Caribbean Stud (CS) is virtually extinct in Nevada. Likewise throughout the U.S. it is rapidly disappearing, giving way to a wealth of alternative poker-style games. The main problems with CS are its very high house edge (5.224% with perfect strategy) and high fold percentage (47.77%). Games like Ultimate Texas Hold’em, Four Card Poker, Texas Hold’em Bonus Poker and others are finding much more traction with domestic players. However, this is not the case internationally.

One of the biggest surprises I encountered during my travel in Asia was the abundance of CS. Part of this is jurisdictional; very few proprietary games are approved for casino use to begin with. The other part of the equation is that players are enjoying CS it in large numbers. Internationally, CS is not going anywhere.

Before continuing, I must fully acknowledge the work James Grosjean shared in his (unavailable) book “Beyond Counting.” Grosjean’s work greatly exceeds what I have reproduced here, and I do not pretend to have duplicated everything he has done. However, everything I present in this post is from my own work. The fact that my results roughly match Grosjean’s results is a fortunate coincidence.

Advantage play against CS comes from three directions. First, edge sorting provides a significant opportunity. As shown in this post, the AP can get an edge of about 6.90% with a sort of Ace/King vs. others. This method has occurred in at least one instance I know about. The strategy is trivial enough that it can be learned in under an hour. The edge is big. The method is easy. If the cards and the dealing/shuffling procedures are just right, edge sorting CS will surely happen again.

The next method of AP play against CS is hole-carding. However, it is not enough to see two dealer cards. With knowledge of the dealer’s up-card and one additional card, the AP is still giving up a house edge of 1.74%. When the AP knows three cards then he gets a big edge over the house (11.57%), but such an opportunity is surely rare.

The final method of AP play against CS is player-collusion. It has long been known that if players share information in CS then they can gain the edge over the house. The general understanding is that if players share the number of Aces, Kings and matches (cards that match the dealer’s up-card) in their hands, then these players can improve their play/fold strategy and get an edge. Beyond this general understanding, however, few specifics are available.

Here is the bottom line. If there are seven collaborating players seated at the table, then they can get an edge over the house. If there are six or fewer collaborating players at the table, then they cannot get an edge over the house by collusion alone; they will need something more.

I ran two simulations of one billion (1,000,000,000) hands of CS each, using computer-perfect collusion. In other words, we are assuming each player at the table is using a computer program (for example, an “app” on a cell-phone) to enter his cards. The program then gets the information for all of the colluding players at the table. The program processes the information and then instructs each player to either play or fold his hand.

Here are the results of these simulations:

  • With seven colluding players, using computer-perfect collusion, the players have a 2.378% edge over the house.
  • With six colluding players, using computer-perfect collusion, the casino has a 0.398% edge over the players.

Casino management frequently over-reacts to the danger that collusion poses to CS. This point needs to be emphasized: if there are six or fewer colluding players, no edge by collusion alone is possible.

With a mind to understanding the meaning of "collusion alone" I ran some simulations where I assume that players were not only colluding, but also managed to see a second dealer card. That is, the player's observe the dealer's up-card, and also know one of the dealer's hole-cards. I conducted computer-perfect play simulations for four, five, six and seven colluding players, when they know two dealer cards. Here are the results:

  • Four players, 1.927% edge over the house.
  • Five players, 3.161% edge over the house.
  • Six players, 4.930% edge over the house.
  • Seven players, 7.456% edge over the house.

I will not begin to consider a practical strategy when two dealer cards are known. I cannot imagine this is knowable in any meaningfully profitable sense. If fewer than seven players appear to be colluding, I suggest this is a strong indicator for secondary dealer hole-card exposure and the use of electronic devices.

Back to the case when only the dealer up-card is known.

With seven colluding players seeing only the dealer up-card, the problem of beating CS in practice is similar to the problem for High-Card Flush (HCF). As shown in this blog post, computer-perfect collusion strategy for HCF is essentially unknowable. For HCF, computer-perfect collusion strategy gives a player edge of 7.33% over the house. A strategy that can actually be learned and implemented at the table yields a much weaker 4.04% edge over the house. By analogy with CS, the 2.38% theoretically possible player-edge for CS is not feasible in practice. A human-feasible collusion strategy for CS will yield a significantly lower return than the 2.38% computer-perfect number.

