Discount Gambling

Bust Bonus BJ Side Bet

Posted in blackjack sidebets by stephenhow on May 14, 2013

A reader pointed me out to Galaxy Gaming’s Bust Bonus blackjack side bet that the dealer will bust, which you make after seeing the dealer’s upcard. I figured I’d run the numbers to see if it was any good, or if it was countable. Well, it might be a fun bet on a few upcards, but it’s kind of expensive for the (offsuit) odds they offer.

Bust Bonus Side Bet (6 Decks)
Dealer Upcard	Probability Suited Bust	Probability Offsuit Bust	Probability No Bust	Payout Suited Bust	Payout Offsuit Bust	Return
Ace*	0.002222	0.199059	0.798719	50	3	-0.090437
Deuce	0.008751	0.347909	0.643339	25	1	-0.076644
Trey	0.011524	0.365435	0.623042	15	1	-0.084754
Four	0.014574	0.383896	0.601530	10	1	-0.071892
Five	0.017883	0.401749	0.580368	5	1	-0.089206
Six	0.020844	0.418415	0.560741	3	1	-0.079793
Seven	0.011716	0.250219	0.738064	15	2	-0.061881
Eight 888	0.011133 0.000207	0.227312 0.005041	0.756307	10 75	2 25	-0.048772
Nine	0.011602	0.217640	0.770758	20	2	-0.103441
Ten/Face*	0.012263	0.217975	0.769761	20	2	-0.088547

*Bust Bonus wagered after dealer peeks for blackjack.

The most countable bet is against a dealer 8 upcard. It has the lowest house edge (4.9%), and has high payouts for the 888o and 888s busts. The EORs are large, and a simple unbalanced count (Eight => -8, Nine, Ten/Face, Ace, Deuce, Trey, Four => +1; bet when running count >= +24) yields an average +7.5% edge/bet on 17.3% of the dealer 8 upcard hands. Of course, a dealer 8 only occurs on 1/13th of the hands, so it’s not a very practical bet. An ideal count (using total shoe composition including suits) yields a theoretical max return of +7.5% edge/bet on 1.6% of the dealt hands.

Sadly, the standard BJ counts (like the unbalanced Knockout count) don’t correlate with the EV of any of these bets, because unlike blackjack, the Ace hurts the Bust Bonus bet. (Ace rich shoe makes it harder to bust.)

1 comment

High Card Flush

Posted in +EV, collusion by stephenhow on April 22, 2013

A couple of readers have asked about Galaxy Gaming’s new High Card Flush game, which has a few placements now, and may be picking up some steam. The game is pretty simple, where each player and the dealer receive 7 cards. Each hand is measured by its highest flush, where a flush is first ranked by its length (number of cards of same suit), then by its card values. Each player must Ante before the hand, then wagers a 1x-3x Play bet (depending on flush size), or folds. The dealer qualifies with a three-card, 9-high flush. If the dealer doesn’t qualify, the Play bets push, and the remaining Antes are paid even-money. If the dealer qualifies, the Ante and Play bets receive even-money action against the dealer hand.

As you would expect, collusion helps in this game. A Monte Carlo analysis shows that with 6 confederates, perfect knowledge of the dealt cards gives each spot at least a +7.3% edge over the house. But practically, you’d be lucky if you could even communicate the suit counts (number of cards of each suit) dealt. If you figure out a non-suspicious way of doing this, then the following simple strategy yields a +3.1% edge over the house:

High Card Flush Advanced Strategy (6 Player Hands Seen)
Flush Size	Play Bet
1 or 2 cards	1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else fold others
3 card, Jack-high or lower	1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else fold others
3 card, Queen-high	1x if lowest suit count is 9 or higher, else fold.
3 card, King-high or better	1x if lowest suit count is 8 or higher, else fold.
4 cards	1x
5 cards	2x
6 or 7 cards	3x

where the suit counts 4-tuple is the sorted number of cards of each suit.

4 comments

Phil’Em Up Poker

Posted in collusion by stephenhow on April 19, 2013

When I playing Mississippi Stud in Vegas last week, I overheard someone mention a game called Phil’Em Up Poker. I looked at the game, to see if collusion would yield an edge. The rules are pretty simple. The game is played with a 52-card deck plus a Joker which may be used for Aces, straights, and flushes. Each player bets an Ante, and receives two hole cards. Two community cards are dealt face up. Each player may either make an additional 1x bet (i.e., “double-up” his action), or check, before the 3rd community card is exposed. If a player makes a pair of Tens or better, he wins according to a paytable. There is no dealer hand. The house edge is a reasonable 3.3%.

