# Discount Gambling

## Simple Collusion Strategy for Caribbean Stud

Posted in caribbean stud by stephenhow on September 20, 2010

(This post was calculated for 6 players; see updated analysis for +EV collusion with 7 players.)

I’ve always wondered how easy and effective collusion at Caribbean Stud would be. People have looked into the theoretical gains of collusion, and conclude its possible to reduce the house edge to 0.4%, using a computer to analyze all 30 confederate cards and the dealer upcard.

I developed a practical collusion strategy for Caribbean Stud, which reduces the house edge from a practical 5.32% down to 1.32% with 6 players. That’s a significant savings, since often players are grinding away, trying to hit a large progressive, or a high hand bonus overlay. On a \$5 bet, a normal strategy costs (\$5)(.0532) = \$0.266/hand. That’s pretty expensive, close to \$11/hr @ 40 hand/hr. Using the simple collusion strategy detailed below, you’d cut your costs by 75%, down to (\$5)(.0132) = \$0.066/hand, or \$2.64/hr. That’s a big deal, especially if you play a lot, or for long sessions. Normally, Caribbean Stud is an expensive game, long-term.

To collude, you’ll need a cooperative table of 6 players (you and 5 others). You don’t need much info, and the strategy implementation is extremely simple. The key is calling a nominal folding hand, when you have enough info to justify a call, in hopes the dealer doesn’t qualify. Also, there are times when you should fold a normal calling hand, because it’s likely that the dealer holds a better hand.

The easiest way to share the needed info among players is by signaling with colored chips. In designated per-player areas, each confederate should place a red (\$5) chip to indicate each copy of the dealer upcard they posses. Additionally, each player should place a blue (\$1) chip for every Ace or King they hold. With this collusion information, you should modify the simple strategy as follows:

Hand Modified Strategy
Junk Raise if see at least 3 red chips and 3 blue chips.
(3 upcard copies and 3 A/Ks.)
A K J 8 3 Fold if one or less red chips seen.
(1 or less upcard copies.)
One Pair 2’s or 3’s Fold if one or less red chips seen, AND upcard is higher than pair.
(1 or less upcard copies.)
One Pair 4’s thru 8’s Fold if no red chips seen, AND upcard is higher than pair.
(No upcard copies.)
Pair 9’s or higher No change.

### Raising with Junk

The following table shows the simulated average value of raising with a junk hand, given the 6 confederates have all 3 copies of the the dealer upcard (not A/K), broken down by the number of Aces and Kings the confederates hold. For example, say the dealer upcard was a 9c, and the players have all the remaining nines (9d, 9h, 9s). Then, the table below says not to raise junk unless the confederates (i.e., 30 cards seen) hold any combination of at least 3 Aces and/or Kings. So if the confederates hold 2 Aces and 2 Kings, the value of raising a junk hand is -0.66649. This is much better than folding (value -1). Or, if the confederates hold 0 Aces and 4 Kings, the value of a junk hand is -0.67487. On the other hand, if the confederates hold only 1 Ace and 1 King, then it’s better to fold the junk hand (-1) than raise it (-1.05500).

```         0 Kings   1 King  2 Kings  3 Kings  4 Kings
4 Aces  -0.69915 -0.55349 -0.50180 -0.51584 -0.69635
3 Aces  -0.84700 -0.67379 -0.54392 -0.49117 -0.51812
2 Aces  -1.05753 -0.84399 -0.66649 -0.54415 -0.48636
1 Ace   -1.29684 -1.05500 -0.84116 -0.66779 -0.53230
0 Aces  -1.54412 -1.29312 -1.04717 -0.85175 -0.67487
```

Note that the table is symmetrical about the diagonal Aces=Kings, and can be reduced using the index Aces+Kings:

 Known Aces+Kings EV(Raise) 0 1 2 3 4 5 6 7 8 -1.54 -1.30 -1.05 -0.85 -0.67 -0.54 -0.50 -0.52 -0.70

The frequency of 6 confederates (30 cards) holding all 3 the dealer upcard outs is a high 19.5% = C(48,27)/C(51,30). That means in one out of five hands, a few confederates will probably be helped. Remember, that raising a junk hand with a value of -0.70 is better than folding (-1), and for a \$5 Ante, that’s a \$1.50 savings. Any time you can play a slow carnival game for around 2% of the Ante, you’re doing well. Caribbean Stud is a fun, relaxing game, but it’s normally expensive. Reducing its house edge from 5.5% to 1.3% makes it a lot more enjoyable.