the value in percent is dependent on the player’s bet size. For instance, if 6,7,8 suited pays a $100 bonus, then the value in percent will be quite different for a player who has a $2 bet and a player who has a $200 bet. The first player would receive a 50:1 payout on his hand, while the second player would receive only an extra half-bet. The $2 bettor would likely be correct in hitting his hand against any dealer upcard, if his hand contained two of the needed suited cards. The $200 bettor would usually be making an error if he hit this hand in violation of his basic/count strategy.
Also, take note of the huge negative effect of “BJ Pays 6-to-5,” a rule now common in many Las Vegas single-deck games. This rule is a killer. And note how much worse yet it is if BJ Pays 1-to-1 (even money), as is standard in all “Super Fun 21” games. All those other “good” rules that the “Super Fun” game allows do not make up for this huge negative. Serious card counters should stick with the traditional “BJ Pays 3:2” games.
Let’s walk through an estimation of our “off the top” expectation in a more typical blackjack game. Consider a standard Atlantic City 8-deck game, which allows double after splits, but no resplits. Our basic strategy expectation is derived by adding together the effects of the number of decks in play, and the rule effects (from the multi-deck column). We get:
Blackjack may be just a card game, but you'd better take it as seriously as the casinos do if you expect to beat them. That means paying attention to the house edge from the number of decks and blackjack rules, crowd conditions, and, above all, penetration. Believe me, the casinos are dead serious about beating you. ♠
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