Optimal Outcomes for Phil’Em Up Poker
Hand	Bet	Combinations	Probability	Payout	Return
FIVE_ACES	2	5	0.00000035	1000	0.000697
natural ROYAL_FLUSH	2	20	0.00000139	250	0.000697
wild ROYAL_FLUSH	2	100	0.00000697	100	0.001394
natural STRAIGHT_FLUSH	2	180	0.00001254	50	0.001254
wild STRAIGHT_FLUSH	2	720	0.00005018	25	0.002509
FOUR_OF_A_KIND	2	4,140	0.00028853	20	0.011541
FULL_HOUSE	2	21,840	0.00152212	15	0.045664
FLUSH	2	39,020	0.00271946	9	0.048950
STRAIGHT	2	77,460	0.00539850	7	0.075579
THREE_OF_A_KIND	2	211,200	0.01471939	3	0.088316
TWO_PAIRS	2	365,640	0.02548294	2	0.101932
High Pair	2	1,562,112	0.10886993	1	0.217740
Low Pair	2	75,648	0.00527222	-1	-0.010544
HIGH_CARD	2	339,708	0.02367563	-1	-0.047351
STRAIGHT	1	25,200	0.00175629	7	0.012294
THREE_OF_A_KIND	1	105,600	0.00735969	3	0.022079
TWO_PAIRS	1	327,360	0.02281505	2	0.045630
High Pair	1	922,608	0.06430030	1	0.064300
Low Pair	1	3,514,752	0.24495734	-1	-0.244957
HIGH_CARD	1	6,755,112	0.47079118	-1	-0.470791
total		14,348,425			-0.033067
expected		14,348,425

Collusion doesn’t help. That’s because only 3.8% of hands are bet on a draw only. Collusion will change few decisions, and result in little gain. With 7-player collusion, perfect play will only reduce the house edge to 3.2%.

6 comments

Double Attack Blackjack

Posted in +EV, counting, double attack blackjack by stephenhow on March 29, 2013

Thanks to reader John A. for pointing out this game to me. The game has been around (mostly in Atlantic City), but it’s new to me. It looks like the predecessor to Triple Attack Blackjack, as it’s based on a Spanish deck (10’s removed, J/Q/K’s remain) and the player may double his bet after the first card is dealt face up to the dealer. After this initial double attack option, the hand plays out normally with the total amount bet as the hand wager. (I.e., doubles and splits are based on the total amount bet after any double attack.)

The rules following the double-attack option are as follows:

Dealer stands on soft-17
Double-down at any time (no re-doubles)
Surrender at any time, including double-down rescue and after splits
No re-splitting of Aces
Blackjack pays even money

The house edge for the game is a reasonable 0.50% on the initial bet. The element-of-risk is even lower, as you double your wager 58% of the time (i.e., you double-attack vs. a dealer 2-8). The return is even lower still if they allow you to surrender after splitting Aces. The EORs are listed in the following table for removing a single card from a 8-deck shoe.

Double Attack Blackjack EORs (8 deck)
Removed Card	EOR	Balanced	Unbalanced
Deuce	+0.0832%	+1	+1
Trey	+0.1127%	+1	+1
Four	+0.1514%	+1	+1
Five	+0.1917%	+1	+1
Six	+0.1184%	+1	+1
Seven	+0.0341%		+1
Eight	-0.0560%
Nine	-0.0895%	-1	-1
Face	-0.1466%	-1	-1
Ace	-0.0937%	-1	-1

Basic Strategy

The basic strategy for the game was auto-generated by my analyzer program. You should double-down rescue 16 and lower against a dealer 8-thru-A, and 17 against an Ace. The strategy simulates at a -0.53% return, averaged over the whole shoe, very close to the analyzer’s calculated -0.50% return.

The unbalanced count in the above table yields 23.8% +EV betting opportunities (count >= +23) in an 6-deck shoe game with 52 cards behind the cut card. The average +EV hand returns +0.52%/bet. Compare this to the “Knockout” unbalanced count for 6-deck standard blackjack with cut card @ 5th deck, where 21.3% of the hands are +EV (count >= +17) with an average yield of +0.30%/bet.

