Discount Gambling

I noticed another new AGS game in my nearby Harrah’s Casino, called Premium Hold’Em, extending their products of 3-hole card poker games against the dealer.  I kind of enjoy squeezing these hole cards, catching minimal glimpses of colors and symbols, and making (correct) decisions without knowing my full hand.  Sometimes, this leaves me with emergency outs after seeing the dealer’s hand.

Plus, once you’ve played enough Ultimate Texas Hold’Em (UTH), all those decisions become automatic, and all you seem to notice is that string of miracle dealer hands which constantly scoop the table.  UTH can be boring *and* painful.

So, what’s new with AGS’s Premium Hold’Em?  Well, first, there’s the novelty and the squeezing of your three hole cards.  Next, the all-in preflop raise is lowered to 3x, which should reduce the variance a bit (from UTH’s 4x).  Then, the entire community board is dealt out at once as a four card “flop”.  Finally, you can either 2x, 1x, or fold after seeing your entire hand!  (Compare to UTH’s 2x decision before the turn+river.)  Of course, the dealer has three unseen hole cards, but the 2x option is still more favorable than in UTH.  Overall, these advantages are offset by the lowered 3x preflop raise, and the higher pair-of-treys Dealer qualifier.  Bottom lime, the house edge of Premium Hold’Em is lower than UTH, at 2.06% of an Ante.

Oh, and the bonus side bet pays down to Jacks Up (two pairs), and has a low house edge of 4.87%.  Again, this offers lower variance than the UTH Trips side bet.

Basic Strategy

I wanted to try out the game, so I worked out a Basic Strategy.  I worked it down to the detail that I like, using hand features that make sense to me.  The table is organized by the two decision points, and divided into mutually exclusive categories for the Player hand, and also for the board.  For each determined sub-category, the selected betting rules should be followed top-down, looking for the first matched condition, or else falling through to the bottom action.

The betting rules are meant to be concise and unambiguous, but I’ll elaborate on some of them below, to clarify with examples.

The Basic Strategy is sub-optimal by only 0.28% from the ideal 2.06% house edge.

You can practice the game for free at the AGS website.

Typical play is actually pretty easy, and the Basic Strategy is intuitive and easy to learn.  The top part of the table tells you what the 3x preflop raising hands are.  They occur about 15.7% of the time, and are easy to remember.

All the river decisions for hands Three-of-a-Kind and higher are simple and intuitive.  Thankfully, you should 2x bet *all* full houses and flushes, independent of the board.  The only time you slow down with a straight is when you 1x it against a 4-flush board.  Notice you still 2x the idiot end straight against an open-ended board, and you still 2x a baby flush against a 4-flush board.  And you still 2x a flush against a double-paired board.

You’ll slow down with Three-of-a-Kind when they’re on board.  You will 2x them with a nut (best) kicker in the hole.  For example, say you’re holding Kh 9s 6h, and there’s trips on board with 8s 8c 8h Ad.  You have the nut (best) kicker in the hole (Kh), so you can still 2x your hand.  However, if your hole cards were Qh 9s 6h, you should only 1x the hand since your Qh is only 2nd nut kicker.  You could still 1x call with Jh 9s 6h (3rd nut kicker), but you would fold with anything lower than the 3rd nut kicker.

If you have Three-of-a-Kind and the board is a scare flush (4-flush), then you can still 1x your hand.  For example, if your hole cards were 7d 3s 3h and the board was 3c Kc Jc 6c, you’d only 1x your trip 3’s.

You’ll generally 2x your two pairs, unless the board is dangerous.  So, against a scare flush (4-flush) or a scare straight (open-ended) board, you’ll just 1x your two pairs.  You should 2x raise your two pairs any time you can beat a double-paired board, or even if you just have the nut kicker.  If the board is paired, you’ll 2x raise your top pair or over pair.  You can 2x raise with bottom pair, just as long as you have 3rd nut kicker or better.  You’ll never fold two pairs when you beat the board.  The only time you’ll fold with two pairs is when you’re playing the board, and don’t have 4th nut kicker or better.

Note: “bottom pair” on a paired board means you’ve paired the lowest singleton on the board.  For example, say the board is 3s 3h Kd 7h.  If you’re holding a 7, you’ve made bottom pair on the board.  If you’re holding a K, you’ve made top pair on the board.  If you have a pair of wired Aces, they’d be an over pair to the board.  If you have a pair of wired Deuces, they’d be an under pair to the board. Also, if you held a pair of wired Fives in the hole, they’d be an under pair to the board too (lowest board singleton is a Seven).

There are a few occasions when you can 2x raise one pair against a safe board (no 4-flush or 4-straight board, including gutshot board).  Against a safe board, you can 2x raise one pair if it’s an over pair, top pair, or second pair.

Generally speaking, you’ll at least 1x call any pair made with the board.  You will fold 3rd or bottom pair against a scare board.  You’ll only call a wired under pair to the board if it’s qualifying, and there’s no 4-flush or 4-straight on board.

You can 1x call playing one pair on board, if you have the nut kicker.  You can also 1x call the pair on board if you hold the 2nd nut kicker, and the board is 2-flush or less.

There’s only one case to 1x bet a no-pair hand on the river.  You should 1x play an Ace-King high hand against a rainbow flop when holding the nut kicker, and the board isn’t 4-cards to a straight.  For example, if your hole cards were Qh 8d 3s and the board was Ac Kh 9d 4s, then you should 1x your hand, because your Qh hole card is the nut kicker.

Collusion Analysis

The game is not vulnerable to even 5 confederates colluding at a full-table sharing perfect information.  Ideal decisions using all known player hole cards (15 of them at a full table) yield only about a +1.4% improvement over Basic Strategy (about +0.5% from counter-strategy pre-flop decisions, and about +0.9% from counter-strategy river decisions).

Bonus Side Bet

An optional bonus on your final hand, or the Dealer’s final hand, is available before the hand begins.  The table below shows the payouts and their frequencies and returns.  The house edge is a reasonable 4.87%, and it pays down to Jacks Up (two pairs), instead of the UTH Trips or better bonus.

Detailed Stats

The total outcomes for every possible starting hand, for every possible flop, and every possible dealer hole cards are shown in the table below, following optimal decisions.  The player will 3x about 15.7% of his hands, will 2x about 37.4% of his hands, will 1x about 23.9% of the hands, and will fold about 23.0% of the hands.