During the hold ‘em poker craze that began when Chris Moneymaker won the World Series of Poker in 2003, there was a rush to bring poker variants to the casino floor. Four Card Poker (FCP) was introduced in the height of the poker craze. For several years it wasn’t clear which poker games would succeed. The market has mostly shaken out in the poker genre and FCP is one of the few survivors. Today it has more than 200 placements.
As its name implies, FCP is based on poker hands that consist of four cards rather than the usual five. It has the familiar Ante/Play bet structure that other poker variants use. This structure forms the basis of the possibility of vulnerability to advantage play. Because the player can make a strategic decision (play/fold) after the dealer is dealt his cards, additional information has value.
The hands are ranked as follows:
-
Four-of-a-Kind
-
Straight Flush
-
Three-of-a-Kind
-
Flush
-
Straight
-
Two Pair
-
One Pair
-
High Card
As usual for poker, the hands are ranked first according to their type and further compared based on the rank of the cards that make up the hand. Here are the rules:
-
The player makes an Ante bet.
-
The player is dealt five cards; the dealer is dealt six cards.
-
One of the dealer’s cards is dealt face up; the other five cards are dealt face down.
-
After consider his five cards and the dealer’s up-card, the player can choose to either fold, make a Play bet of 1 unit (raise 1) or make a Play bet of 3 units (raise 3).
-
If the player folds, he forfeits his Ante bet and the hand is over.
-
If the player makes a Play bet, then the dealer’s hand and player’s hand are compared.
-
The player makes the best four-card poker hand using any four of his five cards.
-
The dealer makes his best four-card poker hand using any four of his six cards.
-
If the dealer’s hand beats the player’s hand, then the player forfeits his ante and play bets.
-
If the player’s hand beats or ties the dealer’s hand, then the player wins 1-to-1 on both his ante and play bets.
There is an additional automatic bonus that is paid to the player, regardless of the outcome of the hand, if he has Three-of-a-Kind or better. The most common pay table for the automatic bonus is as follows:
-
Three-of-a-Kind pays 2-to-1.
-
Straight Flush pays 20-to-1.
-
Four-of-a-Kind pays 25-to-1.
There is some debate on the house edge for this game.
In this article, Elliot Frome gives an "Expert Strategy" he claims leads to a house edge of 1.50%.
Shuffle Master states that the house edge is 1.58% in this document: four-card-poker-SHFL
In Beyond Counting, Grosjean presents an optimal strategy for FCP and claims that by using his strategy, the house edge is 2.840%.
On the Wizard of Odds website it is stated that by using a simple strategy suggested by Stanley Ko, the house edge for FCP is 3.40%. That strategy is to raise 3 with a pair of tens or higher, raise 1 with a pair of twos through nines, otherwise fold.
I wrote a computer program to evaluate the game under computer-optimal strategy. After running a timing test, I determined that this program would take about 14.4 days of computer time to complete a full cycle. I didn’t have the patience to wait around, so I decided to run a Monte Carlo simulation. I choose starting hands at random (players five cards, dealer up-card) to get an approximation of the house edge. I let the simulation run on four threads overnight (about 11 hours). After evaluating 1,505,376 random starting hands, I attained a simulated house edge of 2.91%. This is close to Grosjean’s edge making it clear that Grosjean wins the battle of the house edge for FCP (surprise!).
I also double checked the Ko strategy to raise 3 with a pair of tens or higher, raise 1 with a pair of twos through nines, otherwise fold. He claims this strategy has a house edge of 3.40%. A simulation of 250,000 random starting hands returned a house edge of 3.54%, which supports Ko's result.
There seems to be wide-spread misinformation about the house edge for FCP. The baseline house edge for FCP is not 3.40% (Ko) and it is not 1.58% (Shuffle Master) and it is not 1.50% (Frome). Grosjean gives an exact edge of 2.84%. I get a simulated house edge of 2.91%.
The answer to the (Shuffle Master, Frome, Grosjean, Ko) discrepancy is that Shuffle Master and Frome are considering the house edge based on total player wagers rather than based on the player's initial wager. This is the so-called "element of risk" and is not normally used to describe the house edge. After inquiring, I received an email from Elliot Frome stating that his value is based on the "element of risk." That is, his edge is the fraction of the overall average wagers the plays per round, rather than the fixed initial wager.
Frome also stated that neither GLI or the Nevada Gaming Control Board have had a problem with this value for the edge. While this is no doubt the case, stating the house edge as the element of risk is at best deceptive for the player and useless for the casino when they compute T-Win. My experience with GLI is that they are inconsistent with how they express the edge (for example, blackjack is always based only on the initial bet). My experience with the NGCB is that they do not rigorously double-check GLI.
My expert opinion - GLI and gaming mathematicians should stop using the element of risk to describe the edge. The value of this number is that it gives the player an idea of the experience he will have based on his total average wager. From the casino side, it is at best confusing and at worst can lead to tangible losses for the casino who bases their marketing reinvestment on this number. From the statistical analysis side, it is useless for risk analysis.
I next turned to the question of hole-card play. In this case, the player sees both the dealer’s up-card and one of the dealer’s five face-down cards. In other words, the player has knowledge of two of the dealer’s six cards. In Beyond Counting, Grosjean states his belief that knowing a second dealer card is not enough information to generate a player advantage. He gives a “napkin estimate” of the house edge at about 0.35% in this case.
I wrote a computer program to evaluate the basic hole-card game under computer-optimal strategy. A full cycle would require about 35.9 days of computer time to complete. Again, instead of running a full cycle, I decided to run a Monte Carlo simulation, where I chose starting hands at random to get an approximation to the house edge. After evaluating four million (4,000,000) random starting hands (five player cards, two known dealer cards), I attained a simulated house edge of 0.43%. This is certainly within normal statistical variance of Grosjean’s result. The conclusion is that FCP does not have a hole-card vulnerability if the player only sees one additional card.
I next turned to the situation where the player sees the dealer’s up card and two of the dealer’s hole-cards. In other words, the player has knowledge of three of the dealer’s six cards. There is no discussion of this situation in Beyond Counting. However, it is clear that the player must have an edge over the house in this situation.
I wrote a computer program to completely evaluate the double-hole-card game under computer-optimal strategy. A full cycle would require about 47.8 days of computer time to complete. Again, instead of running a full cycle, I decided to run a Monte Carlo simulation where I chose starting hands at random to get an approximation to the player edge. After evaluating 26,570,000 random starting hands (five player cards, three known dealer cards), I attained the following:
-
The player’s edge is about 4.82%.
-
The player folds about 44.3% of his hands.
-
The player raises 1x on about 24.6% of his hands.
-
The player raises 3x on about 31.1% of his hands.
The question of player strategy in this situation is difficult. Rather than enumerate a strategy, I decided to create a log file and post it. This log file gives full details for one million (1,000,000) randomly chosen starting situations, the EV for optimal play, and the playing decision. In this file, column K contains the evaluation code for the player’s starting hand. This is a value from 1 to 7013 that gives the poker rank of the player’s best four-card hand. I used these codes to greatly speed up the program.
Here is a full description of this file:
-
Columns A, B, C, D, E: player’s five cards.
-
Columns G, H, I: three known dealer cards.
-
Column K: evaluation code for player’s best four card hand.
-
Column L: expected value of best play.
-
Column M: correct player strategy.
Note that the weakest hand the player plays in the log file is:
-
Player: (2D, 6S, TC, JC, KD), Dealer: (2C, 6H, JS), Strategy = raise 1, EV = -0.9302
Finally, I considered the highly unlikely scenario where the player has access to the dealer’s up-card and three dealer hole-cards. After evaluating 56,000,000 random starting hands (five player cards, four known dealer cards), the player’s edge is about 15.82%.
I do not consider FCP to be significantly vulnerable to advantage play. Here are my recommendations for a dealer who is exposing two or more of his hidden cards:
-
Routinely audit dealers for correct procedure.
-
If a game dumps, consider the possibility of dealer/player collusion.
-
A dealer who is exposing multiple hole-cards may be better suited to work in management.
