Discount Gambling

Panda-8 Co-Count with Dragon-7

Posted in +EV, baccarat, panda-8 by stephenhow on December 9, 2012

Screen Shot 2012-12-09 at 7.08.26 PMWith the simplified unbalanced count for the EZ-Baccarat Dragon-7, it’s occasionally fun to count a shoe and find opportunities to bet $25 – $100, to try to win $1000 – $4000. But overall, counting the Dragon gets pretty boring. It only takes a second to see the value of the hand, and to update your count. Then you spend a lot of time watching everyone think deeply about their next bets. Hopefully, the count gets to +32, so you can finally make a bet.

Counting the Dragon-7 would be pretty good if you could make about twice the +EV it offers (+52% of a fixed bet per shoe). Or at least it’d be fun if you could easily track the Panda-8 as well, to add some variety to the game. (I’ve previously posted a complicated Panda-8 count and a RCmin table that yields +22% of a fixed bet per shoe.)

Well, I can’t double the EV of the Dragon-7, nor can I easily get you the full +22% of the Panda. But, here’s an ultra-simplified Panda-8 co-count that you should be able to track alongside the Dragon-7. It’s an unbalanced count, for simplicity. It only has a few taps. The few (4) taps it has are equal to those of the Dragon-7 unbalanced count. Also, these are key taps (you focus on the +2 Nines for the Dragon; it uses the same +1 unbalanced Aces; finally, the -1 Fours and Fives are easy to remember, because they add up to 9). You should be able to track your main Dragon-7 count, then quickly scan the hand for its Panda-8 value.

Simplified Panda-8 Co-Count
Card Count Value
Ace +1
Four, Five -1
Nine +2

Starting from a running count (RC) of 0, you should bet the Panda-8 when its count gets to +35. You’ll get an average of about 2 bets per shoe (when 16 cards are placed behind the cut card), and a profit of around +9.0% of a fixed bet per shoe. It’s not a whole lot, but it’ll make sitting around the baccarat table a little more fun/tolerable. Also, it’ll give you more cred with the degenerates watching their Player lines, Panda lines, and their second bankers 🙂

Thanks to Linus B for his initial work on the Panda co-count. I greatly simplified it here for us script-kiddies.

Unbalanced Dragon 7 Count

Posted in +EV, baccarat, dragon-7 by stephenhow on October 24, 2012

If you’re ever at an EZ-Baccarat table wondering how to properly count the Dragon-7, here’s an easy-to-use unbalanced count that you won’t forget. Unbalanced counts are very handy, because their running counts (RC) approximate true counts, without any division. They’re a nice little trick that everyone should use. I modified the count from my Dragon-7 tracking sheet post into the unbalanced count below.

You simply start the count at -32 for a new shoe, then update the running count for each card dealt, including the exposed burn card. When the running count is > 0, bet the Dragon-7 side bet. This count scheme simulates at a profit rate of +52% of a fixed bet per 8-deck shoe, when 16 cards are placed behind the cut card. You’ll get about 6.8 betting opportunities per shoe.

Unbalanced Dragon-7 Count (Start at -32, bet when RC >= 0)
Card Rank Count Value
Ten/Face 0
Ace +1
Deuce 0
Trey 0
Four -1
Five -1
Six -1
Seven -1
Eight +2
Nine +2

The variance of the bet is very high, and unless you’re heads-up with the dealer, the hand rate is very slow. If you’re wondering if you can grind out a profit from the bet, look at the outcome distribution below for a 500 unit bankroll with a +1000 unit goal, else playing for 500 shoes. While the risk of ruin is only 3.5%, you still have a 24% chance of losing after 500 shoes. Your average win is +250 units. So, if you have a $50k bankroll, can find a heads-up EZ-Baccarat table with a $100 max Dragon-7 bet, are committed to playing for hundreds of hours, and don’t draw any suspicion from casino personnel, then you can win from $50 to $100 per hour, depending on how fast you play. It might be fun for the first hour or two, but only if you hit a dragon. Try playing my Dragon-7 shoe simulator before you head out to the casino.

