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Double Blackjack
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Introduction
Double Blackjack made its casino debut at the Green Valley Ranch in November, 2015. The game could roughly be described as a cross between blackjack and pai gow poker. Much like pai gow poker, Double Blackjack is a slow-playing game with a lot of pushes, so blackjack players searching for lower volatility alternative may find it a fun way to gamble without much risk.
Rules
- The game is played with a single 52-card deck.
- All cards are counted the same way as in blackjack.
- Play begins with the player making equal Two-Card and Three-Card bets. The player may also make an optional Double-21 side bet.
- The dealer shall give each player and himself five cards.
- The player shall arrange his cards into a Two-Card hand and Three-Card hand.
- Both hands are scored as in blackjack, except:
- Any total under 17 or over 21 is a "fouled hand."
- There is no "blackjack." An ace and a 10-point card is worth the same as a three-card 21.
- After the players have set their hands, the dealer shall set his according to the house way below.
- The player's Two-Card hand shall be compared to the dealer's Two-Card hand. Likewise, the player's Three-Card hand shall be compared to the dealer's Three-Card hand.
- Any player fouled hand is an immediate loser.
- If the player did not foul a hand, and the corresponding dealer hand did foul, then the player shall win that hand.
- Otherwise, the Two-Card bet shall be resolved by comparing the player's Two-Card hand to the dealer's Two-Card hand.
- Likewise, the Three-Card bet shall be resolved by comparing the player's Three-Card hand to the dealer's Three-Card hand.
- If the player has two 21-point hands, then the player shall win a bonus equal to half of one of the individual wagers, regardless of whether each hand wins or draws. For example, if the player bets $10 on both the Two-Card and Three-Card bets, and gets two 21-point hands, then he will be paid an automatic bonus of $5, in addition to whatever he wins against the dealer.
- The rules of the Double-21 bet are explained below.
If that was not clear, here is a hierarchy of each hand.
- 21 points (including blackjacks).
- 20 points.
- 19 points.
- 18 points.
- 17 points.
- Fouled hands — Under 17 or over 21
The higher ranking hand shall win, except if both the player and dealer have fouled hands, then the dealer wins.
In case you're wondering where the house advantage is, the answer is the same as in blackjack, positional advantage. If the player has a fouled hand, he immediately loses, even if the corresponding dealer hand also fouls.
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House Way
Following is the dealer house way. A "pat hand" is 17 to 21 points. The dealer should set his hand according to the first applicable rule.
- The dealer will set the best Three-Card pat hand that also has a Two-Card pat hand.
- The dealer will set the best Three-Card pat hand.
- The dealer will set the best Two-Card pat hand.
- The dealer will set any two non-pat hands.
Strategy
The following is the Charlie Patrick strategy that will result in a house edge of 2.1785%, which is only 0.146% greater than optimal strategy.
- If only one valid hand can be made, then maximize the Two-Card hand, subject to these exceptions:
- Make a Three-Card 21 rather than Two-Card 17-19;
- Make a Three-Card 20 rather than a Two-Card 17.
- If you can make two valid hands, then play in a way that maximizes the Three-Card hand, subject to these exceptions:
- If both hands are 19 or less and you can make a Two-Card 21 (while busting the other hand), then do that.
- If both hands are 18 or less and you can make a Two-Card 20 or Three-Card 21 (while busting the other hand), then do that.
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Analysis
The following table shows a detailed breakdown of all possible events under optimal strategy. An "automatic win" is when the opposing hand is fouled.
Detailed Return Table
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Auto-Win Two-Card hand and Auto-Win Three-Card hand with double-21 | 2.5 | 1,041,377,024 | 0.000261 | 0.000653 |
| Auto-Win Two-Card hand and Auto-Win Three-Card hand | 2 | 15,881,915,580 | 0.003984 | 0.007968 |
| Auto-Win Two-Card hand and Beat Dealer Three-Card hand with double-21 | 2.5 | 16,633,130,912 | 0.004172 | 0.010431 |
| Auto-Win Two-Card hand and Beat Dealer Three-Card hand | 2 | 144,815,111,676 | 0.036325 | 0.072650 |
| Auto-Win Two-Card hand and Lose to Dealer Three-Card hand | 0 | 270,232,570,624 | 0.067784 | 0.000000 |
| Auto-Win Two-Card hand and Tie Dealer Three-Card hand with double-21 | 1.5 | 19,262,735,552 | 0.004832 | 0.007248 |
| Auto-Win Two-Card hand and Tie Dealer Three-Card hand | 1 | 171,193,142,652 | 0.042942 | 0.042942 |
