I thought I’d post a thorough description of how to play the PlayCraps game at Viejas, including how to account for the shuffle, using a fair-weighted counts for all the points.
First, here’s the value of each roll, and how it contributes to the counts for each point.
|4||+4 if both cards ≤ 3, -4 if both cards ≥ 4, else 0|
|5||+2 if no Fives or Sixes, -1 if one Five or Six, -4 if two Fives/Sixes|
|6||+1 if no Sixes, -2 if one Six, -4 if boxcars|
|8||+1 if no Aces, -2 if one Ace, -4 if snake-eyes|
|9||+2 if no Ace or Deuce, -1 if one Ace or Deuce, -4 if two Aces/Deuces|
|10||+4 if both cards ≥ 4, -4 if both cards ≤ 3, else 0|
Ideally, you’ll keep a count for each point. Of course, this is hard to do. I just wait for the point to come out, then I try to guess if I saw any key cards lately, and make up an initial count for the point. Then I adjust the count for each roll as in the above table. It’s very easy while the muck accumulates. Then, when the dealer shuffles the muck into the CSM, I keep counting as normal, for about 5 rolls (approx. buffer depth). After these five rolls, I start the count again, based on a guess of what’s in the muck.
simulated session with annotations after the jump! (more…)
I wondered what advantage you could get by sharing hole card information against the bank hand in the house game World Poker Tour All-In 3x. For 6 players, you can net a +1.25% edge against the house. For 3 players, you get a small player edge (+0.05%) on an otherwise 0.74% house game. You can read about the full 3-player collusion strategy in my page for the game.
Using the improved don’t pass counting system, using a trailing six roll window, as described in the previous post:
|Point||Conditions to lay (max) don’t pass odds|
|4||running count over the last 6 rolls <= -2|
|5||seen at most 2 fives or sixes in the last 6 rolls|
|6||seen at most 1 six in the last 6 rolls|
|8||seen at most 1 ace in the last 6 rolls|
|9||seen at most 2 aces or deuces in the last 6 rolls|
|10||running count over the last 6 rolls >= 2|
where the running count is incremented when both die are high (>= 4), and decremented when both die are low (<= 3).
Applying 10x don't pass odds, I simulated the game using my model for the CSM (continuous shuffling machine), and I got the following results:
Macintosh:Debug show$ ./playcraps -a -m14 -n100000000 -r max muck depth: 14, CSM buffer depth: 10, rolls: 1.0e+08 net: +584729, EV: +0.58% per roll Macintosh:Debug show$ ./playcraps -a -m20 -n100000000 -r max muck depth: 20, CSM buffer depth: 10, rolls: 1.0e+08 net: +621717, EV: +0.62% per roll Macintosh:Debug show$ ./playcraps -a -m20 -b6 -n100000000 -r max muck depth: 20, CSM buffer depth: 6, rolls: 1.0e+08 net: +646693, EV: +0.65% per roll, +2.18% per come out
meaning that the dealer shuffles the muck back into the CSM when it’s more than 14 cards deep. The CSM is modeled with a buffer depth of 10, meaning that the earliest a card can come back out of the shoe is 10 rolls after any shuffle.
The results show that you’ll win +0.58% of your don’t pass bet, on average, per roll. So, for a $5 don’t pass bet, you’ll make $.029/roll, when laying 10x don’t pass odds according to the above table. Note the results improve a little if the dealer allows the muck to collect a little longer (+0.62%/roll for a 20 max card muck).
Note how the count scheme is insensitive to the buffer depth modeled in the CSM. When I decreased it to 6 rolls (12 cards), the return actually improved a little. In the last simulation, I also calculated the return per come out, which came out to +2.18% of the don’t pass bet. Again, that’s only about $0.10 per $5 don’t pass bet.
It’s not a lot of money. Even at a fast 500 roll/hr, you’re only making $14.50/hr. The bankroll requirements for this strategy are large, because you’re laying $100 to win $50 against the 4/10. It’s probably not an option to try and grind this game out. However, if you like playing don’t pass craps, then at least you’re getting the psychological benefit of a $14.50/hr tailwind :)
Note that just playing blind 10x don’t pass odds gives you the same ~2% EV. Employing a count scheme is just reducing your 10x odds variance a little from ~35 to ~32 (its still huge). I enjoy varying my odds with every roll. It only takes a small amount of effort, and it makes me feel like I’m in control.
I wanted to quantify the edge of a count system for the “dice” dealt out of the CSM for the PlayCraps game I’ve been talking about. I tried a few simple ideas, based on how I actually play the game at the table. A good method needs to be practical and not mentally taxing. After all, we’re playing craps, and we want to have fun.
I know when a large run of high rolls (both die are >= 4) occurs, the distribution for the next “roll” is skewed towards the 4/5/6, and away from the 8/9/10 points. If the point is on 10, I jack up my don’t pass odds. (See below how I play the 5/9 and 6/8 points.)