Hand	Dealer Upcard
Hand	2	3	4	5	6	7	8	9	10	A
Soft Totals
soft 21	S	S	S	S	S	S	S	S	S	S
soft 20	S	S	S	S	S	S	S	S	S	S
soft 19	S	S	S	S	S	S	S	S	S	S
soft 18	S	S	S	D	D	S	S	H	H	H
soft 17	H	H	D	D	D	H	H	H	H	H
soft 16	H	H	H	H	D	H	H	H	H	H
soft 15	H	H	H	H	H	H	H	H	H	H
soft 14	H	H	H	H	H	H	H	H	H	H
soft 13	H	H	H	H	H	H	H	H	H	H
Hard Totals
hard 20	S	S	S	S	S	S	S	S	S	S
hard 19	S	S	S	S	S	S	S	S	S	S
hard 18	S	S	S	S	S	S	S	S	S	S
hard 17	S	S	S	S	S	S	S	S	S	R
hard 16	S	S	S	S	S	H	H	H	H	H
hard 15	S	S	S	S	S	H	H	H	H	H
hard 14	H	H	S	S	S	H	H	H	H	H
hard 13	H	H	H	H	H	H	H	H	H	H
hard 12	H	H	H	H	H	H	H	H	H	H
hard 11	D	D	D	D	D	D	D	D	H	D
hard 10	D	D	D	D	D	D	D	H	H	H
hard 9	H	H	H	H	H	H	H	H	H	H
hard 8	H	H	H	H	H	H	H	H	H	H
hard 7	H	H	H	H	H	H	H	H	H	H
hard 6	H	H	H	H	H	H	H	H	H	H
hard 5	H	H	H	H	H	H	H	H	H	H
Pairs
A-A	P	P	P	P	P	P	P	P	P	H
10-10	S	S	S	S	S	S	S	S	S	S
9-9	S	S	P	P	P	S	P	P	S	S
8-8	P	P	P	P	P	P	P	P	P	P
7-7	P	P	P	P	P	P	H	H	H	H
6-6	H	H	P	P	P	H	H	H	H	H
5-5	D	D	D	D	D	D	D	H	H	H
4-4	H	H	H	H	H	H	H	H	H	H
3-3	H	P	P	P	P	P	P	H	H	H
2-2	H	P	P	P	P	P	H	H	H	H

1 comment

Blackjack Switch

Posted in blackjack switch by stephenhow on March 24, 2013

Ok, I just spent way too much time working out a Blackjack Switch strategy. At first, I just wanted to calculate the EORs for Blackjack Switch, to see if it was more countable than regular blackjack. (The EORs are about the same as regular blackjack, so it’s probably not more countable.) Then I got carried away making a switching strategy, trying to keep it simple and intuitive (i.e., real). Hopefully, this post will save someone the bother of going through all this again.

House Edge

I put the Blackjack Switch rules in my blackjack analyzer program, which found the best switch decision and combined EV for each 4-card starting hand (100x more hands than standard BJ). I calculated the value of each 2-card hand assuming the other 2-card hand hadn’t yet played out (a simplifying assumption to make calculations practical). For a 6-deck shoe and Las Vegas rules (H17, DAS, re-split all pairs up to 4 hands, no LS, no BJ after switch), I got a combined return of -1.00% for both hands, which is a house edge of 0.50% per hand.

Effect of Removed Cards (EORs)

The sensitivity of the game’s EV to the removal of a given card rank is called the “effect of removal” (EOR). If Blackjack Switch was highly vulnerable to counting, you’d see it in the EORs. This is the first place to look. The table below lists the effect on the optimal EV of the game by removing one card of a given rank from a 6-deck shoe.

6-Deck Blackjack Switch EORs
Removed Rank	EOR	Balanced	Unbalanced
Deuce	0.0959%	+1	+1
Trey	0.0711%	+1	+1
Four	0.0891%	+1	+1
Five	0.1086%	+1	+1
Six	0.1077%	+1	+1
Seven	0.0379%
Eight	0.0166%
Nine	0.0564%	-0.5
Ten/Face	0.0925%	-1	-1
Ace	0.0561%	-0.5

The EORs are very similar to those of regular blackjack. The sum of the 2-6 EORs is about 0.45% in both cases. However, the Ace is half as powerful compared to regular blackjack. In Blackjack Switch, an Ace plus a Nine is comparable to an Ace in regular blackjack. You could probably use your normal counting system for BJ Switch, but note that it’s overestimating the power of Aces, and underestimating the power of Nines.

I also broke out separate EORs for the switch EV (the nominal 9.25% advantage obtained through the player switch option). If these values were large, it’d indicate an exploit through an indexed switch strategy. However, these switch EORs are very low, about 5x lower than the overall EORs. So an indexed switching strategy would not yield much benefit.

Basic Strategy

Basic strategy for Blackjack switch consists of the initial switch decision, and the post-switch basic strategy table.

Switching Strategy

Blackjack Switch has a pretty big following, and probably no one follows any published strategy. I’ve played about 6 hands of this game IRL, and when I hit my 12 against a dealer deuce upcard, the other player at the table repeatedly pleaded with me, asking (rhetorically) “Why would you hit that?!”.

Judging by the game’s popularity, people don’t have any problems making their switch decisions. The analysis shows a handful of intuitive rules returns almost all the switch advantage, and is suboptimal by only 0.13% (per hand). The table below summarizes the prioritised switching strategy, with the frequencies and costs for each rule for a 6-deck shoe.