Easy Six Baccarat

Posted in +EV, baccarat by stephenhow on September 12, 2012

A reader just asked about a no-commission game called Easy Six Baccarat. I’ll keep this post short, for those in-the-know. Use the simple taps (6 => -7, 7 => +3, 8 => +2, 9 => +2) and a true count threshold of 5.0. For an 8-deck shoe with 52 cards behind the cut card, you’ll net +49% of a fixed bet per shoe, on an average of 12 bets/shoe. For an 8-deck shoe with 16 cards behind the cut card, you’ll net +84% of a fixed bet per shoe, on an average of 15 bets/shoe.

For simplicity, you can use the RCmin thresholds in following table:

hand # Min RC Threshold
burn
1 40 40 39 39 38 38 37 37 36 36 35 35
13 34 34 33 33 32 32 31 31 31 30 30 29
25 29 28 28 27 27 26 26 25 25 24 24 23
37 23 22 22 21 21 21 20 20 19 19 18 18
49 17 17 16 16 15 15 14 14 13 13 12 12
61 11 11 11 10 10 9 9 8 8 7 7 6
73 6 5 5 4 4 3 3 2 2 2 1 1

Lunar Poker @ Pechanga Casino

Posted in +EV, collusion, lunar poker by stephenhow on August 24, 2012

Well, someone is finally bringing the infamous Lunar Poker (aka Russian Poker) to the US, starting at my nearby Pechanga Casino. The game is a very interesting version of the old Caribbean Stud Poker, with a lot more options like drawing cards, buying an extra card, buying insurance, and forcing the dealer to draw (all for a price).

The game has been infamous, because the many player options result in an incalculable number of possible hand combinations (6.27x 10^20 according to the WoOs), and because of the absence of a published strategy. It sounds like people have played this game by the seat of their pants for years in Europe and Asia. But a lot of us won’t play a game without first knowing the basic strategy and house edge. So I grinded out the analysis, just in case you run across this game.

Rules

The rules follow the basic structure of Caribbean Stud Poker. You place an Ante before the hand starts, and the players and dealer each receive five cards. The dealer exposes one of his cards. You eventually decide to either Raise 2x, or fold your Ante. The dealer turns up his hand, and needs Ace-King or better to qualify. If the dealer doesn’t qualify, then the remaining Antes are paid even-money, and the Raise bets push. If the dealer qualifies, then the Antes push, and the Raise bets are paid according to a paytable.

So far, these rules are just like Caribbean Stud, except here, the Ante only pays when the dealer doesn’t qualify.

Now, Lunar Poker offers the following player options before the Player makes his 2x Raise decision:

  • The player may either receive an extra (6th) card, or may replace 2-5 of his cards, for the cost of 1x the Ante.
  • With three-of-a-kind or better, the player may take even-money insurance against the Dealer not qualifying (up to 1/2 the amount of the winning payout).

The players make their 2x Raise or Fold decision, then the dealer turns up his hand. If the dealer doesn’t qualify, the Antes and Insurance pay even money. If the dealer qualifies, then the player must beat the dealer to win his Raise bet and push his Ante. Else, the player loses his Ante and Raise. Insurance loses if the dealer qualifies and the player wins. If the dealer qualifies and the player loses, Insurance pushes. (Note: Pechanga lets you can take Insurance on up to the full amount your potential win.)

Finally, if the dealer doesn’t qualify, the player has an option to:

  • Pay 1x Ante to force the dealer to replace his highest card with a draw from the deck.

If the dealer qualifies after the draw, then the player’s Ante and Raise resolve as before. If the dealer doesn’t qualify, then the Ante and Raise push. Note: if you decide to Force the dealer to draw, then you forfeit the pay on the Ante you would normally receive. (It is expensive to Force the dealer; you forfeit your win on the Ante, AND you have to pay 1x!)

