Practical Collusion For Four Card Poker
I continue to look at the effects of collusion on the carnival games, and today I worked on Four Card Poker. In Four Card Poker, each player is dealt five cards, from which they play their best 4 card poker hand. The dealer has the advantage of starting with 6 cards, but gives back some edge by allowing the player to raise 1-3x after seeing the hand. Also, one of the dealer cards is exposed.
The simplest and most common strategy does not take into account the dealer upcard. The player simply calls 1x with a pair of 2’s thru 9s, and raises 3x with a pair of Ten’s or higher. This strategy yields a house advantage of 3.4%.
I worked out a very simple collusion strategy, which only requires communication of dealer upcard copies amongst the confederates. As I’ve pointed out in my posts on practical collusion for Caribbean Stud, it’s very easy to communicate how many copies of the dealer upcard the confederates hold (signal with chips, coughs, etc.). It’s very easy to know how many copies of the dealer upcard are left for the dealer to pair with. Using a simple strategy based on this upcard copies count, the house edge is cut in half down to 1.7% (on a full table).
The table below summarizes the collusion strategy when playing on a full (6 player) table. Once you know how many copies of the dealer upcard are held by the players, you play your one-pair hand according to the table. Additionally, you should 1x call AK-high when the players hold all 3 copies of the dealer upcard.
| Pair | Dealer Upcard | |
|---|---|---|
| Upcard > Pair | Upcard ≤ Pair | |
| 2’s | fold if 0-2 copies bet 1x if 3 copies |
bet 1x |
| 3’s thru 5’s | fold if 0-1 copies bet 1x if 2-3 copies |
bet 1x |
| 6’s thru 7’s | fold if 0 copies bet 1x if 1-3 copies |
bet 1x |
| 8’s | fold if 0 copies bet 1x if 1-2 copies bet 3x if 3 copies |
bet 1x |
| 9’s | fold if 0 copies bet 1x if 1-2 copies bet 3x if 3 copies |
bet 1x if 0 copies bet 3x if 1-3 copies |
| T’s thru J’s | bet 1x if 0-2 copies bet 3x if 3 copies |
bet 1x if 0 copies bet 3x if 1-3 copies |
| Q’s | bet 1x if 0-1 copies bet 3x if 2-3 copies |
bet 1x if 0 copies bet 3x if 1-3 copies |
| K’s | bet 1x if 0-1 copies bet 3x if 2-3 copies |
bet 3x |
| A’s | bet 3x | bet 3x |
Examples
You have a pair of 4’s. The dealer upcard is a 6, but nobody holds a 6 in their hands (0 copies). Fold your 4’s.
You have a pair of Jacks. The dealer upcard is a Ten, but noboby holds a Ten in their hands (0 copies). Just 1x call your Jacks.
You have a pair of Jacks. The dealer upcard is a Ten, and you and your 5 friends hold one Ten between your hands (1 copy). You should 3x raise your Jacks.
You have a pair of 8’s. The delaer upcard is a Jack, and you and your friends hold all 3 remaining Jacks (3 copies). You should 3x raise your pair of 8’s.
Simple +EV Collusion For Caribbean Stud
This is an update to my previous collusion analysis for 6 players. I found that Caribbean Stud games actually seat 7 players, and I constructed a simple collusion strategy that yields a +1.1% player advantage.
If you enjoy playing Caribbean Stud, and have a set of friends that can fill up a table, there’s a simple way to play together to obtain a +1.1% advantage against the house. Here’s what each player (i.e., you and your 6 confederates) need to do:
- For each copy of the dealer up card a teammate holds, he places a $5 chip (red) in a designated signaling area.
- For each remaining Ace or King a teammate holds, he places a $1 chip (blue) in the designated signaling area.
- Each player now knows how many total copies of the dealer upcard the confederates hold, as well as additional Aces and Kings.
- Each player adjusts the basic strategy decision according to the table below (i.e., confederates know when to Raise with junk, and fold weak hands).
| Hand | Modified Strategy |
|---|---|
| Junk | Raise if see at least 3 red chips and 4 blue chips. (3 upcard copies and 4 A/Ks.) |
| A K J 8 3 | Fold unless 3 red chips seen. (All upcard copies held by friends.) |
| One Pair 2’s thru 6’s | Fold if upcard is higher than pair, unless 2 or more red chips seen. (2 or more upcard copies.) |
| One Pair 7’s thru K’s | Fold if upcard is higher than pair, unless 1 or more red chips seen. (1 or more upcard copies.) |
While this collusion strategy is non-verbal, and no one will really care if you do it for $10 bets, the casino would very quickly catch on to a table full of $100 Ante players all calling with junk at the same time. Otherwise, a table full of players would make on average (7 players)($100/player)(1.1%/hand)(40 hands/hr) = $308/hr. And of course the variance would swamp the average for any reasonable time frame (i.e., shorter than months).
