Discount Gambling

To graphically demonstrate the large +EV advantage obtained by a simple Ace count in Triple Attack Blackjack, I compared the EV frequencies at the 1/2 shoe depth to normal blackjack using the hi-lo count. The graph below shows the Triple Attack edge dominates that of normal blackjack. The edges obtained in Triple Attack are both larger, and more frequent than in standard blackjack. The Ace count in Triple Attack is much simpler to track, compared to the hi-lo count, because you only have to identify one card — the Aces. The graph clearly shows that if you’re going to count at a shoe game, you should play Triple Attack. (Compare the areas under the curves, to the right of EV=0.)

And for the last hand in the shoe:

For anyone interested in grinding out a profit from Triple Attack Blackjack, I thought I’d post a few graphs showing the distribution of outcomes for playing 100 shoes. I assumed a penetration of 7/8ths of the shoe, and for a heads-up player shamelessly betting 10x/20x when the Ace count is good, but otherwise playing basic strategy. These graphs should not be used to estimate bankroll requirements, but just give an idea of what to expect in the medium-run, and what kind of average profit rate to expect.

For a 10x big bet when the Ace count is good (+2 after 2 decks):

Cumulative Distribution of 100 Shoe Session Outcomes, 10x Bet for +EV Ace Count

Note the bias in the distribution clearly favors the player counting Aces. The x-axis is the net win/loss in small (1x) bets. The curves clearly cross the 50% cumulative frequency in positive net territory.

For a 20x big bet when the Ace count is good (+2 after 2 decks):

Cumulative Distribution of 100 Shoe Session Outcomes, 20x Bet for +EV Ace Count

Of course, you can also see the potentially large swings, which discourages all but extremely serious gamblers from trying this sort of thing. For example, varying your bet 20x against a unit $5 minimum bet means betting nickels when the count is bad, and black ($100) when the count is good. Over a 100 shoes, there’s a 10% probability of ending up stuck $6000 or more. Of course, there’s a greater chance of winning $6000 or more, and you’ll “average” a win of (250 units)($5/unit) = $1250 for this 100 shoe “session”.

In the next post, I’ll compare Triple Attack to normal 6-deck blackjack, showing how much more effective counting is, as far as shoe games go.

My nearby Barona Casino opened up a new Triple Attack Blackjack game last month, and I tried it out this weekend. I really like this game, because of the weird, aggressive hitting like Spanish 21. But more importantly, the game allows you to double your initial bet after seeing your first card, and bet again after seeing the dealer upcard. You’re immediately paid on 21 or any 6-card total, and you can double down at any time. The house gets its advantage by paying even money on blackjack, and pushing on dealer 22.

For the full rules and basic strategy, see the Wizard’s analysis, which yields a 1.18% house edge. But for most cases, his basic strategy charts boil down to the following simple rules:

• If your first card is an Ace, triple your bet.
• If your first card is a Face, triple your bet against 2-9, else double your bet.
• 3rd Attack 2 against a 6.
• 3rd Attack 8 against 6,7.
• 3rd Attack 9 against 5,6,7,8.
• Double 10 against 3-6.
• Double 11 against 7 and under.
• The only soft total double is soft-18 against a 6.
• Double all 5-card 14’s and under.
• Double-for-less (split aces rescue) soft-17 and under, and soft-18 against 9 and A.
• Hit all 12’s and 13’s.
• Hit 14 except against a 6.
• Hit 15 against a 2 or 3.
• Hit 16 against a 2.
• Hit 17 against an Ace, and for some 5-card totals.
• Never split 4’s, 5’s, 6’s, 10’s.
• Split 2’s and 3’s against 6,7.
• Split 7’s against 4-7.
• Split A’s and 8’s against everything except A.
• Split 9’s against 6,8,9.

I really enjoyed the game. I also won about $400 in 3 sessions, which is unheard-of for me, since I flat bet for the minimum $5. I never vary my bet. However, in Triple Attack, you often end up betting 3 units, or even 6 or more, if you split hands like Aces. I don’t think I’ve ever won that much money playing blackjack before. The players all loved it, because they make more decisions, and their initial $20 bet can easily become a $120 bet on a good hand, or just remain a $20 bet on a weak hand.

Of course, it immediately occurred to me that card counting should be more effective in Triple Attack compared to standard blackjack. While you might dramatically increase your bet for a good count in regular blackjack, in Triple Attack, you only have to commit 1/3 of this amount on the 1st Attack, then another 1/3 only if you like your first card (an Ace or Face), and another 1/3 only if you like the dealer upcard. So, even though the count may be good, you can get away from a bad hand on the first card, and also if the dealer has a good upcard. Compare that to normal blackjack, where you bet huge on a good count, then get a 6, then the dealer shows an Ace.

Using basic strategy, I looked at the EV sensitivities of each of the card ranks (i.e., the effect of removing one card from the shoe). Interestingly, the effect of removing a Face card was very small. Removing a 5 improved the hand EV the most, but not by much compared to a 2, 3, 4, 6, or even a 7. By far, the Ace was the most powerful card, making the other cards insignificant.

So, I tried out a simple scheme, which only relies on the Ace count, compared to the expected number dealt. You subtract the number of actual Aces seen from the expected number of Aces dealt to get the Ace count. For example, after dealing 1/2 the shoe (192 cards), you should have seen 16 Aces. If you’ve only seen 12 Aces, then the count is +4. The Ace count is a measure of the “extra” Aces left in the shoe. Using a basic strategy simulator, I generated the curves showing the effectiveness of this simple Ace-Count system (including double-1-unit-for-less as Ace-split rescue):

Triple Attack Blackjack EV vs. Ace-Count and Shoe Depth

This effect is huge! Look at the green curve, which shows the effect of the Ace count after 3/4 of the shoe has been dealt (288 cards dealt). At this point, 24 Aces should have been dealt, on average. But if only 20 Aces were dealt, then the count is +4, and the next hand has a 4% player advantage. Similarly, a negative count tells you to bet the minimum, or even Wong (sit out) until the next shoe. The blue curve shows the effect after 1/4 of the shoe dealt (96 cards), and the red curve shows the half-way point (192 cards). You can see that generally speaking, it’s good to increase your bet any time the count is +2 or more (the shoe is “loaded” with 2 or more extra Aces).

We don’t need to look at the distribution of Ace counts to know that +1, +2, +3, +4, +5 counts happen all the time, especially at the end of shoe, where the effect if largest. Note that the green curve shows about a 1.25% EV improvement for each surplus Ace left in the shoe.

A simple counting strategy bets the minimum for the first 2 decks of the shoe, then a large bet if the Ace count is +2 or better. Using basic strategy, and assuming the cut card at 48 cards left in the shoe, the following returns are obtained:

On the other hand, the shoe is +EV about 20% of the time (+2 or better count) after the first 1/4 of the shoe is dealt. So, if you just like varying your bet on a good count, you’ll really enjoy this game. Overall, if you Wong and only bet +EV Ace counts, then you’ll extract about +17% total EV out of a shoe (i.e., +0.17 bets/shoe). The Ace-count system is very easy to implement (e.g., use the height of muck cards to estimate the expected Ace count), and is very fun. Some people have a good idea of when the shoe is Ace-rich or not. That’s all you need to know for this Triple Attack game!

And there’s no time to (possibly) bet big like the last hand of the shoe (purple curve, cut card @ 1 deck left):

Note that unless the Ace count is 0 (unlikely, only 22% of the time), you’re either a big favorite, or a big underdog on the last hand. It’s easy to get the Ace count right on the last hand, assuming the cut card was placed with 48 cards behind it. At the last hand, you should have seen 28 Aces. If you only saw 24, then you have a +4 count, and the EV of the next hand is almost 10% in your favor (a $100 1st Attack bet will return a $9.80 profit on average, including possible 2nd and 3rd Attacks, doubles, etc.). But, if you saw 30, then the count is -2, and the house has a 6.2% edge on the last hand, so bet the minimum, or Wong.

I suggest you simply count the number of Aces seen, even using chips if it helps you keep track. Compare your Ace count to the number of decks (48 cards/deck) seen in the muck rack. There are 4 Aces per deck. If the number of Aces seen is less than 4*(mucked decks), then the Ace count is positive. If your count is +2 or better, you’re +EV, and can increase your 1st Attack bet.

Get an idea of what a deck (48 cards) looks like in the discard rack. Ask the dealer for their estimate of the number of decks in the muck. You’ll find it’s pretty easy to estimate the number of dealt decks. Thus, you should be able to determine the Ace count very easily. Also, by watching the level of cards in the discard rack, you’ll know how powerful your count is (see above graphs for 1/4, 1/2, 3/4, and 7/8 shoe dealt depths).

Example: You look at the discard rack, and it looks like 3 decks have been dealt. This means that 4*3 = 12 Aces should have been dealt. Your actual Ace count is 15. The Ace count is -3, and you’re -EV, so bet the minimum. However, if the actual Ace count is only 9, then the Ace count is +3, and you’re +EV, so you should bet more.

This counting system couldn’t be any easier, or any more powerful. It doesn’t get much better than this.

Finally, I saw that at request, Barona increased the limits on the Triple Attack game to $25-$1000. I was locked out until the guy left, and the table returned to $5-$1000. In general, Barona is very flexible about increasing table limits.