Calculating the House Edge for any Blackjack Game
Calculating the House Edge for Any Number of Decks and Blackjack Rules Set
(From Blackbelt in Blackjack, 3rd Edition, Cardoza Publishing 2005)
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:
|Effects in Percent|
|Double on 10-11 only:||-0.26||-0.21||-0.18|
|Double on 9-10-11 only:||-0.13||-0.11||-0.09|
|Hits Soft 17:||-0.19||-0.20||-0.21|
|Double After Splits:||+0.14||+0.14||+0.14|
|Draw to Split Aces:||+0.14||+0.14||+0.14|
|Late Surrender (H soft17):||+0.03||+0.06||+0.09|
|Less Common Rules|
|Double on 8-9-10-11 only:||-0.13||-0.11||-0.09|
|Double on 11 only:||-0.78||-0.69||-0.64|
|Double 3 or More Cards:||+0.24||+0.24||+0.24|
|Double after Ace splits:||+0.10||+0.10||+0.10|
|Double on 3+ cards:||+0.24||+0.23||+0.23|
|No Ace Splits:||-0.16||-0.17||-0.18|
|Early Surrender (H soft17):||+0.70||+0.71||+0.72|
|Early Surrender v. 10 only:||+0.19||+0.21||+0.24|
|BJ Pays 6-to-5:||-1.74||-1.71||-1.71|
|BJ Pays 1-to-1:||-2.32||-2.28||-2.26|
|BJ Pays 2-to-1:||+2.32||+2.28||+2.26|
|Suited BJ Pays 2-to-1:||+0.58||+0.57||+0.56|
|21 Pushes Dlr. 10-up BJ:||+0.20||+0.20||+0.20|
|No Hole Card (European):||-0.10||-0.11||-0.11|
|5-card 21 Pays 2-to-1:||+0.20||+0.20||+0.20|
|6-card 21 Pays 2-to-1:||+0.10||+0.10||+0.10|
|Suited 678 Pays 2-to-1:||+0.01||+0.01||+0.01|
|7-7-7 Pays 3-to-2:||+0.01||+0.01||+0.01|
|6 Cards Unbusted Wins:||+0.10||+0.10||+0.10|
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:
|Double After Splits:||+0.14|
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. ♠
For more information on winning at blackjack with card counting or other professional gambling methods, see Arnold Snyder's Blackbelt in Blackjack.
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