The human-feasible collusion method that Grosjean described in Beyond Counting is called the “MAK” count. Here’s how it works. After all the cards are dealt in a round, the players examine their cards and take note of three pieces of information. These are the total number of:

  • M = “Matches.” Each player notes how many cards they have that match the dealer's up-card. The total number of matches the team holds is 0, 1, 2 or 3.
  • A = “Aces.” Each player notes how many Aces are in his hand. The number of Aces the team holds is 0, 1, 2, 3 or 4.
  • K = “Kings.” Each player notes how many Kings are in his hand. The number of Kings the team holds is 0, 1, 2, 3 or 4.

The players then share their hand information with each other by some method (signaling with chips, speaking in a foreign language, etc.). Each player then knows how many matches, Aces and Kings their group holds altogether: this is called the MAK count. By knowing the MAK count, his own cards and the dealer’s up-card, each player can then alter his play/fold strategy to optimally use the available information.

By way of comparison, the HCF collusion strategy (see this post) is easily grouped into about 10 different strategy decisions based on the signal group. That’s not the case with the MAK count. The amount of information that the MAK count provides is very large. A simple strategy that gives the player an edge is all but impossible. Here's why.

Observe that there are 4 possible values for M, there are 5 possible values for A, there are 5 possible values for K and there are 13 possible values for the dealer’s up-card. This gives 4x5x5x13 = 1300 possibilities for these four values overall. However, some of these are impossible (for example M = 2, A = 1, K = 0 and dealer up-card = Ace). After getting rid of all the impossible situations, it turns out that there are 1140 combinations (M, A, K, up-card) that can actually happen. The MAK collusion strategy depends on giving a minimally playable hand for each of the 1140 possibilities for (M, A, K, up-card).

By simulating one billion (1,000,000,000) rounds of CS, I was able to generate the minimally playable hand for each of these 1140 possibilities. I note that a quick check of that data shows that some of my results have an “off-by-one error” (OBOE) error.  I'll leave it to the serious AP's to fix these OBOEs so they can get the resulting 0.0001% gain in their edge over the house.

The following Excel spread sheet gives the minimally playable hand for each of the 1140 possibilities for (M, A, K, up-card), as determined by the simulation:

CS_MAK_Collusion_Strategy

This MAK collusion strategy is by no means easy to use in the form it is provided. In Beyond Counting, Grosjean manages to reduce these 1140 entries to a single page by gathering like situations together and efficiently collating the distinct possibilities into a matrix. If you feel like improving on Grosjean’s presentation, you have everything you need in the Excel spread sheet I provided. I’m not going to even try. Nevertheless, there are a couple of intuitively obvious simplifications:

  • If M = 0, then the player needs a pair higher than the dealer’s up-card (or better) to continue.
  • If M = 3, and A + K = 7 or 8, then the player should play every hand.

Here are the results of using the MAK collusion strategy, based on a simulation of one billion (1,000,000,000) hands of CS:

  • With a team of seven colluding players, using the MAK collusion strategy, the players have a 1.34% edge over the house.
  • With a team of six colluding players, using the MAK collusion strategy, the house has a 1.88% edge over the players.

CS collusion requires a full table. CS is not beatable by collusion alone using a team of six or fewer players. If six or fewer players appear to be colluding at the table, then they are either playing a losing game or they are using some additional piece of information (for example, seeing a second dealer hole-card).

CS collusion is tough. In the scheme of what’s possible, this method requires an intense amount of work. First, the strategy each member of the team must master is very complex. It is also challenging to find a way of sharing the information efficiently and accurately at the table. Doing this in real-time will require a lot of talent and a lot of practice.

CS collusion is small. The edge of 1.34% is small compared to many other advantage play opportunities. Within the scope of CS, it would be much stronger to try to find a game that could be edge-sorted. The upside profit-potential from collusion is much lower than for many methods that are both easier to learn and easier to implement.

The conclusion is that it is highly unlikely a collusion team will target CS. If a team appears to be working together to beat CS, they may be sharing information, but they are almost certainly taking advantage of a secondary weakness as well.

The following are my game protection suggestions for CS collusion:

  • Don’t let players share information about their cards.
  • Turn folded hands face-up and spread the hand before putting the hands into the discard tray.
  • Limit the table to at most six-spots.
  • Watch for secondary opportunities players can combine with collusion.
About the Author
By

received his Ph.D. in Mathematics from the University of Arizona in 1983. Eliot has been a Professor of both Mathematics and Computer Science. Eliot retired from academia in 2009. Eliot Jacobson