Simple Blackjack Switch Strategy
Rule	Frequency	Cost
Switch doesn’t change hands, else	26.94%	0%
Switch improves both hands¹, else	28.60%	0.021%
Play desired² hand(s) over no desired hands, else	20.03%	0.003%
vs. 2-6 dealer upcard
Play double and split, else	0.19%	0.004%
Maximize desired hand, else	7.90%	0.038%
ignore others	0.52%	0.002%
vs. 7-A dealer upcard
Play two strong³ hands against upcard, else	1.05%	0.002%
Play two non-weak⁴ hands over any weak⁵ hand(s), else	0.95%	0.004%
Play strong hand over non-strong hand, else	4.19%	0.005%
Play strong hand and non-bustable⁶ hand, else	1.20%	0.021%
Maximize desired hand if alternative is weak, else	4.72%	0.017%
Play 7/17 if no desired hand, else	0.97%	0.013%
ignore others	2.75%	0.011%
Total	100%	0.13%

¹Or improves one hand without hurting other. See hand rankings defined below.
²Desired hands are defined by Cindy Liu, as (in descending order): BJ, 21, 20, 19, AA, 11, 10, 9, 8/18, and 8-8 vs. 2-6 upcard.
³Strong hand = desired hand with last digit of total greater than dealer upcard. For splits, use split card value.
⁴Non-Weak hand = desired hand with last digit of total greater than or equal to dealer upcard.
⁵Weak hand = desired hand with last digit of total less than dealer upcard.
⁶Non-bustable = stand, or hand that won’t bust on next card (splits, totals <= 11, soft-totals).

The hand ranking used for comparing two candidate hands against a fixed upcard is as follows. Hands in the same level are equal, except for sub-ranks in parenthesis.

Desired hand. (Compare two desired hands by their rank.)
Split hand.
Any soft total.
any hard total <= 7
17 vs. 2-6
12+ hitting hard total (lower is better)
standing hand

Hopefully, the switching strategy is intuitive enough to understand without any detailed description of the rules. I’ve posted a whole bunch of example hands of the switching strategy that should clarify how it works in practice.

Post-Switch Strategy

Here’s the basic strategy for playing your hand after the switch. The strategy is auto-generated by my blackjack analyzer program for a 6-deck shoe game.

Hand	Dealer Upcard
Hand	2	3	4	5	6	7	8	9	10	A
Soft Totals
soft 21	S	S	S	S	S	S	S	S	S	S
soft 20	S	S	S	S	S	S	S	S	S	S
soft 19	S	S	S	S	S	S	S	S	S	S
soft 18	S	S	S	D	D	S	S	H	H	H
soft 17	H	H	H	D	D	H	H	H	H	H
soft 16	H	H	H	H	D	H	H	H	H	H
soft 15	H	H	H	H	H	H	H	H	H	H
soft 14	H	H	H	H	H	H	H	H	H	H
soft 13	H	H	H	H	H	H	H	H	H	H
Hard Totals
hard 20	S	S	S	S	S	S	S	S	S	S
hard 19	S	S	S	S	S	S	S	S	S	S
hard 18	S	S	S	S	S	S	S	S	S	S
hard 17	S	S	S	S	S	S	S	S	S	S
hard 16	S	S	S	S	S	H	H	H	H	H
hard 15	S	S	S	S	S	H	H	H	H	H
hard 14	S	S	S	S	S	H	H	H	H	H
hard 13	H	S	S	S	S	H	H	H	H	H
hard 12	H	H	H	H	S	H	H	H	H	H
hard 11	D	D	D	D	D	D	D	D	H	H
hard 10	D	D	D	D	D	D	D	H	H	H
hard 9	H	H	H	D	D	H	H	H	H	H
hard 8	H	H	H	H	H	H	H	H	H	H
hard 7	H	H	H	H	H	H	H	H	H	H
hard 6	H	H	H	H	H	H	H	H	H	H
hard 5	H	H	H	H	H	H	H	H	H	H
Pairs
A-A	P	P	P	P	P	P	P	P	P	P
10-10	S	S	S	S	S	S	S	S	S	S
9-9	S	S	P	P	P	S	P	P	S	S
8-8	P	P	P	P	P	P	P	P	H	H
7-7	S	P	P	P	P	P	H	H	H	H
6-6	H	H	P	P	P	H	H	H	H	H
5-5	D	D	D	D	D	D	D	H	H	H
4-4	H	H	H	H	H	H	H	H	H	H
3-3	H	H	H	P	P	P	H	H	H	H
2-2	H	H	H	P	P	P	H	H	H	H

2 comments

Knockout Baccarat

Posted in baccarat by stephenhow on February 15, 2013

I was browsing the web tonight, and saw one of the game publishers that I follow placed their new Knockout Baccarat game at a casino in the UK. It’s a variation on baccarat, where you post an Ante on the Player or Banker, then they deal the first player card face up. You then decide to 2x raise your Ante, or fold. The hand is then played out following standard drawing rules. If your hand wins, then your Ante is paid even-money, and your Raise bet is paid odds depending on the losing hand total.