Paytable

For winning hands against a qualified dealer hand, the Raise bet pays according to the following paytable. More importantly, you are paid on a second hand from the paytable, when the second hand uses at least one different card from your first payout hand. (Note: “hands” do not include kickers; e.g., a three-of-a-kind hand contains only 3 cards for purposes of the paytable.) I’m not going to provide examples of the second payout, as this is described elsewhere.

Hand Payout
Royal Flush 100-to-1
Straight Flush 50-to-1
Four-of-a-Kind 20-to-1
Full House 7-to-1
Flush 5-to-1
Straight 4-to-1
Three-of-a-Kind 3-to-1
Two Pairs 2-to-1
One Pair 1-to-1
AK 1-to-1

Basic Strategy

I worked out a simple strategy for the game that simulates at a 1.43% house edge. That’s not bad as far as carnival games go, but it looks like their claim of “House Advantage Under 1%!” is false.

Draw Decision

The first decision on what to hold and draw is presented in the table below.

Draw Decision for Lunar Poker
5-Card Hand Decision
Royal Flush
Straight Flush
Flush
Straight
Always buy 6th card.
Four-of-a-Kind Stand.
Full House Buy 6th card unless dealer upcard copies you.
Three-of-a-Kind Stand if 4-of-a-kind not possible,
else hold trips and exchange 2 cards.
Two Pairs Stand.
One Pair w/ AK Discard 2’s or 3’s (hold AK and exchange 3) against higher upcard, Queen or lower,
else stand.
One Pair Buy 6th card for open-ended, flush draw, or gutshot.
Hold pair and exchange 3 if pair below upcard,
else stand.
AK Buy 6th card for open-ended, flush draw, else
Buy 6th card for perfect gutshot to 6-card straight, else
Buy 6th card for gutshot straight draw against A or K upcard, else
Hold AKs and royal cards higher than dealer upcard, else
Hold AK and exchange 3
Nothing Buy 6th card for open-ended or flush draw, else
Buy 6th card for perfect gutshot to 6-card straight, else
Hold AKs and any Royal cards, else
Hold two or more Royal cards higher than the dealer upcard, else
Hold three straight flush cards higher than the dealer upcard, else
Hold A against K upcard or lower, else
Hold K against J upcard or lower, else
Hold Q against copied J upcard or lower, else
Hold Q against 5 upcard or lower,
Else fold.

where open-ended straight draws include double-gutshot straight draws.

Insurance

It’s only correct to take insurance in a few cases. Never insure your hand against an Ace or King upcard. Otherwise, take insurance when you copy the dealer upcard 2 or more times. If you only copy the dealer upcard once, then take insurance when you also hold 2 or more Aces or Kings in your hand.

2x Raise / Fold

You should 2x Raise any pair or better. Fold any non-qualifying hand. Otherwise, play AK according to the table below.

2x Raise Decision
Hand Decision
Pair or better Raise 2x.
AK Call with any copies of the dealer upcard, Q or lower, else
Call with AKJ83 or better with any copies of the dealer upcard (including A, K), else
Fold all others.
non-qualifying Fold.

Force Dealer Bet

Your potential Raise payout and the possible dealer outs determine when you should try to force the dealer to draw. The table below tells you when to pay 1x to replace the highest dealer card with one from the deck. Remember, you’re forfeiting your instant Ante win by Forcing the dealer to draw. Plus, you’re paying 1x for the Force, so you need at least a 4:1 payout to make it profitable (i.e., don’t Force trips-only hands).

Force Draw Strategy
Potential Payout Conditions
3-to-1
or lower
Never force.
4-to-1 Don’t force dealer flush or open-ended draws that beat you unless all dealer pair outs are available, else
Don’t force if you hold 2 or more of the dealer’s pair outs,
else force.
5-to-1 Force unless you hold 4 or more of the dealer’s pair outs.
6-to-1
or higher
Always force.