| Auto-Win Two-Card hand and Auto-Lose Three-Card hand | 0 | 606,652,463,296 | 0.152171 | 0.000000 |
| Beat Dealer Two-Card hand and Auto-Win Three-Card hand with double-21 | 2.5 | 10,085,759,200 | 0.002530 | 0.006325 |
| Beat Dealer Two-Card hand and Auto-Win Three-Card hand | 2 | 18,971,124,968 | 0.004759 | 0.009517 |
| Beat Dealer Two-Card hand and Beat Dealer Three-Card hand with double-21 | 2.5 | 26,090,109,056 | 0.006544 | 0.016361 |
| Beat Dealer Two-Card hand and Beat Dealer Three-Card hand | 2 | 131,933,704,648 | 0.033094 | 0.066188 |
| Beat Dealer Two-Card hand and Lose to Dealer Three-Card hand | 0 | 128,148,972,648 | 0.032145 | 0.000000 |
| Beat Dealer Two-Card hand and Tie Dealer Three-Card hand with double-21 | 1.5 | 11,602,471,584 | 0.002910 | 0.004366 |
| Beat Dealer Two-Card hand and Tie Dealer Three-Card hand | 1 | 68,091,636,552 | 0.017080 | 0.017080 |
| Beat Dealer Two-Card hand and Auto-Lose Three-Card hand | 0 | 595,514,623,868 | 0.149377 | 0.000000 |
| Lose to Dealer Two-Card hand and Auto-Win Three-Card hand | 0 | 86,215,332,672 | 0.021626 | 0.000000 |
| Lose to Dealer Two-Card hand and Beat Dealer Three-Card hand | 0 | 86,188,573,640 | 0.021619 | 0.000000 |
| Lose to Dealer Two-Card hand and Lose to Dealer Three-Card hand | -2 | 64,951,660,696 | 0.016292 | -0.032585 |
| Lose to Dealer Two-Card hand and Tie Dealer Three-Card hand | -1 | 56,198,751,560 | 0.014097 | -0.014097 |
| Lose to Dealer Two-Card hand and Auto-Lose Three-Card hand | -2 | 85,454,009,044 | 0.021435 | -0.042870 |
| Tie Dealer Two-Card hand and Auto-Win Three-Card hand with double-21 | 1.5 | 108,290,880 | 0.000027 | 0.000041 |
| Tie Dealer Two-Card hand and Auto-Win Three-Card hand | 1 | 55,743,953,408 | 0.013983 | 0.013983 |
| Tie Dealer Two-Card hand and Beat Dealer Three-Card hand with double-21 | 1.5 | 615,864,960 | 0.000154 | 0.000232 |
| Tie Dealer Two-Card hand and Beat Dealer Three-Card hand | 1 | 67,181,537,388 | 0.016852 | 0.016852 |
| Tie Dealer Two-Card hand and Lose to Dealer Three-Card hand | -1 | 49,660,086,572 | 0.012457 | -0.012457 |
| Tie Dealer Two-Card hand and Tie Dealer Three-Card hand with double-21 | 0.5 | 1,491,651,840 | 0.000374 | 0.000187 |
| Tie Dealer Two-Card hand and Tie Dealer Three-Card hand | 0 | 33,458,667,776 | 0.008393 | 0.000000 |
| Tie Dealer Two-Card hand and Auto-Lose Three-Card hand | -1 | 205,300,374,932 | 0.051497 | -0.051497 |
| Auto-Lose Two-Card hand and Auto-Win Three-Card hand | 0 | 126,967,955,096 | 0.031848 | 0.000000 |
| Auto-Lose Two-Card hand and Beat Dealer Three-Card hand | 0 | 370,165,268,436 | 0.092851 | 0.000000 |
| Auto-Lose Two-Card hand and Lose to Dealer Three-Card hand | -2 | 137,607,591,068 | 0.034517 | -0.069034 |
| Auto-Lose Two-Card hand and Tie Dealer Three-Card hand | -1 | 284,358,619,664 | 0.071328 | -0.071328 |
| Auto-Lose Two-Card hand and Auto-Lose Three-Card hand | -2 | 38,827,063,968 | 0.009739 | -0.019479 |
| total | 3,986,646,103,440 | 1.000000 | -0.020325 |
The lower right cell shows an expected loss of 2.03% of both wagers. Since the house edge is traditionally defined as the ratio of the expected loss to the initial bet, and the initial bet is two units, then I would put the house edge at 1.02%.
If the above table is more than you wanted to know, then the following table shows a summary of all possible net wins.
Abbreviated Return Table
| Net Win | Combinations | Probability | Return |
|---|---|---|---|
| 2.5 | 53,850,376,192 | 0.013508 | 0.033769 |
| 2 | 311,601,856,872 | 0.078161 | 0.156323 |
| 1.5 | 31,589,362,976 | 0.007924 | 0.011886 |
| 1 | 362,210,270,000 | 0.090856 | 0.090856 |
| 0 | 2,303,544,428,056 | 0.577815 | 0.000000 |
| 0.5 | 1,491,651,840 | 0.000374 | 0.000187 |
| -1 | 595,517,832,728 | 0.149378 | -0.149378 |
| -2 | 326,840,324,776 | 0.081984 | -0.163968 |
| Total | 3,986,646,103,440 | 1.000000 | -0.020325 |
Double-21
The following table shows the probability and return of all possible outcomes of the Double-21 side bet. The lower right cell shows a house edge of 6.54%.
Double-21 Side Bet
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| AAKKK | 500 | 24 | 0.000009 | 0.004617 |
| Double 21 — Full House | 100 | 72 | 0.000028 | 0.002770 |
| Double 21 with Soft Three-Card Hand | 50 | 5,440 | 0.002093 | 0.104657 |
| Double 21 with Hard Three-Card Hand | 10 | 51,136 | 0.019676 | 0.196756 |
| Both hands at least 20 | 5 | 261,608 | 0.100659 | 0.503294 |
| All other | -1 | 2,280,680 | 0.877536 | -0.877536 |
| Total | 2,598,960 | 1.000000 | -0.065442 |
Acknowledgements
I would like to thank:
- Dan Lubin and Al Lwin, the game inventors, for supplying to me the math report by Stephen How. This saved me the time of analyzing it myself.
- Charlie Patrick, for the use of his strategy above.
External Links
This game is discussed at my forum at Wizard of Vegas.
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