Raise Paytable
Losing Total	Raise Payout
6-8	2:1
4-5	1:1
0-3	1:2

I decided to analyze the game, to see if any part of it was countable. You never know, people make vulnerable games, and you might find some double-digit exploit. Well, to cut to the chase, this new game isn’t very countable. I did find the game was kind of pointless, and that the Player bet is a lot worse than with regular baccarat.

I ran the numbers, and there’s no decision. You always 2x Raise your Ante. You just pick Banker or Player before the hand begins, and that’s it. Folding is never even a close option. The Banker bet has a 4.72% house edge, and the Player bet has a 10.98% house edge. Before you roll over in shock, you should really divide these edges by 3, since you’re always 2x raising your Ante. So, the element-of-risk is 1.57% for Banker, and 3.66% for Player.

Update: Oh, I see. Sometimes you win more than even-money, sometimes you win less. That’s different, I guess. Still, that could be done without the needless extra step of the 2x raise. The raise becomes silly once everyone realises you always make it. (Even against a player 9, you still make the 2x Banker Raise. Folding is a mistake costing at least 31% of the Ante.)

Banker Outcome	Net	Frequency	Probability	Return
win vs. 8	5	55,279,842,324,480	0.011060	0.055298
win vs. 7	5	158,356,022,816,768	0.031681	0.158407
win vs. 6	5	257,091,239,776,256	0.051435	0.257174
win vs. 5	3	209,913,277,599,744	0.041996	0.125988
win vs. 4	3	248,966,126,419,968	0.049809	0.149428
win vs. 3	2	287,320,282,048,512	0.057482	0.114965
win vs. 2	2	307,130,524,108,800	0.061446	0.122892
win vs. 1	2	327,378,644,353,024	0.065497	0.130993
win vs. 0	2	440,816,606,990,336	0.088192	0.176383
lose	-3	2,230,518,282,592,260	0.446247	-1.338740
push	0	475,627,426,473,216	0.095156	0.000000
total		4,998,398,275,503,360	1.000000	-0.047213

Player Outcome	Net	Frequency	Probability	Return
win vs. 8	5	55,165,968,408,576	0.011037	0.055184
win vs. 7	5	155,954,129,788,928	0.031201	0.156004
win vs. 6	5	239,782,224,326,656	0.047972	0.239859
win vs. 5	3	246,959,201,783,808	0.049408	0.148223
win vs. 4	3	267,343,809,949,696	0.053486	0.160458
win vs. 3	2	268,582,958,172,160	0.053734	0.107468
win vs. 2	2	280,469,279,004,672	0.056112	0.112224
win vs. 1	2	301,505,544,974,336	0.060320	0.120641
win vs. 0	2	414,755,166,183,424	0.082978	0.165955
lose	-3	2,292,252,566,437,890	0.458597	-1.375792
push	0	475,627,426,473,216	0.095156	0.000000
total		4,998,398,275,503,360	1.000000	-0.109777

I checked the EORs for these bets, and they’re on the order of blackjack EORs relative to the Ante. So you’re not likely to overcome the 4.72% Banker Ante edge. Thus the main bets are not very countable.

There’s a bunch of side bets you can make, but they’re not very good, nor very countable. The Ideal Countability column lists the average return of a fixed bet per shoe, made using an ideal count (perfect shoe composition info), for an 8-deck shoe with 16 cards behind the cut-card.

Knockout Baccarat Sidebets
Sidebet	Payout	Return	Ideal Countability
Player Natural Winner	4:1	-0.187055	~1.6% bet/shoe
Banker Natural Winner	4:1	-0.187055
Winner w/ 1,2,3,4	7:1	-0.162387	~30% bet/shoe
Winner w/ 5,6	3.5:1	-0.170433	~8% bet/shoe
Winner w/ 7	4.5:1	-0.128165	~16% bet/shoe
Winner w/ 8	3:1	-0.131132	~16% bet/shoe
Winner w/ 9	2.5:1	-0.159784	~7% bet/shoe

1 comment

Dealer Bluff Collusion Strategy (+EV)

Posted in +EV, dealer bluff by stephenhow on January 28, 2013

When you play ShuffleEntertainment’s Dealer Bluff 6-Card Poker, you can feel at a disadvantage. The dealer makes the first bet (1x to 3x the Ante), and of course, you don’t know what he has. He might be bluffing. It’s easy enough to fold your weak hand to a strong bet, or to raise your strong hand against a weak bet. But the in-between decisions aren’t obvious, and you’re left blindly following basic strategy.