Simple Two Player Collusion

If you’re friendly with your table-neighbor, you can slightly modify basic strategy to get a +EV return of +0.43% on the Ante. The drawing decision is modified accordingly:

Buy/Exchange Decision for Two Player Collusion
5-Card Hand Decision
Three-of-a-Kind Stand pat if your neighbor holds your quad out, else
hold trips and exchange 2 cards.
One Pair
w/o AK
Buy 6th card for open-ended or flush draw, else
Buy 6th card with over-pair (above dealer upcard) and gutshot if all straight outs remain, else
Buy 6th card with under-pair (below dealer upcard) and gutshot if any straight outs remain, else
Stand pat against dead upcard (3 copies) Q or lower, else
Hold under-pair (below dealer upcard) and draw 3 if all outs remain, else
Stand pat for all others.
AK Buy 6th card for open-ended or flush draw, else
Buy 6th card with 2+ outs to perfect gutshot (6-card straight), else
Buy 6th card with 3+ outs to gutshot against A/K upcard, else
Stand pat against dead upcard (3 copies), Q or lower, else
Hold two or more royal cards, exchange rest, else
Buy 6th card with at least 2 gutshot draws to AKQJT, else
Hold AK and exchange 3 cards.
Nothing Buy 6th card for open-ended or flush draw, else
Buy 6th card with 2+ outs to perfect gutshot (6-card straight), else
Stand pat against dead upcard (3 copies), Q or lower, else
Hold two or more royal cards, exchange rest, else
Hold your highest card, 9 or better, higher than the upcard and not copied by your neighbor, else
Hold 3 straight flush cards higher than the upcard, else
Fold all others.

Only take insurance when you and your neighbor hold 3 total copies of the upcard, Queen or lower. Never insure against an Ace or King upcard.

Finally, modify the 2x Raise decision:

  • Call any 2:1 pay or better, else
  • Fold pair deuces against uncopied upcard 3 thru Q, else
  • Call any other pair, else
  • Call any hand when you and your neighbor hold all 3 copies of the dealer upcard Queen or lower, else
  • Call AKJ83 or better when you and your neighbor hold any copies of the upcard, else
  • Call AK when you and your neighbor hold 2 copies of the dealer upcard Queen or lower, else
  • Fold all others.

Improved Six-Card Poker Collusion Strategy (+EV)

Posted in +EV, six card poker by stephenhow on August 11, 2012

I’ve been playing a lot of Six Card Poker at my local Viejas Casino, which gave me the chance to think about a better collusion strategy. When I first posted about this game, I was disappointed that the theoretical limit for collusion would yield only around +1.2% on the Ante bet. So I didn’t try too hard to make a good collusion strategy.

But it’s a pretty fun game, since the dealer shows half his hand, and with a full table, you’ll know 39 of the 52 cards. You can get the rules of the game from the WoOs.

After thinking it through, I boiled down the 6-way collusion strategy to the following three rules:

  • fold any hand already beat by dealer
  • fold any qualifying hand when there are 4 or more remaining single-card outs that beat you
  • fold any non-qualifying hand when a kicker out remains that beats you, or there are 3 or more remaining pair outs for the dealer

This collusion strategy simulates at +0.43%, which isn’t bad. It’s pretty easy to count remaining dealer outs among the confederates. People just have to chime-in on how many copies of the dealer cards they have. The strategy is extremely simple, and the variance is pretty low given the 1x call, and the help in folding -EV hands.

Counting CSM Blackjack (+EV)

Posted in +EV, blackjack, csm by stephenhow on July 27, 2012

A few people have asked me about the countability of blackjack dealt from a constant shuffle machine (CSM). I’m a big advocate of counting the CSM, especially for card craps, because of the ease of the windowed count. Even if the dealer collects no muck (i.e., immediately shuffles completed hands back into the CSM), you’ll still be +EV more than 8% of the time for good BJ rules. It’s a lot easier to count a CSM than a shoe. I call it counting for the ADHD crowd. All you have to do is pay attention to the last 16 cards (or the collected muck) fed into the CSM before the hand is dealt. Plus, you’ll probably never get backed off from CSM blackjack, even when wildly varying your bets.

