Last Updated: Oct. 21, 2014

Four Card Poker

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Introduction

Four Poker is a new poker variation invented by Roger Snow and marketed by Shufflemaster. The game is similar to Three Card Poker but as the title suggests, four cards are used instead of three. Also, there is no dealer qualifying hand and the player can raise up to three times his ante. However, the dealer gets one extra card to form his best hand.

Rules

  1. Two initial bets are available: The Ante and the Aces Up.
  2. All players get five cards each and the dealer gets six cards. One of the dealer cards is placed face up, and five face down.
  3. Players making the Ante bet must decide to fold or raise.
  4. If the player folds he forfeits all bets1.
  5. If player raises, then he must raise at least the amount of the Ante and at most, three times the Ante.
  6. The player keeps his best four cards and discards one.
  7. Following is the ranking of hands from lowest to highest: high card, pair, two pair, straight, flush, three of a kind, straight flush, four of a kind.
  8. After all decisions have been made, the dealer will turn over his cards and select the best four out of six.
  9. The player's hand shall be compared to the dealer's hand, the higher hand winning.
  10. If the dealer's hand is higher, then the player shall lose the Ante and Raise.
  11. If the player's hand is higher or equal then the Ante and Raise shall pay one to one.
  12. If the player has at least a three of a kind, then he shall also be paid a Bonus, regardless of the value of the dealer's hand. Two different pay tables are available for the Bonus, as displayed below, and are based on the ante bet. Pay Table 1 is the only one I know of to be actually used.
  13. Another bet is available (similar to the Pairplus in Three Card Poker), based only on the player's four card hand, called the Aces Up. Seven pay tables are available as indicated below. The only one I know of to be actually used is pay table 5.

Notes:
1 I have an unconfirmed report that in Atlantic City if the player folds his Aces Up bet will still remain alive.

Bonus Pay Table

Hand Table 1 Table 2
Four of a kind 25 30
Straight flush 20 15
Three of a kind 2 2

Aces Up Pay Table

Hand Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7
Four of a kind 50 to 1 50 to 1 50 to 1 50 to 1 50 to 1 50 to 1 50 to 1
Straight flush 40 to 1 40 to 1 30 to 1 30 to 1 40 to 1 40 to 1 40 to 1
Three of a kind 9 to 1 7 to 1 9 to 1 7 to 1 8 to 1 8 to 1 7 to 1
Flush 6 to 1 6 to 1 6 to 1 6 to 1 5 to 1 6 to 1 5 to 1
Straight 4 to 1 5 to 1 4 to 1 5 to 1 4 to 1 4 to 1 4 to 1
Two pair 2 to 1 2 to 1 2 to 1 2 to 1 3 to 1 2 to 1 3 to 1
Pair of aces or better 1 to 1 1 to 1 1 to 1 1 to 1 1 to 1 1 to 1 1 to 1

Analysis

The following return table is based on optimal player strategy under the 2-20-25 Ante Bonus pay table. The lower right cell shows a house edge of 2.79%.

Return Table Based on Optimal Strategy

Player Hand Raise/Fold Win/Loss Combinations Probability Pays Return
Four of a Kind 3 Win 40,182,878,736 0.000240 +29 0.006960
Four of a Kind 3 Lose 18,594,576 0.000000 +21 0.000002
Straight Flush 3 Win 133,224,330,456 0.000796 +24 0.019096
Straight Flush 3 Lose 265,177,080 0.000002 +16 0.000025
Three of a Kind 3 Win 3,675,379,352,400 0.021951 +6 0.131703
Three of a Kind 3 Lose 103,559,138,928 0.000618 -2 -0.001237
Flush 3 Win 6,599,621,152,728 0.039415 +4 0.157660
Flush 3 Lose 784,564,849,080 0.004686 -4 -0.018743
Straight 3 Win 5,257,469,039,688 0.031399 +4 0.125597
Straight 3 Lose 1,301,555,952,216 0.007773 -4 -0.031093
Two Pair 3 Win 5,539,444,298,496 0.033083 +4 0.132333
Two Pair 3 Lose 2,420,447,417,280 0.014456 -4 -0.057823
One Pair 3 Win 14,764,551,298,548 0.088179 +4 0.352714
One Pair 3 Lose 10,806,299,820,804 0.064539 -4 -0.258155
One Pair 1 Win 13,535,004,289,296 0.080835 +2 0.161671
One Pair 1 Lose 22,887,448,286,136 0.136691 -2 -0.273382
One Pair Fold Fold 5,495,692,732,992 0.032822 -1 -0.032822
High Card 1 Win 148,058,445,132 0.000884 +2 0.001769
High Card 1 Lose 422,493,233,796 0.002523 -2 -0.005047
High Card Fold Fold 73,523,856,056,112 0.439108 -1 -0.439108
Totals 167,439,136,344,480 1.000000 -0.027879

The average final bet under optimal strategy is 2.142342 units, making the element of risk, -0.027879/2.142342 = 1.30%. The standard deviation, relative to the original bet, is 2.71.

Beginner Strategy

A simple strategy to this game, first proposed by Stanley Ko, is as follows.

  • Raise 3X with a pair of tens or higher.
  • Raise 1X with a pair of twos to nines.
  • Fold all other.

According to the second edition of "Beyond Counting" by James Grosjean, this "simple strategy" results in a house edge of 3.396%.

Intermediate Strategy



The following intermediate strategy was created to balance power and simplicity by our own JB.

  • Pair of Aces or better: Bet 3X
  • Pair of Js, Qs, Ks: Bet 3X if dealer's upcard is lower than your pair or matches a rank in your hand, otherwise bet 1X
  • Pair of 9s, 10s: Bet 1X if dealer's upcard outranks your pair, otherwise bet 3X
  • Pair of 8s: Bet 3X if dealer's upcard is a 2, otherwise bet 1X
  • Pair of 3s, 4s, 5s, 6s, 7s: Bet 1X
  • Pair of 2s or AKQ: Bet 1X if dealer's upcard matches a rank in your hand, otherwise fold
  • All other: Fold

Against the 2-20-25 Ante Bonus pay table, the house edge is 2.8526% and the element of risk is 1.3233%.

Advanced Strategy



I'm proud to present the following advanced strategy, also created by my sidekick JB.

  • Pair of Aces or better: Bet 3X
  • Pair of Ks: Bet 3X, except bet 1X against an Ace and you don't have an Ace nor 4.
  • Pair of Js or Qs: Bet 3X, except bet 1X if the dealer's card outranks pair your pair rank and does not match a singleton in your hand.
  • Pair of 9s or 10s: Bet 3X, except bet 1X if dealer card outranks your pair rank.
  • Pair of 8s: Bet 1X, except bet 3X against a 2
  • Pair of 4s thru 7s: Bet 1X
  • Pair of 3s: Bet 1X, except fold against a Jack if your highest kicker is a 10 or lower
  • Pair of 2s or AKQ: Fold, except bet 1X if dealer card matches a rank in your hand
  • AKJT: Fold, except bet 1X against a Jack
  • AKJ9 or lower: Fold

Against the 2-20-25 Ante Bonus pay table, the house edge is 2.8498% and the element of risk is 1.3216%. Here is a house edge comparison of various known strategies.

  • Simple: 3.396%
  • Intermediate: 2.853%
  • Advanced: 2.850%
  • Optimal: 2.788%

To put it another way, here are the cost of errors:

  • Simple: 0.606%
  • Intermediate: 0.065%
  • Advanced: 0.062%
  • Optimal: 0.000%

Aces Up Analysis



The next table shows the probability of each hand and the return under pay table 5 of the Aces Up side bet. I chose table 5 because that is the only one I knew to be in play at the time of this writing. The lower right cell shows a house edge of 3.89%.

Return for Aces Up Pay Table 5

Hand Combinations Probability Pays Return
Four of a kind 624 0.00024 50 0.012005
Straight flush 2072 0.000797 40 0.03189
Three of a kind 58656 0.022569 8 0.180552
Flush 114616 0.044101 5 0.220504
Straight 101808 0.039173 4 0.15669
Two pair 123552 0.047539 3 0.142617
Pair of aces 81096 0.031203 1 0.031203
Nothing 2116536 0.814378 -1 -0.814378
Total 2598960 1 -0.038917

The next table shows the house edge according to all four Aces Up pay tables.

Aces Up House Edge

Pay Table House Edge
1 1.98%
2 2.58%
3 2.78%
4 3.37%
5 3.89%
6 4.24%
7 6.15%

Note: There is also a similar game called Crazy Four Poker.

Acknowledgments

I would like to recognize:

  • JB for the analysis of the optimal strategy.
  • Stanley Ko for the simplified strategy.
  • James Grosjean for the unpublished advanced strategy.

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