Interestingly, full-table collusion (6 players) makes this game +EV against the dealer. After all, the confederates’ 36 cards give some indication of what the dealer holds. (E.g., the dealer can’t have a pair of Aces when the confederates hold three of them.) So, you can get better idea of when to call, raise, or fold your hand. I worked out the collusion strategy details, hoping for a big edge (some games, ahem, yield near double-digit edges with collusion; you never know). Alas, I only came up with a +0.66% +EV 6-way collusion strategy 😦

The full rules and game details are available from the WoO. Briefly, you post the familiar ShuffleEntertainment Ante = Blind bets before the hand starts, and each player and the dealer receives 6 cards. The shuffler reads the dealer hand, and bets 1x to 3x against the players. The player, in turn, must either call (wager a Play bet equal to the dealer bet), raise (wager a Play bet twice the amount of the dealer bet), or fold his Ante and Blind. The dealer will always call any raise. The hands are then turned over, and the bets are resolved. The remaining Antes push if the dealer doesn’t qualify with a pair or better. The Play bets always receives even-money action against the dealer hand. The Blind bets only pay for winning player hands of trips or better, according to a paytable.

The dealer follows a simple table that dictates the 1x, 2x, and 3x betting frequencies for each type of hand (nothing, low pair (2-5), mid pair (6-9), high pair (T-A), two pairs, etc.). This betting table completely describes “how the dealer plays”, and basic strategy is a nearly optimal counter-strategy (based on your hand only).

My collusion strategy tracks the “strong ranks” available to the dealer. Strong ranks are defined as card ranks (2 thru A) that the confederates only hold 0 or 1 copies of. These ranks are “strong”, because of the dealer’s chance of holding a pair of them. For example, the Seven is a strong rank for the dealer if the 6 confederates hold 1 or less Seven’s in total. But if the confederates hold 2 Aces, then the Ace is not a strong rank for the dealer. When you hold a pair, you’re usually interested in the number of strong ranks that are higher than your pair. When you hold 22’s or less, you’re interested in the total number of strong ranks.

Dealer Bluff 6-Way Collusion Strategy
Dealer Bet	Basic Strategy	Collusion Strategy
1x	2x pair 3’s or better	2x two pairs or better
		2x pair 7’s thru A’s when 0-2 higher strong ranks 2x pair 3’s thru 6’s when 0-1 higher strong ranks 2x pair 2’s when 0 strong ranks
		1x pair when 3 or less higher strong ranks
		fold pair when 4+ higher strong ranks
	1x KJ8 or higher	2x AK when 0 strong ranks
		1x A-high when 0-2 strong ranks
		1x K-high when all Aces seen and 0-2 strong ranks 1x K-high when 3 Aces seen and 0-1 strong ranks
	fold others	fold others
2x	4x pair J’s or better 4x pair T’s w/ 0-2 cards under T	4x two pairs or better 4x pair 9’s thru A’s when 0 higher strong ranks
	2x pair 7’s thru T’s 2x pair 6’s w/ 0 cards under 6	2x pair 8’s thru K’s when 1 higher strong ranks 2x pair 5’s thru 7’s when 0 higher strong ranks
	fold other pairs	fold other pairs
3x	6x with pair K’s with A-kicker, or better	6x two pairs or better 6x pair A’s or K’s when 0 higher strong ranks
	3x with pair T’s thru K’s	3x pair 8’s thru Q’s when 0 higher strong ranks
	fold pair of 9’s or less	fold others

For each dealer bet (1x-3x), the strategy is listed in priority from the top down. Yes, the strategy says to fold a pair of K’s against a 3x dealer bet if 1 or less Aces are held among the 6 confederates. There are undoubtedly better collusion strategies out there. As I said, I was hoping for a big edge, especially since you have 6x, 4x, and 2x raise opportunities. But I couldn’t find much more than the above +0.66% strategy, so I kept it simple and published it for reference’s sake.

How would you use this strategy in practise? Well, I guess you’d find a table full of friendly, helpful players. Then you start betting black ($100 Antes), and start asking questions when you need help. Say you’re holding AK, and the dealer bet is 1x. You start asking around if anyone has any deuces, treys, fours, etc. You count the dealer strong ranks (when the players have 1 or less cards of the rank), and play accordingly. When the floorman asks you not to discuss your hands during play, just tell him it’s not going to help much. You should be able to play for an hour before they ask you to leave.

