Discount Gambling

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: