Six Card Poker

Last Update: May 4, 2012

Rules

1. Six Card Poker is a poker variant game played heads up against the dealer. The object of the game is to win by having a better five-card poker hand than the dealer using six cards.
2. The game is played using one standard 52-card deck. The game uses standard poker rules for scoring and comparing hands.
3. To begin, the player makes an Ante Bet.
4. The dealer deals six cards face down to the player. The dealer deals to himself, three cards face up and three cards face down.
5. The player examines his cards and must either fold the hand, losing his Ante, or raise by making an additional wager that is equal to his Ante. Players may not share information about their cards with other players.
6. The dealer reveals his hole cards and compares his best five-card hand to the player’s best five-card hand. The dealer must qualify with ace-king or higher.
7. If the dealer does not qualify, then the player’s Ante bet is a push. The raise bet is resolved as follows:
   • If the dealer's hand beats the player's hand, then the player loses his raise bet.
   • If the player's hand beats the dealer's hand, then the player is paid 1 to 1 on his raise bet.
   • In the event of a tie, the player’s raise bet is a push.
8. If the dealer does qualify:
   • If the dealer's hand beats the player’s hand, then the player loses his Ante and raise bets.
   • If the player's hand beats the dealer’s hand, then the player is paid 1 to 1 on his Ante and raise bets.
   • In the event of a tie, the player’s Ante and raise bets push.
9. There are also two optional side bets, the Aces Up and Two-Way Bad Beat, which are explained below.

Analysis

The game was analyzed through brute force, combinatorial analysis. The following table summarizes the analysis results. The bottom left cell shows the house edge of 1.2717%. With an average bet size of 1.6987, the element of risk is 0.7486%.

Ante Bet Analysis
Outcome	Dealer Qualifies	Net Win	Combinations	Probability	Frequency	Return
Player Wins	yes	2	1,040,023,036,938,352	0.272693	1 in 3.7	0.545387
Player Wins	no	1	755,514,209,224,548	0.198095	1 in 5.0	0.198095
Ties	yes	0	210,847,746,240	0.000055	1 in 18,088	0.000000
Ties	no	0	50,992,983,876	0.000013	1 in 74,792	0.000000
Fold	-	-1	93,119,328,972	0.301316	1 in 3.3	-0.301316
Dealer Wins	no	-1	2,937,744,249,924	0.000770	1 in 1,298	-0.000770
Dealer Wins	yes	-2	865,968,968,971,212	0.227057	1 in 4.4	-0.454113
Total				1		-0.012717

Strategy

Two basic playing strategies are presented below.

Intermediate basic strategy:

1. Fold if dealer has a higher ranked hand using his three up cards alone. This rule supersedes all other rules.
2. Fold with ace-10 or lower.
3. Raise with ace-king or higher.
4. With ace-queen, fold if the dealer is showing an ace or king.
5. With ace-jack, fold if the dealer is showing an ace, king, or queen.

Simple basic strategy:

1. Fold if dealer has a higher ranked hand using his three up cards alone. This rule supersedes all other rules.
2. Raise with ace-king. Fold otherwise.

• With optimal play, the house edge is 1.27%.
• Using the intermediate basic strategy, the house edge is 1.48 %.
• Using the simple basic strategy, the house edge is 1.63%.

Aces Up

Aces Up is a side bet based on the poker-value of the player's hand only. The following table shows what each hand pays, the probability, and contribution to the return. The lower right cell reflects a house edge of 3.30%.

Aces Up Analysis
Event	Pays	Combinations	Probability	Return
Royal flush	500	188	0.000009	0.004617
Straight flush	100	1,656	0.000081	0.008134
Four of a kind	30	14,664	0.000720	0.021609
Full house	10	165,984	0.008153	0.081530
Flush	8	205,792	0.010108	0.080867
Straight	6	361,620	0.017763	0.106576
Three of a kind	4	732,160	0.035963	0.143853
Two pair	2	2,532,816	0.124411	0.248821
Aces	1	751,332	0.036905	0.036905
Loser	-1	15,592,308	0.765886	-0.765886
Total		20,358,520	1.000000	-0.032973

Two-Way Bad Beat

The "Two-Way Bad Beat" pays if either you or the dealer has at least a pair of aces and loses. The higher the losing hand, the more the bet pays. In other words, it pays based on the worse hand between you and the dealer, as long as they don't tie.

The following table shows all the odds. The zeros for the probability and return of a losing straight flush are not a mistake. The probability of that is about 1 in 25 trillion. Such a situation was the basis of the movie Honeymoon in Vegas, but I'm getting off topic. The lower right cell of the table below shows a house edge of 10.84%.

Two-Way Bad Beat Analysis
Event	Pays	Probability	Return
Straight Flush	10000	0.000000	0.000000
Four of a Kind	5000	0.000001	0.006250
Full House	500	0.000099	0.049370
Flush	200	0.000307	0.061440
Straight	100	0.000972	0.097168
Three of a Kind	35	0.004056	0.141965
Two Pair	10	0.034513	0.345127
Pair of Aces	9	0.015036	0.135328
Loser	-1	0.945016	-0.945016
Total		1.000000	-0.108367

Acknowledgments

Usually, I like to do my own math. However, in this case ShuffleMaster , who is marketing the game, kindly provided the math report by mathematician Cindy Liu, which the Ante bet analysis is based on. The Two Way Bad Beat is based on the work of Elliot Frome. I did the analysis of Aces Up.