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One of the most common blackjack side bets in California card rooms is known as Buster

Blackjack (BBJ). I found 🛡 this documentation online. The player who makes the BBJ wager

wins if the dealer busts. The payout is then based 🛡 on the total number of cards in the

dealer’s busted hand. There are a lot of details to share, so 🛡 I’ve decided to break up

the discussion into the six-deck case and the two-deck case. This article covers card

counting 🛡 BBJ in a six-deck shoe game.

When I was asked about the vulnerability of BBJ

to card counting by one of 🛡 California’s largest banking corporations, I told them “not

very.” After all, the dealer needs low cards to generate hands with 🛡 lots of cards, he

needs mid-cards to give stiff totals of 15 and 16, and he needs high cards to 🛡 bust

hands. My intuition was both right and wrong: the difference is the pay table. There

are eight different pay 🛡 tables for BBJ that are in use or recommended. I investigated

all eight and found that five of these eight 🛡 are vulnerable and three are not. There is

also a progressive version of BBJ that I did not investigate at 🛡 all.

To begin with, I

am going to focus on one specific vulnerable pay table for BBJ. This is the most 🛡 common

pay table used by card rooms banked by one of the largest corporations in California.

The pay table is 🛡 as follows:

Dealer busts with 8+ cards pays 200-to-1.

Dealer busts

with 7 cards pays 100-to-1.

Dealer busts with 6 cards pays 30-to-1.

Dealer 🛡 busts with 5

cards pays 6-to-1.

Dealer busts with 4 cards pays 3-to-1.

Dealer busts with 3 cards

pays 1-to-1.

Otherwise, the player 🛡 loses the BBJ bet.

Here is the combinatorial

analysis for this pay table:

In particular,

The house edge is 4.724%.

The standard

deviation is 🛡 2.724.

The hit frequency is 28.576%.

The following table gives the effect

of removal (EOR) for each card. The EOR tells the 🛡 change in house edge by removing a

single card of that rank and re-computing the house edge. A positive EOR 🛡 means that the

edge moves towards the player side if the card is removed. A negative EOR means that

the 🛡 edge moves towards the house side if the card is removed.

Notice how much larger

the EOR is for the cards 🛡 A, 2, 3 than for any other card. With an abundance of these

three cards in the shoe, the dealer 🛡 is more likely to have hands with a large number of

cards. Whether the dealer’s hand busts or not is 🛡 beside the point to beat this pay

table: the primary goal is to have multi-card hands. Removing a single deuce 🛡 from a

six-deck shoe moves the edge towards the house by 0.77%. That’s big. Also, an Ace never

busts with 🛡 fewer than four cards. Removing an Ace from the six-deck shoe moves the edge

towards the house by 0.35%.

In the 🛡 table above, under the column Sys #1, I give tags

for the optimal card counting system #1. I simply use 🛡 the EOR’s to two decimal places,

scaled to make the index for a ten-valued card equal to 1. System #1 🛡 is not intended to

be used in practice. I include it only for purpose of determining an upper bound for

🛡 the profit possible by card counting BBJ.

Under the column Sys #2, I give a balanced

card counting system that can 🛡 be used in practice. System #2 shows the importance of

the cards A, 2, 3 to the card counter, with 🛡 deuces the most valuable card. The betting

correlation for this system is 0.977. For math geeks, this value is the 🛡 cosine of the

angle between the vectors given by system #1 and system #2. The closer this value is to

🛡 1, the more accurate the system.

To analyze these two systems, I simulated one hundred

million (100,000,000) six-deck shoes, with the 🛡 cut card placed at 260 cards (1 deck

from the end), for each system. The following table summarizes the results 🛡 of these

simulations:

Note that system #2 returns 95.0% of the profit that optimal system #1

returns.

These numbers indicate a moderate 🛡 vulnerability to advantage play. If the AP

uses system #2 against BBJ on a fast heads-up game where he gets 🛡 200 rounds per hour,

and the AP makes aR$100 wager on BBJ whenever he has the edge, then the AP 🛡 can

earnR$140 per hour on the BBJ side bet. The AP using system #2 will have an average

edge over 🛡 the house of 5.28% and will make the BBJ bet on 13.29% of his hands.

As I

mentioned at the start 🛡 of this post, there are eight different pay tables that I

considered for BBJ. The following table gives the combinatorial 🛡 analysis for another

pay table in common use:

In particular

The house edge is 4.691%.

The standard deviation

is 1.989.

The hit frequency is 🛡 28.576%.

Because a hand that contains an Ace does not

bust with fewer than four cards, the increase in payout for 🛡 busting with three cards

does not come into play with a hand that includes an Ace. Moreover, busting with four

🛡 or more cards has greatly reduced payouts. These two factors eliminate the

effectiveness of Aces to the card counter.

The following 🛡 table gives the EOR’s for this

pay table, along with a card counting system based on these EOR’s. In particular,

🛡 notice how much the EOR for the Ace has changed.

System #3 is a reasonable choice,

based on the EOR’s. The 🛡 betting correlation for this system is 0.973.

I was very

surprised at how different the tags for this pay table are 🛡 from those developed for the

previous pay table. This highlights the profound difference in strategy involved in

beating these two 🛡 pay tables. I know of no other blackjack side bet that has as

significant a strategy change for the AP 🛡 based purely on the pay table being used.

To

analyze this pay table and system #3, I ran a simulation of 🛡 twenty million (20,000,000)

six-deck shoes, with the cut card placed at 260 cards. The results were very

disappointing for the 🛡 AP:

The trigger count is +8.

The average edge is 2.33%.

The bet

frequency is 3.64%.

The number of units won per 100 hands 🛡 is 0.085.

These numbers

indicate a very low vulnerability to advantage play. If the AP uses system #3 on a fast

🛡 heads-up game where he gets 200 rounds per hour, and the AP makes aR$100 wager on BBJ

whenever he has 🛡 the edge, then the AP can earn aboutR$17 per hour.

There are two other

pay tables that are similar to this 🛡 one, paying 2, 2, 4 units respectively for busting

with 3, 4 or 5 cards. Simulations against these other pay 🛡 tables showed an even lower

vulnerability to card counting. Because of their low vulnerabilities, I will not

include these pay 🛡 tables in any further discussion of card counting BBJ.

The following

table summarizes the combinatorial analysis for six different pay tables 🛡 for BBJ:

The

following table gives the results of card counting BBJ against each of these six pay

tables.

For pay tables 🛡 #1, #2, #3, #4 and #5, I used system #2 with tags (-2, -3, -2,

0, 0, 0, 1, 1, 🛡 1, 1).

For pay table #6, I used system #3 with tags (1, -3, -2, 0, 1,

-1, 2, 1, 1, 🛡 0).

The only difference between pay tables #2, #3 and #4 is the top award.

For pay table #4, the AP 🛡 can earn about 0.737 units per 100 hands by card counting BBJ.

This is as good as it gets for 🛡 the AP using system #2 in a six-deck shoe game.

The

relative weakness of card counting using system #2 against pay 🛡 table #5 surprised me; I

thought it would do better. These results may be because system #2 was designed for 🛡 pay

table #2. A different count may work better against pay table #5. I’ll leave the

development and evaluation of 🛡 that card counting system as an exercise to the

reader.

Normally, at this point, I would simulate various penetration levels in 🛡 the

shoe to demonstrate the effectiveness of cut-card placement on game protection. I would

then make recommendations for protecting BBJ 🛡 that follow a standard game protection

model for card counting blackjack side bets. In this case, however, it is completely

🛡 clear how to protect BBJ.

I have only one game protection recommendation for BBJ, and

it’s easy:

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