Edge Sorting Caribbean Stud

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The table game Caribbean Stud (CS) used to be among the most popular of all proprietary table games in this country. In 1994, there were 167 tables of CS in Nevada.  By the year 2000, that number dropped to 97 tables. In 2009 there were only 15 tables left. According to the latest data from the Nevada Gaming Control Board, as of November, 2012, there are no tables left in Nevada. Internationally, however, the game is still very popular. For example, there are hundreds of tables of CS in Macau.

In CS, players are not permitted to share their hands with each other. Player collusion for CS is covered extensively in Beyond Counting, where advantage player savant James Grosjean describes the "MAK" count (MAK = "Matches", "Aces", "Kings"). Grosjean states that: "at a full table of seven players, the edge would be nearly 2% using the provided 36-card strategy if the players saw the burn card."

Grosjean also considers the situation where the AP knows one or more hole-cards in addition to the dealer up-card. Grosjean gives this hole-card edge data:

• If the AP knows 1 card (the dealer up-card), then the house edge is 5.23%.
• If the AP knows 2 cards (including the dealer up-card), then the house edge is 1.74%.
• If the AP knows 3 cards (including the dealer up-card), then the player edge is 11.57%.
• If the AP knows 4 cards (including the dealer up-card), then the player edge is 35.21%.
• If the AP knows all five dealer cards, then the player edge is 64.27%.

I won't repeat Grosjean's hole-card strategies here because I have not duplicated his work. CS was probably a great opportunity in the late 1990's, when it was among the few proprietary games offered: casinos didn't yet understand the dangers of collusion and hole-card exposure. Those are the days of yore.

Recently, I was told about an incident when a team was discovered using edge sorting to beat CS. According to my source, this team sorted Aces and Kings in one direction and all the other cards in the opposite direction. After they were discovered and backed off, there was an unanswered question: just how strong was their edge?

Before I tackle this question, a quick review is in order. CS is played with a single deck of cards. The play of the game is similar to Three Card Poker, with the common ante/raise/qualify structure. Here are the rules for CS:

1. The player makes an “ante” wager.
2. The player and dealer each get five cards. Four of the dealer’s cards are dealt face-down, and one dealer card is turned face-up.
3. After inspecting his cards and the dealer’s up-card, the player can fold and forfeit his ante wager, or he can make a “raise” wager to stay in the hand. The raise wager equals twice the ante wager.
4. The dealer then exposes his cards. The dealer’s hand must be Ace/King or higher to qualify.
5. If the dealer does not qualify, then the player wins even money on his ante wager and his raise wager is a push.
6. If the dealer qualifies, then the player competes against the dealer.
7. If the dealer qualifies and beats the player, then the player loses his ante and raise wagers.
8. If the dealer qualifies and ties the player, then the ante and raise wagers push.
9. If the dealer qualifies and the player beats the dealer, then the player wins even money on his ante wager, and his raise wager is paid according to a bonus pay table.

Here is the most common bonus pay table for the raise wager:

1. Royal Flush pays 100-to-1.
2. Straight Flush pays 50-to-1.
3. Four-of-a-Kind pays 20-to-1.
4. Full House pays 7-to-1.
5. Flush pays 5-to-1.
6. Straight pays 4-to-1.
7. Three-of-a-Kind pays 3-to-1.
8. Two Pairs pays 2-to-1.
9. All others pay 1-to-1.

A very complicated optimal player strategy yields a house edge of 5.224%. The player doesn’t give up that much by using the simpler strategy of raising with AKJ83 (off-suit) or higher. In this case, the house edge is 5.316%. The full mathematical analysis for CS is given here.

Next we consider an AP who is edge sorting CS. There are many ways to sort the cards, but the Ace/King sort is intuitively obvious.  Assume the AP sorts the cards into these two groups:

• High cards = {K, A}
• Low cards = {2, 3, 4, 5, 6, 7, 8, 9, T, J, Q}.

Assume that the entire deck is perfectly sorted so that the Aces and Kings are sorted in one direction and all other cards are sorted in the opposite direction. Because of the way CS is dealt, it is likely that the AP will be able to view all five cards and therefore know High/Low information about each dealer down-card together with knowing the specific up-card. I decided to use the sort information in its most basic form. Namely, the AP will simply know how many of the dealer’s cards are high cards, and how many are low cards.

The AP looks at his hand. He then looks at the board and counts the number of high cards, which will be one of the numbers: 0, 1, 2, 3, 4, or 5. This total includes the dealer's up-card, which may be an Ace/King as well. Based on the AP’s hand and the number of dealer high cards, the AP decides to raise or fold. My next step was to find the raise/fold strategy points for each number of high cards, 0, 1, 2, 3, 4 or 5. I did this by a combinatorial program that slowly converged to the correct answer for each number of dealer high cards. It was "math geek" fun to watch the program find the answers.

Here is optimal raise/fold strategy based on the number of high cards in the dealer's hand.

• If the dealer has 0 high cards, raise with any hand that is 3/3/4/5/6 or better.
• If the dealer has 1 high card, always raise.
• If the dealer has 2 high cards, raise with any hand that is A/K/J/T/7 or better.
• If the dealer has 3 high cards, raise with any hand that is A/A/2/3/4 or better.
• If the dealer has 4 high cards, raise with any hand that is A/A/K/K/5 or better.
• If the dealer has 5 high cards, raise with any hand that is 2/2/2/2/3 or better.

For example, if the AP knows that 3 of the 4 dealer's down-cards are high cards, and the dealer's up-card is a Deuce, then the dealer has a total of 3 high cards. By the strategy above, the AP plays A/A/2/3/4 or higher. Compare this to the situation when the AP knows that 2 of the 4 dealer's down-cards are high cards, and the dealer's up-card is an Ace. Again, the AP should play A/A/2/3/4 or higher. These are very different situations, but the strategy above treats them the same. Clearly there are improvements that could be made to the strategy by incorporating the specific value of the dealer's up-card.

A simulation of one billion (1,000,000,000) rounds of CS using this strategy gives a player edge of 6.90%. The edge-sorting AP raises 68.83% of his hands, and folds 31.17% of his hands. By comparison, a non-AP who uses optimal CS strategy raises 52.23% of his hands.

Here is a combinatorial breakdown of the possible outcomes for the AP who is edge sorting CS, based on my simulation:

Game protection for edge sorting relies on one tactical decision and one strategic decision:

• Tactical decision point: Make sure there is a turn included in every shuffle procedure, even when the game is being dealt from an automatic shuffler.
• Strategic decision point: Use safe cards.