EV vs. Windowed Count

I use my model of the ShuffleMaster 126 (source) CSM in the blackjack simulations for this post. I’ve talked in detail about this model before, in my posts on card craps. Basically, there’s a buffer of at least 16 cards in the chute (so the dealer never waits for a card), which introduces state into the system. If the dealer collects no muck, you simply use the running count of the last 16 cards fed into the shuffler. Use the simple hi-lo count (2-6 are +1, 10-A are -1). If the dealer collects a lot of muck, and feeds it all into the shuffler right before the next hand starts, then use the count of the entire muck.

For these simulations, I used 6 decks in the CSM, and typical-good H17 blackjack rules (3:2 BJ, late surrender, re-split Aces 3-times, double-after-split). My blackjack analyzer calculates the ideal EV for these rules at -0.445% for 6 decks. I ran the simulator head’s up against the dealer, and kept track of the 16-card windowed count and the subsequent hand outcome. I plotted the next-hand EV vs. the windowed hi-lo count in the graph below.

The graph shows a very linear relationship between the 16-card windowed hi-lo count and the EV of the next hand. When the running windowed count is +5 or more, the next hand from the CSM is +EV. The windowed count is ≥ 5 about 8.2% of the time.

Frequencies of 16-Card Windowed Counts
Count Frequency Approx. BJ EV
0 11.5% -0.44%
1 10.8% -0.34%
2 9.36% -0.24%
3 7.46% -0.16%
4 5.46% -0.05%
5 3.67% +0.04%
6 2.25% +0.14%
7 1.25% +0.22%
8 0.627% +0.29%
9 0.280% +0.43%
10 0.111% +0.49%

Conclusions

Serious card counters will tell you you can’t count a CSM. But the data above shows that a CSM goes +EV more than 8% of the time. Plus, it’s infinitely easier to count a CSM than it is to count a shoe. You can lose track of the count for a hand or two. As soon as you regain attention, you’ll know what the count is. You can probably vary your bets wildly without attracting any attention or interest from the floor. You can probably even Wong hands when the count is bad. Or less than +5.

Counting a CSM is great for the casual counter. It’s basically short-attention span counting. If you see the last 16 cards into the CSM prior to the deal are low (have a running count of 5 or better), then you’re +EV for the next hand. Even if you just see a net +3 count for the last 16 cards, you still know the next hand will be better than average. You can start/stop paying attention on a per-hand basis (unlike a shoe, where you have to wait for the next shoe if you lose the count).

At it’s simplest, CSM counting will tell you when the next hand will be better-than-average (half the time), or worse-than-average (half the time). So, if you Wong half of the time, you’ll only play the better-than-average hands (EV better than -0.45%; the above curve to the right of count=0), and miss the bad hands. That’s a quick way to reduce the house edge from 0.45% to 0.22% (only play 53% of the hands; wait until the count is ≥ 0).

Field of Gold (+EV)

Posted in +EV, blackjack sidebets by stephenhow on June 12, 2012

Everybody thank Eliot Jacobson for working for you. He emailed me this morning telling me the Field of Gold blackjack side bet was crushable. And it is. For real. The count is simple, and for a 6-deck shoe game, you’ll be able to bet 19% of the time, with an average edge of +6.5%/bet. That’s a profit of ($25 bet)(19% bet/hand)(+6.5% EV/bet) = $0.31/hand. For double-deck, the numbers are even better, yielding 27% betting opportunities with an average edge of +8.2%/bet. For $25 bets, that yields a profit of $0.55 per hand dealt. That adds up pretty quickly. It’s more than $100/hr.

This is about as good as it gets, as it’s a fast blackjack game. Oh, by the way, Eliot also brought you the equally crushable Lucky Lucky bet, which I probably understated in my post. These are probably the best two countable side bets out there, and nobody but Eliot has noticed them. Oh, and he’s the one that pointed out the Dragon-7 vulnerability.