In the graphs below, don’t worry about the negative parts of the curves. These are times you’re not laying don’t pass odds. Your flat bet is still a 6:5 favorite on the 6/8, a 3:2 favorite on the 5/9, and a 2:1 favorite on the 4/10, minus the delta shown in the graph. Plus, these are good times to take pass odds on the point (e.g., your friend is pass line, and he jacks up the odds when you take them down).
I formalized the strategy by keeping a running count of the last 6 rolls. A roll is high if both die are ⚃, ⚄, or ⚅. A roll is low if both die are ⚀, ⚁, or ⚂. All other rolls are neutral. Then I just keep the hi/lo total for the last 6 rolls. It’s pretty much what we naturally do in our heads anyway. Since every low or high roll significantly distorts the distribution, you get pretty excited to see one. (My eyes perk up every time I see a ⚀ ⚀, or ⚀ ⚁, or ⚁ ⚁, etc. I start looking to lay the no-4. Conversely, if I see ⚅ ⚄ then ⚃ ⚄ then ⚄ ⚄ I get pretty excited about lay no-10, or jacking up my don’t-10 odds.)
This above plot shows the results of simulating this count system, and tracking the distribution of the next “roll” out of the CSM. It clearly shows that the player gains a huge advantage by increasing his don’t pass odds when the count is good, and taking down the don’t pass odds when the count is bad. (The converse applies to the pass line bet. Just flip the graph to get the advantage of taking odds on the point for a given count. When laying don’t pass odds are bad, taking pass odds are good, and visa-versa.) Sometimes, I adjust my don’t pass odds bet on every roll.
This makes for a very good craps game, since this dirt-simple count system buys you from 0.5% to 1.5% on your 4/10 odds bet. The 5/9 aren’t too bad either. See below for the count system for the 6/8 points. The 6-roll window does not need to be exact, by any means. A 5,6,7, or 8 roll wide window produces similar results. They say people can remember about 7 numbers (e.g., telephone numbers). So you’ll end implementing this naturally anyways. Note that the simulation assumed the dealer let about 7 rolls accumulate in the muck before shuffling it into the CSM.
This strategy lets you lay/take odds only when you get an advantage for doing so. Normally, people lay/take odds on the point, and wait until the roll ends (hit the point, or 7-out). But with this counting method, you watch the “rolls” out of the shoe, and change your pass / don’t pass odds accordingly. You get to predict the future, and you actually have a little insight into it.
Update: Here’s how to play the 6/8 points.
The key card for the 6 point is the ⚅. This card can make a 7, but cannot make the point. Just keep track of how many of them you’ve seen in the last 6 or so rolls. Use this count as an index into the below graph. You’ll see that as long as the count is below 3, you still have an edge laying odds against the 6. For counts >= 3, you’d be laying odds at a disadvantage. Take them down, and wait for the count to go back below 3.
A similar approach is used for the 8 point. Here, the key card is the ⚀. This card can make a 7, but cannot make the point. Again, keep track of how many you’ve seen in the last 6 rolls or so. The more that are out, the worse off your don’t-8 odds bet is. See how the return changes from about +0.3% to -1.2% as the count increases from 0 to 7. Again, back off your don’t-8 odds bet when you see too many ⚀ come out.
2nd Update: Here’s better way to play the 5/9 points.
The key cards for the 5 point are the ⚄ and ⚅. These cards can make a 7, but not the point. So count the number of these cards you see in the last 6 rolls. When the count gets to 5, take your don’t pass odds down. Wait for the neutral (other) cards to flush down the count to 4 or below, then lay your don’t-5 odds again.
The situation is similar for the 9 point, where the key cards are the ⚀ and ⚁.
My nearby Barona Casino has been spreading the ShuffleMaster game Mississippi Stud face up for some time now. I finally looked at the game, as it’s played there. Here’s a strategy yielding a 1.5% player edge for a full game (6 seated players). Do not use this strategy unless playing at a full table. Otherwise, play basic strategy (4.91% house edge).
