Collusion in Caribbean Stud Poker
Caribbean Stud Poker Collusion/Cheating
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.
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:
The expected value of a game-state and associated strategy can be calculated as follows:
The overall expected value of a n-known card Caribbean Stud Poker game is calculated as follows:
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.
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.
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:
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
Graph 1 - Player Advantage(%) vs Known Cards(n)
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.
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