Anyway, here’s what you’re looking for:

Use the following count:

Field of Gold Count
Card Value
Ace -3
Deuce -1
Nine, Ten, Queen, King +1
others 0

Bet when the true count (= runningCount/decksRemaining) is 2.2 or better. You can see from the graph below that the EV is strongly correlated with the true count.

This side bet becomes highly profitable when the shoe is rich in aces and deuces, and lean in high cards. During some shoes, the deck can easily get really good, or really bad for the Field of Gold bet. The distribution of the side bet EV is plotted below over the course of the 6-deck shoe and the double-deck game. Note the peak of the distribution is centered about the nominal -5.66% return of the bet right after the shuffle. Then, especially towards the end of the shoe, the side bet return can vary wildly. Note all the area under the long tail of the distributions to the right of the y-axis (EV>0). This is why the game is crushable.

(5 deck penetration of 6-deck shoe; 29 cards behind cut card in double-deck.)

Card Craps Simple Explanation

Posted in +EV, card craps, csm by stephenhow on June 10, 2012

I love the card craps at Viejas, not because I’m ever going to win any money there, but because it’s so obviously countable. However, it’s almost impossible to explain to anyone why the odds are different than dice, or why the game is countable. After all, they use a Constant Shuffle Machine (CSM) with 312 cards, right? So, once again, I’m going to explain how the card buffering in the exit chute of the CSM makes the game easily countable.

A picture is worth a thousand words. Example code and simulations are the proof of the pudding. All the code used in this example is available on github, where you can browse or download it.

You can read up on the details of card craps @ Viejas. Here’s how they play it. They use a normal craps layout, but replace the dice with two cards (1 thru 6), dealt out of a 312-card CSM. They take two cards out of the shuffler, call the roll, then muck the two cards back into the CSM. They allow 10x pass/dont odds on all points.

The reason why the CSM screws up the game (favors the dont’s) is that on the comeout, the two cards that just made the point have no chance of coming out on the next roll. Nor do they have any realistic chance of coming out in the next few rolls. This is because a CSM buffers a dozen or more cards in the chute where the dealer pulls the cards from. This buffer is necessary to deal blackjack. (Imagine the dealer waiting for the machine to drop one shuffled card at a time.)

Dice Baseline

Ok, so download the example code, compile and run it with the -d option for normal dice. The results are just as you’d expect. The pass line returns -1.42%, and the dont pass returns -1.36%, and odds and counting don’t make any difference:

>./cardcraps -d
using normal dice
...
1665000000 games:
pass flat: -0.0142, pass10x: -0.0144, pass w/count: -0.0142,
dont flat: -0.0136, dont10x: -0.0134, dont w/count: -0.0135

It takes billions of games to settle out the averages (especially when playing 10x odds), so don’t worry about the 1/100th of percents.

A) 36-Card Deck Is Same As Dice

At Pala Casino, they use a 36-card deck (one card per roll), and a simple deck shuffler. No buffer. Each card has a picture of two dice. The shuffler spits out one card from the red deck, one card from the blue deck. The player “roll” chooses between the blue or red card. Exact same odds as craps. At Pala, no one ever says anything like “How many cards are in there?”, or “This machine deals a lot of sevens!”.

B) 2-Card Roll Hurts Pass Odds

Now, let’s try the case B in the above diagram. We use the -c option to select an ideal shuffler, and -m 0 option to indicate no buffered cards in the chute.

>./cardcraps -m 0 -c
using CSM with 52 dice sets, and minBufferDepth of 0 cards
...
1265000000 games:
pass flat: -0.0137, pass10x: -0.0266, pass w/count: -0.0170,
dont flat: -0.0137, dont10x: -0.0053, dont w/count: -0.0075

This shows that even without a buffer, making a dice roll from two cards out of a perfectly shuffled 312-card shoe favors the don’t pass odds. You can use a simple spreadsheet to show this. The point is that you’ll distort the well-known dice roll distribution by using 2 cards dealt from a shoe. It’s a simple exercise to prove (a simple spreadsheet will give you the exact numbers).