7 comments

Two-Person Panda-8 Co-Count

Posted in +EV, baccarat, dragon-7, panda-8 by stephenhow on January 17, 2013

There are times when you’re at a casino with a friend, and you want to count the EZ-Baccarat Dragon-7. Normally, it’s kind of boring, and you certainly don’t need two people to do it. While it’s a good advantage play, it’d be better and a lot more fun if your friend could help with the Panda-8. I’ve posted a very complicated Panda-8 count that yields about 22% of a fixed bet per shoe. I’ve also posted a super-simple Panda-8 co-count that only yields about 9% of a fixed bet per shoe, but is meant as a single-person add-on to the Dragon-7 count.

In this post, I’ve worked out a better Panda-8 co-count that can be easily tracked by a second person. You add its running count to the Dragon-7 RC to determine when to bet the Panda. The idea exploits the common values between the two counts, resulting in a simple Panda-8 co-count. I worked this out, because I plan to use it.

Here’s the taps for the Panda-8 co-count. You add its running count to the unbalanced Dragon-7 running count, and bet when the total count is +35 or higher. You’ll get about +13.4% of a fixed bet per shoe, on an average of 3.6 bets per shoe.

Panda-8 Co-Count (Add to Dragon-7 Unbalanced RC, Bet When Combined RC >= 35)
Rank	Count
Six, Seven, King, Queen	+1
Trey	-1
Eight	-3

Unbalanced Dragon-7 Count (Bet for RC >= 32)
Rank	Count
Four, Five, Six, Seven	-1
Eight, Nine	+2
Ace	+1

5 comments

Bust It Blackjack Side Bet

Posted in +EV, blackjack sidebets by stephenhow on December 24, 2012

I ran across the Bust It blackjack side bet last weekend at the Palazzo in Las Vegas. It seemed countable, so I ran the numbers today. The bet is simple. You make the side bet before the hand begins, and if the dealer busts on 3 cards, you win according to the paytable. If the dealer doesn’t bust on 3 cards, you lose. The basic house edge for a 6-deck shoe game is -6.91%. The EORs are fairly high, as listed below.

EORs for 6-Deck Bust It Side Bet
Card	EOR	Balanced Count	Unbalanced Count	Simplified Count
Deuce	+0.006589	+2	+2	+2
Trey	+0.005042	+2	+2	+2
Four	+0.002963	+1	+2	+2
Five	+0.000256	0	0	0
Six	-0.006910	-2	-2	-1
Seven	-0.001608	-1	0	0
Eight	-0.003443	-1	-1	-1
Nine	-0.003001	-1	-1	-1
Ten/Face	-0.002231	-1	-1	-1
Ace	+0.009038	+3	+3	+2

If the cut card is placed after the 5th deck, then an ideal count (using perfect shoe composition) yields 14.7% betting opportunities, with an average +6.73% advatange per bet. That’s an average return of about 1.0% per dealt hand.

Practically, you’d use the unbalanced count in the table above and bet with a running count of +25 or more. This practical count yields 14.4% betting opportunities, with an average +6.1% edge per bet. That works out to an average return of +0.88% per dealt hand.

Depending on the side bet limits, counting this bet could be profitable. But, more likely, they’ll limit you to a $25 max bet. So your profit rate would be (100 hands/hr)(14.4% bets/hand)(+6.1% profit/bet)($25/bet) = $22/hr. Of course, you’ll almost certainly have to make the main bet too (e.g., the Cosmopolitan wouldn’t let me make bonus bets on my friend’s blackjack hand). If it’s only $5, and you get good rules @ -0.6%, then your cost would be (100 hands/hr)($5/hand)(-0.6%) = $3/hr, leaving you with a $19/hr job.

The unbalanced count is fairly complicated, with its multi-level taps. Unless your a very skilled counter, you’ll be better off using the simplified count above. It only uses +2 and -1 taps, and it still performs well, yielding 13.5% betting opportunities, with an average +5.3% edge per bet. Bet when the running count is +24 or more.

Also, the standard blackjack counts don’t work for this bet (there’s no correlation, I checked). You can tell that blackjack counts are very different than this specialised count, because Aces are +3 and Sixes are -2. Those are opposite to blackjack values, and they make sense. Ace-rich shoes are bad for 3-card busts. Also, sixes are valuable because of the 15:1 payouts.

Note: a reader says the Palazzo/Venetian deals out of 8-deck shoes. If that’s the case, and they place the cut card @ 6 decks, then the ideal return decreases to 10.7% frequency at an average +4.7% edge. The simplified count return decreases to 8.9% opportunities @ +3.5% edge per bet. You would bet for an RC of +32 or higher.

