Mississippi Stud Collusion

Mississippi Stud (MS) certainly has its share of game protection issues. For the savvy advantage player, just about everything works. There is one more method to beat MS that I have yet to discuss in this blog: player collusion. It's not a big play, but the fact that collusion is even possible comes as a surprise to many who operate the game. Indeed, some casinos deal MS face-up.

 

Normally, collusion requires that a significant number of cards be known to the players. For example, in Caribbean Stud, it requires 7 players to get a small edge (each player knows 30 cards in addition to his own five cards). In Three Card Poker, full information sharing with 7 players does not yield an edge (each player knows 18 cards in addition to his own cards). What is remarkable about MS is that collusion works at all. With 6 players, each player knows only 10 cards in addition to his own cards. But those 10 cards are plenty. By being able to wager up to 9 additional units, this limited amount of additional information can be leveraged to obtain a healthy edge over the house.

 

There is a long history of casinos dealing MS face-up. While this way of dealing MS protects it from card marking and edge sorting and increases game pace, a face-up game also makes collusion at a full table a triviality. Some APs just camp out at these games.

 

My effort here is to give results for collusion against MS using “perfect play.” With no additional information, the player knows 0 additional cards. This player is simply playing at the “house edge” of 4.91%. Playing a face-up game yields 10 additional cards. Maybe the AP also sees the burn card and therefore knows 11 cards. Maybe the table has spots for 7 players and he knows 13 cards. And so on …

 

To get the edges for all of these situations, I ran 16 simulations, consisting of the player knowing anywhere from 0 to 15 cards in addition to his own two initial cards. For each number of additional cards, I simulated a minimum of five million (5,000,000) rounds. Given the extremely high volatility for MS, even after 5M rounds the results I obtained are still only accurate to within about 0.2%.

 

Here are the results of my simulations:

 

 

For example, perfect collusion in a six-player face-up game gives the AP an edge of 2.29% over the house.

 

The following plot gives a graphical representation of the data in the previous table. In particular note that, on average, each additional card adds about 0.77% to the player's side:

 

 

This graph may look linear, but clearly it is not. For example, if the edge gained by seeing additional cards was a linear function, then with 47 cards known the player's edge would be about -4.91% + 0.77%*47 = 31.3%. However, knowing 47 cards is the same as having perfect knowledge of the player's five cards, which gives the player an edge of 279.02% over the house (see this post).

This work is all very good in theory (and I am a theoretician at heart), but it leaves the question of a feasible strategy hanging unresolved. I don’t know the practical issues surrounding MS collusion. However, it is clear that perfect collusion strategy is not possible without computer assistance. The AP must give up a part of his edge to create a workable strategy.

 

Fortunately for the AP who wants to collude at MS, Stephen How created effective strategies for the six-player game. On his blog Discount Gambling, How has the following posts:

 

• Mississippi Stud @ Barona Casino, CA In this post, How gives a human-feasible six-player collusion strategy that gives the AP an edge of 1.5%

• Simplified Collusion for Mississippi Stud In this post, How gives an easy strategy for six-player collusion that gives the AP an edge of 0.5%.

 

I have no interest in recreating How's excellent work noted above. I contacted Stephen How and he graciously gave me permission to reprint his 1.5% strategy in this post.

 

How’s strategy uses the following definitions:

 

• “High outs” -- the number of cards remaining in the deck that will give you a high pair (Js thru As)

• “Mid outs” -- the number of cards remaining in the deck that will give you a mid pair (6s thru Ts)

• “Low outs” -- the number of cards remaining in the deck that will give you a low pair (2s thru 5s)

• “Suit outs” – for a suited hand, the number of cards remaining in the deck that are the same suit

• “Straight outs” – on the river bet, the number of cards remaining in the deck that complete a straight.

 

Here is Stephen How's six-player MS collusion strategy:

 

Flop bet:

 

• 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

 

Turn bet:

 

• 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

 

River bet:

 

• 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

 

Nobody said it was going to be easy. For the AP who wants to play against MS, it would surely be more worthwhile to seek out a dealer who is exposing a hole-card. But, if the situation is just right (e.g. the game is dealt face-up), there is no reason to expect that a savvy player wouldn't be just camping out.