Note the pass line player loses more by taking odds. The don’t pass player improves his return by laying 10x odds. That doesn’t happen in a regular dice game. In a dice game, taking or laying odds is fair (0 EV).

C) CSM Is Countable

At Viejas, they use a ShuffleMaster 126 CSM loaded with 312 cards. If you ever open the top (used to happen a lot when they had jams), you’ll see a buffer of approximately 16 cards in the exit chute. This distorts the game, and in general favors the Don’t Pass odds. Sometimes, a good count makes the pass odds +EV.

We’ll run the simulator for the CSM with a minimum buffer depth of 16 cards:

>./cardcraps -m 16 -s
using model of ShuffleMaster 126 CSM with 52 dice sets, and minBufferDepth of 16 cards
using window size of 6 rolls
...
2083000000 games:
pass flat: -0.0147, pass10x: -0.0420, pass w/count: -0.0011,
dont flat: -0.0126, dont10x: +0.0042, dont w/count: +0.0130

Now you see the pass line player is severely penalised for taking odds. I don’t think someone taking 10x odds on every point would think they’ve increased the house edge from a nominal 1.4% to a whopping 4.2% (of the flat bet). And we see that a don’t pass player laying 10x odds on every point now has a small 0.4% advantage over the house. Of course, there’s a lot of variance laying 10x odds to win an average (0.4%)(flat bet). Using a simple (and fun!) count, the don’t player has a 1.3% advantage over the house.

You can use the -v option in the cardcraps program to generate the statistics on the odds bet vs the count for each point. I ran the program, and plotted the results (don’t pass odds advantage; pass odds are inverted):

The correlation between the count and the next roll out of the CSM is clear. The count is simple and important! Quite often, you have a +/- 1-2% advantage in laying odds or taking odds. Where else can you play a craps game where the previous 6 rolls have a significant effect on the next roll?! The graph was generated with a fair simulator (using a Mersenne Twister 64-bit PRNG with a period of 2^19937-1).

Even though the game is +EV, the edge is small relative to the variance. No one will grind out any money from this game. However, it is a lot of fun to watch the rolls, know the count, and guess the outcome. Plus, the game is dealt on a table, so you get to sit and watch the rolls. And it’s probably 10x faster than a craps game with dice. You could get a roll every 5 seconds if you’re heads up with the dealer.

The count provides a fun, small predictor of the next roll out of the CSM. If you like counting, and/or predicting the next roll in craps, then you have to check out the card craps game. Here’s a video that shows how I play the game @ Viejas:

Six Card Poker @ Venetian, Las Vegas

Posted in +EV, collusion, six card poker by stephenhow on May 11, 2012

On my trip to Vegas last month, I saw this new game at the Venetian, and all I could think of was collusion. I figured it had to be beatable, since the dealer shows half his hand (3 upcards), which should exploitable given confederate card information. Well, I finally got around to looking at it, and of course, its not as exploitable as I hoped.

The game is pretty simple, where both dealer and player get 6 cards to make a 5-card poker hand. There’s only an Ante, and a 1x Play bet. The dealer shows 3 upcards, and you decide to either 1x Play or fold your hand. If the dealer doesn’t qualify with Ace-King, then the Ante pushes regardless of the player hand. The 1x Play bet always receives even-money action against the dealer hand. The Wizard of Odds provides a basic strategy, and lists the house edge at 1.27%.

I figured 6-player collusion would help you know when to play Ace-high, and maybe help you fold a pair when a lot of dealer outs remain that beat you. But first, I simulated a bunch of hands finding the optimal decision given confederate card info. This gave me a very close approximation to the ideal edge obtained by perfect collusion. This 6-player edge amounted to only +1.17%. This isn’t much, especially since any actual collusion strategy approaching this limit would be impractically complex.

