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Triple Action Hold ’Em

Last Update: Aug 05, 2010

Introduction

I noticed Triple Action Hold 'Em at Binion's Horseshoe on May 1, 2009. It is also offered at Internet casinos using Cryptologic software. It is a fairly simple poker based game, where you go against the dealer. What makes this game different from the recent batch of poker variants is this one is based on five cards, and uses only a 28-card deck.

Rules

A modified 28-card deck is used, consisting only of the eights to aces.
Play starts with three bets available to the player, the Ante, Flop Bet, and Bonus Bet.
The dealer will deal each player, and himself, two hole cards. In addition, there is one three-card flop, which is dealt face down. The players' hole cards are dealt face down, while the dealer's cards are dealt one up and one down.
The player must either make a Raise bet or fold. The Raise bet must be equal to the Ante bet. If the player folds, then he loses his Ante wager.
The dealer turns exposed his other hole card and the flop.
If the dealer's five-card poker hand is a pair of nines or greater, then he shall qualify.
If the dealer does not qualify, then the ante bet shall pay even money, and the raise shall push.
If the dealer does qualify, then the player's five-card poker hand will be compared to the dealer's; the higher hand wins. If the player has the higher hand, then the Ante and Raise shall pay even money. If the dealer has the higher hand, then the Ante and Raise shall lose. An exact tie results in a push.
If the player did not fold, and his hand is a full house or better, then he shall be paid an Ante Bonus, according to the pay table below, which is based on the amount of the Ante wager. It is not required to beat the dealer for purposes of the Ante Bonus.
Regardless of who wins between the player and dealer, the Flop and Bonus bets shall pay according to the pay tables below. I presume that if the player folds, he still can win the Bonus bet.

Ante Bonus Pay Table
Five-Card Hand	Pays
Royal flush	50 to 1
Straight flush	20 to 1
Flush	6 to 1
Four of a kind	4 to 1
Full house	2 to 1
All other	None

Five-Card Hand	Pays
Flop Bet Pay Table
Straight flush	10 to 1
Three of a kind	10 to 1
Flush	4 to 1
Straight	2 to 1
Jacks or better	1 to 1
All other	Loss

Five-Card Hand	Pays
Bonus Bet Pay Table
Royal flush	100 to 1
Straight flush	50 to 1
Flush	25 to 1
Four of a kind	15 to 1
Full house	8 to 1
Straight	6 to 1
Three of a kind	3 to 1
Two pair	1 to 1
All other	Loss

Strategy

The optimal strategy on the Ante bet is to always raise. Of the three bets, the element of risk is lowest on the Ante bet, at 1.82%. I often get asked about the optimal mixture of bets for games like this. In this case, it is 100% on the Ante, and 0% on the Flop and Bonus bets.

Ante Bet Analysis

The return table below shows the return before factoring in the Ante Bonus.

Event	Pays	Combinations	Probability	Return
Ante Bet Analysis
Win	2	68426184	0.275193	0.550385
Dealer doesn't qualify	1	65881200	0.264957	0.264957
Push	0	4053024	0.0163	0
Lose	-2	110287992	0.44355	-0.8871
Totals		248648400	1	-0.071758

The next table shows the contribution to the return of the Ante Bonus.

Event	Pays	Combinations	Probability	Return
Ante Bonus Analysis
Royal flush	50	4	0.000041	0.002035
Straight flush	20	8	0.000081	0.001628
Flush	6	72	0.000733	0.004396
Four of a kind	4	168	0.001709	0.006838
Full house	2	1008	0.010256	0.020513
All other	0	97020	0.987179	0
Total		98280	1	0.035409

The overall return on the Ante bet is -0.071758 + 0.035409 = -0.036349. So the house edge is 3.63%. The player should always raise, so the average final bet is two units, making the 3.63%/2 = 1.82%.

Flop Bet Analysis

Five-Card Hand	Pays	Combinations	Probability	Return
Flop Bet Analysis
Straight flush	10	20	0.006105	0.06105
Three of a kind	10	28	0.008547	0.08547
Flush	4	120	0.03663	0.14652
Straight	2	300	0.091575	0.18315
Jacks or better	1	576	0.175824	0.175824
All other	-1	2232	0.681319	-0.681319
Total		3276	1	-0.029304

Bonus Bet Analysis

Five-Card Hand	Pays	Combinations	Probability	Return
Bonus Bet Analysis
Royal flush	100	4	0.000041	0.00407
Straight flush	50	8	0.000081	0.00407
Flush	25	72	0.000733	0.018315
Four of a kind	15	168	0.001709	0.025641
Full house	8	1008	0.010256	0.082051
Straight	6	3060	0.031136	0.186813
Three of a kind	3	6720	0.068376	0.205128
Two pair	1	15120	0.153846	0.153846
All other	-1	72120	0.733822	-0.733822
Total		98280	1	-0.053887

Appendix

A large table of expected value figures for each initial combination of player cards and dealer card can be found in my Triple Action Hold 'Em appendix.

Acknowledgment

My thanks to J.B. and Charles R. Mousseau, a casino games analyst in Winnipeg, Canada for their help with the analysis on this one.