“high outs” means the number of cards in the deck that will give you a high pair (Js thru As)
“mid outs” are the number of cards in the deck that will give you a mid pair (6s thru Ts)
“low outs” are the number of cards in the deck that will give you a low pair (2s thru 5s)
- 3x raise a high or mid pair
- 3x raise with 6 high outs, or with 5 high outs and suited
- 3x raise a small pair with both trips outs still in the deck
- 1x call with 3 or more high outs
- 1x call with 2 high outs if suited, OR at least 2 mid outs, OR at all 3 low outs
- 1x call with 1 high out if suited AND at least 2 mid outs
- 1x call with at least 5 mid outs, OR suited and 4 mid outs
- 1x call with all 3 mid outs and all 3 low outs
- else fold
- 3x raise any made hand
- 3x raise a low pair if no outs seen (no cards of your hand are out)
- 3x raise if suited and 8 or more high outs
- 3x raise if suited and 7 or more high outs AND 7 suit outs
- 3x raise if suited and 6 or more high outs AND 8 suit outs
- 1x call if suited
- 1x call if low pair and 2 trip outs
- 1x call if low pair and 1 trip out and at least 2 other pair outs
- 1x call if no-gap straight draw > 456
- 1x call if 1-gap straight draw and at least 5 mid outs
- 1x call if 2-gap straight draw and at least 6 mid outs
- 1x call if at least 4 high outs
- 1x call if 3 high outs AND (at least 2 mid outs, OR 1 mid out and all 3 low outs, OR all 6 low outs, OR wheel draw and 3 low outs)
- 1x call if 2 high outs AND (at least 4 mid outs, OR all 3 mid outs and 2 low outs)
- 1x call if 1 high out AND 5 mid outs
- 1x call if 7 mid outs, OR all 6 mid outs and all 3 low outs
- else fold
- 3x raise any made hand
- 3x raise any flush draw
- 3x raise 8 straight outs
- 3x raise 7 straight outs AND (at least 3 high outs, or at least 4 mid outs)
- 3x raise 6 straight outs AND (at least 4 high outs, or at least 9 mid outs)
- 3x raise 5 straight outs AND (at least 6 high outs, or at least 12 mid outs)
- 3x raise 4 straight outs AND at least 8 high outs
- 3x raise 3 straight outs AND at least 10 high outs
- 1x call all other straight draws
- 1x call any low pair
- 1x call 5 or more high outs
- 1x call 4 high outs AND at least 2 mid outs
- 1x call 3 high outs AND at least 4 mid outs
- 1x call 2 high outs AND at least 6 mid outs
- 1x call 1 high outs AND at least 9 mid outs
- 1x call with all 12 mid outs, or at least 6 mid outs and a previous 3x raise
- else fold
I’m finally getting to a reader’s request about a triple-down blackjack game at the Texas Station casino in North Las Vegas. Here are the relevant rules:
- triple-down on first two cards totals of 9, 10, 11 (including soft totals)
- blackjack pays even money (1:1)
- blackjack is an automatic winner, if you stay
- normal double-down rules
- no triple after split
The overall house edge for this game is 0.83%. The triple-down rule gives the player a +1.62% boost, but it’s not enough to overcome the even money blackjacks. The basic strategy is the same as double-deck, except that doubles on hard 9, 10, 11 are replaced by triples. Also, 9 vs a 7 upcard, and A-9 vs 4/5/6 are triples.
Interestingly, tripling a blackjack against a dealer 6 upcard isn’t too bad of an option. The EV is .998, instead of the 1.0 automatic winner for staying. It’s worth a gamble, if you feel like it. Tripling a blackjack against a dealer 5 upcard isn’t as good, as the EV here is only .946. That’s giving up 5.4% of your original bet, on average.
Someone should petition Texas Station to allow triple-down on any two cards. This still leaves the house edge at 0.59%, which is about equal to a liberal shoe game.
Macintosh:Debug show$ ./laycraps -n 100000000000 -r -t 2 -m 15 max muck depth: 15, CSM buffer depth: 10, threshold: 2, seed: 1249092576 ... ... roll: 61970000, net: 40131.900, return: +0.15% roll: 61980000, net: 40184.350, return: +0.15% roll: 61990000, net: 40218.950, return: +0.15% roll: 62000000, net: 40254.550, return: +0.15% roll: 62010000, net: 40231.850, return: +0.15% roll: 62020000, net: 40243.700, return: +0.15%
Where the 0.15% edge is on the total action, which includes $41 for each roll the lay is ON. This is a pretty conservative way to state the return.
Another way to look at it is the edge for any given roll:
|Running Count||Lay 10 Player Edge|
So, the easy way to play this is to lay the 4 and 10 when the count is good (at Viejas, you pay the vig up front). Then, while the count is good (i.e., RC >= 2 for the lay 10, and RC <= -2 for the lay 4), you leave the lay bet ON. When the count isn't good, you turn the appropriate lay bet OFF. Usually, this means both the lay bets are OFF, then when one of the counts gets good, that bet goes ON. When the count goes bad, both bet OFF. When the count is neutral (0), the distribution shows the odds are greater then 2:1 to hit the 4/10. However, the odds aren't good enough to overcome the vig. But, you can gamble, and turn both bets ON, and if 7 comes up, you win both bets.
It’s a little strange to have both lay bets up there, and turning them ON/OFF with every roll. The dealers might get a little irritated, and you’re only picking up a small edge. (While a lay bet is ON, you’re picking up from approx. 0.25 – 1.0% edge.) Too bad it’s not an electronic game :(