At this point, I only made a half-hearted attempt at finding a practical collusion strategy. There’s so many cards involved, its difficult to come up with a workable signalling system. Also, I looked over the collusion decision points, and it wasn’t simple to identify the conditions for making a counter decision to basic strategy. For what it’s worth, I came up with the following “simple” 6-player collusion strategy that simulates at +0.15%:

  • Call two pairs or better, else
  • Call one pair unless there are 7 or more dealer one-card outs remaining that beat you, else
  • Call Ace-high when 2 or more Aces and Kings seen with 9 upcard copies, else
  • Call Ace-high with 4 or more Aces and Kings seen with 8 upcard copies, else
  • Call Ace-high with 6 or more Aces and Kings seen with 7 upcard copies,
  • else fold

Update: I worked out an improved 6-way collusion strategy that yields a +0.43% return with only a couple simple rules.

Lucky Lucky Blackjack Sidebet (+EV)

Posted in +EV, blackjack sidebets by stephenhow on May 3, 2012

Well, here’s another massively countable side bet that some people might be interested in (advantage players, casino floor supervisors, and the game publisher), but that I’ll never play. I think after this one, designers will know to check their games for vulnerabilities, especially when there’s oversized items in the paytable. And we’ll remember, “It’s not a sucker bet if the count is good.”

Again, Eliot Jacobson pointed this one out to me. (But, if Barona had this side bet, I’d have already looked at it.)

The Lucky Lucky blackjack side bet is played with your first two dealt cards, and the dealer upcard. On these three cards, you get paid for various ways to make 21, and for any 20 and 19 total. The most countable version of this side bet is for the double-deck version with the paytable below. The game is also countable for the 6 deck shoe game, but it’s only 60% as profitable.

Lucky Lucky Side Bet(Double Deck)
Hand Frequency Probability Payout Return
suited 678 32 1.757238E-4 100 0.01757238
777 56 3.075166E-4 50 0.01537583
other 678 480 0.00263586 30 0.07907569
suited 21 936 0.00513992 15 0.07709880
other 21 14904 0.08184334 3 0.24553003
any 20 13792 0.07573694 2 0.15147388
any 19 13344 0.07327680 2 0.14655362
others 138560 0.76088389 -1 0.76088389
total 182,104 1.00000000 -0.02820366

As usual, I program a function that tells me the EV for any given shoe composition. Then I simulate millions of hands, calculating the ideal EV of the side bet at the beginning of each hand. I sum up the times when the side bet is +EV, and find the average +EV bet and +EV frequency. For the double-deck Lucky Lucky, I got

double-deck, cut card @ 75th card
ideal +EV frequency: 0.2769, ideal EV/bet: +0.0591

which is not a practical counting scheme, but the theoretical limit if you used a computer that took into account all info (suits, etc.).

Then I calculated the Effect-of-Removal (EORs) of a given card on the EV, in order to make counting tags. (Outside the gambling world, people would call “EORs” sensitivities, and “tags” coefficients.)

EORs and Tags for Double-Deck Lucky Lucky
Card EOR Tag
Deuce +0.007853 +1
Trey +0.006066 +1
Four +0.004099 +1
Five +0.003171 0
Six -0.010422 -2
Seven -0.017270 -2
Eight -0.012616 -2
Nine +0.002515 0
Ten/Face +0.006270 +1
Ace -0.008477 -1

So, setting the trueCount threshold to 2.4 (bet Lucky Lucky when the trueCount is >= 2.4), you get the practical results in double deck:

practical frequency: 0.2640, average EV/bet: +0.0561

6 Deck Shoe Version

The 6 deck shoe paytable is better than the double deck version, as it pays 200:1 for a suited 777. The EORs are similar, and I came up with the same count tags as the double deck game. Using a trueCount threshold of 2.1, the practical counting scheme yields:

ideal +EV frequency: 0.2311, ideal average EV/bet: +0.0432
practical frequency: 0.2217, practical average EV/bet: +0.0409

which is only 61% of the profit rate as the double-deck game.