Examples
Let’s say your hand is a pair of Kings. The dealer upcard is an Ace. Your confederates do not hold any Aces (i.e., they place no red chips in their signaling areas). You should fold the hand.
Say you have junk. The dealer upcard is a Trey. Your confederates indicate they have all the Treys in their hands (they place 3 red chips in the signaling areas). The confederates also place 5 blue chips, signaling they have 5 Aces/Kings between them. Everyone should call (2x) no matter what they hold.
The dealer upcard is a King. The confederates place 3 red chips, indicating they have all the King “outs”. They also place 4 blue chips, indicating they have all 4 Aces between them. Everyone should call, no matter what they have, since there are at least 3 red chips and 4 blue chips seen.
You have a pair of Treys. The dealer upcard is a Ten. The confederates place 2 red chips, indicating they have 2 Tens between them. You should call your hand. Junk hands should fold.
Double Baccarat @ Sycuan Casino
My local Sycuan Casino offers a unique game that’s a simplification of Pai-Gow tiles. Like the tile game, the players and the bank are dealt 4 cards each. Each hand is set into a front hand of 2 cards, and a back hand of two cards. The back hand must be greater than the front hand. Hand values are ranked by poker pairs, followed by Baccarat totals (0 thru 9). After the player hands are set, the bank turns over its hand and sets it according to “House Way”.
| Hand | Rule |
|---|---|
| Two Pairs | Set large pair behind, small pair in front. (Pair-Pair; never break pair). |
| Pair Aces | Pair Aces behind (never split Aces.) |
| One Pair | Pair behind if 5 or higher front, else Split pair if can form (8,8), (7,9), (8,9) or (9,9), else Pair behind. |
| No Pair | Set (5,9) if possible, else Set hand with highest front + back total, with minimum back – front gap. |
Once all hands are set, the player or banker wins the wager if hands win/win, tie/win, or win/tie. In the case of tie/tie, the bank wins the wager. All other hands push the wager.
The player posts a minimum 1% collection before each hand. If all players push their bets, all collections are returned (“free collection”). This means in a heads-up game against the house, the player only pays the collection for a win or a loss, and gets free collection on a push.
I worked at optimizing a heads-up player strategy against the a house way bank, out of curiosity at what the house edge was. Of course, its an uphill battle against the collection (even when free for pushes), and worst, losing tie/tie. Using exhaustive combinatorics, I came up with the following near-optimal strategy (I only looked at the no-pair cases):
| Hand | Rule |
|---|---|
| Two Pairs | Set large pair behind, small pair in front. (Pair-Pair; never break pair). |
| Pair Aces | Pair Aces behind (never split Aces.) |
| One Pair | Pair behind if 5 or higher front, else Split pair if can form (8,8), (7,9), (8,9) or (9,9), else Pair behind. |
| (6,9), (7,9), (8,9), (9,9) | |
| front + back = 14 | (7,7), (6,8) |
| (5,9) | |
| front + back = 10 | (5,5) |
| (9,1) | |
| (4,6), (3,7), (2,8) | |
| front + back = 9 | (0,9) |
| (1,8) | |
| (4,5), (2,7) | |
| front + back = 8 | (0,8) |
| (1,7) | |
| (3,5), (2,6) | |
| front + back = 7 | (3,4) |
| (0,7) | |
| (2,5), (1,6) | |
| front + back = 6 | (2,4) |
| (0,6) | |
| (1,5) | |
| Set hand with highest front + back total, with minimum back – front gap. | |
This strategy simulates at -1.46% heads up against house way, when minimizing collection to 1% of the bet amount. The frequency of ties simulates at 1.12%. So even if they eliminated the bank wins tie/tie rule, you’d still lose because of collection. As an additional note, if a heads-up player also plays the same House Way as the bank, the house edge increases to 2.0%.
Overall, the head’s-up game is about as good as a free-collection Pai-Gow game. (There are a few free-collection games at the card rooms town.) However, since the casinos don’t offer free-collection Pai-Gow, the head’s-up Double Baccarat game has better odds than the Pai-Gow game, for what it’s worth.
Discount Gambling 
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