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# Collusion in Caribbean Stud Poker

## Caribbean Stud Poker Collusion/Cheating

### A) Introduction

Caribbean Stud Poker is a popular gambling game found in both online and land based casinos. In many versions of the game multiple players are dealt seperate hands and play simultaneously against the dealer. The rules of the game require that each player views his own cards only and sharing of information between players is prohibited. This article examines the change of player advantage in Caribbean Stud Poker if players ignore the rules and share information about what cards they were dealt with each other.

### B) Method

Note: All calculations used in this article assume an ante bet of 1 and results are given in terms of the ante bet as opposed to total amount bet (ante plus raise bet).

Exhaustive Analysis

In Caribbean Stud Poker the only decision the player needs to make is whether to raise or fold the ante bet. At that point in time the total information available to the player (i.e. game-state) is:

• 1 dealer up card
• 5 player cards in their own hand
• n cards in other player's hands

The expected value of a game-state and associated strategy can be calculated as follows:

• Calculate expected value of raising by cycling through all (46-n)C4 possible dealer down cards and summing outcomes.
• Expected value of folding equals -1.
• Expected value of game-state equals maximum of expected value from raising and expected value from folding. Therefore if expected value of raising is greater than -1, the correct strategy is to raise.

The overall expected value of a n-known card Caribbean Stud Poker game is calculated as follows:

• Cycle through all possible game-states of 52C1 dealer up cards, 51C5 player cards and 46Cn other player cards.
• Calculate expected value of each game-state as described above, summing results.

Below are the formulas for calculating the number of game-states and combinations that need to be enumerated to calculate the expected value for a n-known card Caribbean Stud Poker game. Also given is a table for various n values.

 Total Game States = 1 Dealer Upcard x 5 Player Cards x n other player cards = 52C1 x 51C5 x 46Cn
 Exhaustive Combinations = Total Game States x 4 Dealer Down Cards = Total Game States x (46 - n)C4
Table 1 - Number of combinations.
Known
Cards (n)
Total
Game States
Exhaustive
Combinations
01.2215 x 1081.9933 x 1013
51.6744 x 10141.6957 x 1019
104.9793 x 10172.9331 x 1022
156.2509 x 10191.9669 x 1024
206.8505 x 10201.0242 x 1025
258.4816 x 10205.0762 x 1024
301.2111 x 10202.2042 x 1023

The number of calculations required for an exhaustive analysis for n = 0 is well with-in the reach of modern computers. However due to the large number of combinations for n > 0 another method must be used, in this case by random sampling of strategy gains.

Strategy Gains

Rather than directly calculating the expected value of a n known card Caribbean Stud Poker game, an indirect method of calculation can be used based on the n = 0 game. Since the player is unable to vary the size of the ante bet after obtaining the game-state information the only gains available compared to the n = 0 game come from strategy variations (i.e. fold instead of raise, or raise instead of fold).

The expected value of a n known card Caribbean Stud Poker game equals the expected value of the n = 0 game plus the average gain from strategy variation of all n known card game-states. The expected value for n = 0 can be calculated exhaustively and equals a 5.224% player dis-advantage. The average strategy gain can be estimated by random sampling and the sufficient accuracy obtained by having a large enough sample size.

Calculate the strategy gain of a game-state as follows.

1. Determine the optimal strategy of the game-state ignoring the knowledge of the n other player cards.
2. Calculate expected value of the game-state (including the n other player cards) using the strategy determined in 1.
3. Calculate expected value of the game-state using the knowledge of the n other player cards.
4. Strategy gain equals expected value calculated in 3. minus expected value calculated in 2.

Examples:

Dealer
up card
Player
cards
Other
player
cards
1*2 3* 4
9 A K 7 5 2 9 9 Q K A Fold (-1.187)-1.000-0.7860.214
9 A K 7 5 2 6 5 4 3 2 Fold(-1.187)-1.000-1.000(-1.288)0.000
A 8 8 8 J 5 2 2 3 3 7 Raise(4.915)5.0235.0230.000
J 2 2 3 4 8 K 5 5 3 2 Raise(-0.965)-1.094-1.000(-1.094)0.094
*Expected value of raise in brackets.

Random Sampling

Sampling is a statistical method to estimate a population value by measuring a sub-set of the population. In this case the population is the set of all game-states and the value being measured is the average strategy gain achieved by knowing n other player cards. The game-states on which the calculations are performed on are selected randomly with equal probability from the whole population.

The relevant statistics are calculated as follows:

1. Calculate strategy gain given a random selection of 1 dealer up card, 5 player cards and n other player cards.
2. Repeat 1. k times, storing each seperate result.
3. Calculate average strategy gain, standard deviation and standard error of sample from results obtained in 2.

### C) Results

The following results were obtained using the methods described above. An exhaustive analysis was performed for n = 0 and random sampling with a sample size of 100000 was used for all other n.

Table 2 - Expected Value Results
Known
Cards(n)
Sample
Size
Average
Gain
Standard
Deviation
Standard
Error
Overall
0****-0.05224
51000000.005560.027430.00009-0.04668
101000000.012610.047900.00015-0.03963
151000000.020100.068040.00022-0.03214
201000000.032010.096510.00031-0.02023
251000000.048520.133780.00042-0.00372
301000000.075440.188590.000600.02320

Graph 1 - Player Advantage(%) vs Known Cards(n) ### D) Conclusion

In ideal conditions of knowing 30 extra cards dealt to other players and with computer perfect play it is possible for players to gain a 2.3% edge over the house. In practice this would be difficult to achieve due to casino rules prohibiting the sharing of card information and the fact that using using computer devices in land based casinos is illegal in all jurisdictions. It may be more practical to collude online and use computer assistance to determine strategy, though I know of no online casino that deals 7 handed multi-player Caribbean Stud Poker.