6-Deck Bust It Blackjack Side Bet
Dealer Outcome	Frequency	Probability	Payout	Return
bust with 888 suited	240	0.001596%	200	0.003191
bust with 888 coloured	1,080	0.007181%	50	0.003590
bust with 6	73,440	0.488299%	15	0.073245
bust with 7	157,536	1.047450%	9	0.094270
bust with 8	245,232	1.630536%	7	0.114138
bust with 9	342,720	2.278729%	5	0.113936
bust with 10	1,782,144	11.849393%	3	0.355482
no 3 card bust	12,437,568	82.696816%	-1	-0.826968

total	15,039,960	100.000000%		-0.069115

2 comments

Push Your Luck Blackjack Side Bet

Posted in blackjack sidebets by stephenhow on December 20, 2012

I came across the Push Your Luck (PYL) blackjack sidebet yesterday (while browsing, not IRL), and I wondered if it was exploitable in any way. PYL is a simple side bet. You make the bet before you start your blackjack hand, and if you end up pushing your main bet, you win 10:1 on the side bet. The max bet is 1/2 your main bet, and its usually limited to $25.

PYL has been out there for a while, but it’s new to me. It’s pretty simple to code up in my analyzer, which finds the optimal play for the combined (main + side) bet. I don’t know why, but I always expect these bets to be +EV, or somehow exploitable. I’m kind of optimistic that way.

Well, I was very surprised to find the house edge of the side bet is very low. Even when you max the side bet (@ 1/2 your main bet), the house edge of the combined (main + side) bet is only 0.76% for a 6 deck game with good rules (DAS, SP4, SPA4, H17). That’s like a cost of 0.25%, and you’re getting 10:1 odds! Here’s the auto-generated strategy table:

Push Your Luck (PYL) Max Bet Basic Strategy
Hand	Dealer Upcard
Hand	2	3	4	5	6	7	8	9	10	A
Soft Totals
soft 21	S	S	S	S	S	S	S	S	S	S
soft 20	S	S	S	S	S	S	S	S	S	S
soft 19	S	S	S	S	S	S	S	S	S	S
soft 18	S	S	S	S	S	S	S	H	S	S
soft 17	S	S	S	S	D	S	H	H	H	H
soft 16	H	H	H	D	D	H	H	H	H	H
soft 15	H	H	H	H	D	H	H	H	H	H
soft 14	H	H	H	H	H	H	H	H	H	H
soft 13	H	H	H	H	H	H	H	H	H	H
Hard Totals
hard 20	S	S	S	S	S	S	S	S	S	S
hard 19	S	S	S	S	S	S	S	S	S	S
hard 18	S	S	S	S	S	S	S	S	S	S
hard 17	S	S	S	S	S	S	S	S	S	S
hard 16	H	H	S	S	S	H	H	H	H	H
hard 15	H	H	H	H	S	H	H	H	H	H
hard 14	H	H	H	H	H	H	H	H	H	H
hard 13	H	H	H	H	H	H	H	H	H	H
hard 12	H	H	H	H	H	H	H	H	H	H
hard 11	D	D	D	D	D	D	H	H	H	H
hard 10	D	D	D	D	D	H	H	H	H	H
hard 9	H	H	H	D	D	H	H	H	H	H
hard 8	H	H	H	H	H	H	H	H	H	H
hard 7	H	H	H	H	H	H	H	H	H	H
hard 6	H	H	H	H	H	H	H	H	H	H
hard 5	H	H	H	H	H	H	H	H	H	H
Pairs
A-A	P	P	P	P	P	P	P	P	P	P
10-10	S	S	S	S	S	S	S	S	S	S
9-9	S	S	S	S	S	S	S	P	S	S
8-8	P	P	P	P	P	P	P	P	P	P
7-7	P	P	P	P	P	P	P	P	P	H
6-6	P	P	P	P	P	P	H	H	H	H
5-5	D	D	D	D	D	D	H	H	H	H
4-4	H	H	H	H	H	H	H	H	H	H
3-3	P	P	P	P	P	P	P	H	H	H
2-2	P	P	P	P	P	P	P	H	H	H

With a max PYL bet, the game gets a little wild where you’re hitting the majority of your under-17 hands, even against low dealer upcards. People must have a fit at this game. But it looks kind of fun, because you’re trying to get a nice 10:1 payout. It’s a good reason to play crazy.

I thought with such a low house edge, and with a 10-to-1 multiplier, the game would be easily countable. However, the EORs are pretty tame, and are very similar to standard blackjack:

Max Bet Push Your Luck (PSL) EORs
Card	EOR
Deuce	+0.076%
Trey	+0.069%
Four	+0.056%
Five	+0.082%
Six	+0.122%
Seven	-0.066%
Eight	-0.057%
Nine	-0.008%
Ten/Face	-0.031%
Ace	-0.142%

You’ll probably need the proper index plays for (not) hitting your under-17 hands on +EV counts. I looked at the strategy for a small +EV count, and the borderline decisions shift towards standard plays. If I get around to learning the lingo for index plays, I’ll post them here for PYL.

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