Discount Gambling

Premium Hold’Em

Posted in Uncategorized by stephenhow on June 6, 2018

Premium-Holdem-thumb

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.

Premium Hold’Em Basic Strategy
Decision Player Hand Board Prioritized Betting Rules
Preflop

(3x All-In

or

Check)

Trips 3x all Three-of-a-Kind hands.
Pair 3x pair of FIVE’s or better, else

check all others.

3-Suited 3x A-T-x all suited or higher, else

3x three suited cards all NINE or higher, else

check all others.

other 3x A-J-T ranks or all higher, else

check all others.

River

(2x, 1x, or Fold)

Royal Flush,

Straight Flush

2x all hands
Four-of-a-Kind
quads on board 2x with 1st or 2nd nut kicker, else

1x with 7th nut kicker or better, else

fold all others

other 2x all hands
Full House 2x all Full Houses
Flush 2x all Flushes
Straight
4-flush 1x Straight against 4-flush board
other 2x all Straights
Three-of-a-Kind
trips on board 2x trips on board with nut kicker in the hole, else

1x trips on board with 2nd or 3rd nut kicker in the hole, else

fold all others

4-flush 1x Three-of-a-Kind against scare flush
other 2x all hands
Two Pairs
double-paired 2x two better pairs, else

2x nut kicker, else

1x 2nd or 3rd nut kicker, else

fold all others

paired 2x “top pair” or “overpair”, else

2x “bottom pair” with 3rd nut or better kicker, else

1x all others

4-flush or

open-ended

1x all hands vs scare board
other 2x all hands
One Pair
pair on board 1x with nut kicker in the hole, else

1x with 2nd nut kicker in the hole and 2-flush max board, else

fold all others

4-flush or

open-ended

1x 2nd pair or better, else

fold all others

gutshot 1x bottom pair or better, else

fold all other underpairs

other 2x 2nd pair or better, when SIX’s or better, else

1x bottom pair or better, else

1x qualifying underpair, else

Fold all other underpairs

High Card
4-flush or

4-straight

Fold all hands
other 1x Ace-King with nut kicker in the hole vs rainbow board, else

fold all others

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.

Outcome Combinations Frequency Payout Return
ROYAL_FLUSH

4,324

0.000032

50

0.001616

STRAIGHT_FLUSH

37,260

0.000279

30

0.008355

FOUR_OF_A_KIND

224,848

0.001681

10

0.016807

FULL_HOUSE

3,473,184

0.025961

6

0.155766

FLUSH

4,047,644

0.030255

4

0.121020

STRAIGHT

6,180,020

0.046194

3

0.138581

THREE_OF_A_KIND

6,461,620

0.048299

2

0.096597

Two Pairs (J’s Up+)

173,854,08

0.129951

1

0.129951

other

95,970,252

0.717349

-1

-0.717349

total

133,784,560

1.000000

-0.048656

expected

133,784,560

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.

Outcome Combinations Frequency Net Return
Win 3x Play w/ ROYAL_FLUSH vs. qualified dealer

879,320,772

0.000013

504

0.006670

Win 3x Play w/ STRAIGHT_FLUSH vs. qualified dealer

2,149,315,348

0.000032

104

0.003364

Win 3x Play w/ FOUR_OF_A_KIND vs. qualified dealer

55,353,208,088

0.000833

14

0.011663

Win 3x Play w/ FULL_HOUSE vs. qualified dealer

602,898,278,304

0.009074

7

0.063516

Win 3x Play w/ FLUSH vs. qualified dealer

237,450,980,932

0.003574

5.5

0.019655

Win 3x Play w/ STRAIGHT vs. qualified dealer

191,445,164,564

0.002881

5

0.014406

Win 3x Play w/ THREE_OF_A_KIND vs. qualified dealer

575,993,062,524

0.008669

4

0.034675

Win 3x Play w/ TWO_PAIRS vs. qualified dealer

2,246,740,264,764

0.033814

4

0.135256

Win 3x Play w/ ONE_PAIR vs. qualified dealer

1,066,642,147,548

0.016053

4

0.064213

Lose 3x Play vs. qualified dealer

3,231,042,644,372

0.048628

-5

-0.243140

Tie 3x Play vs. qualified dealer

30,011,251,504

0.000452

0

0.000000

Win 3x Play w/ ROYAL_FLUSH vs. non-qualified dealer

201,445,188

0.000003

503

0.001525

Win 3x Play w/ STRAIGHT_FLUSH vs. non-qualified dealer

430,781,892

0.000006

103

0.000668

Win 3x Play w/ FOUR_OF_A_KIND vs. non-qualified dealer

3,718,604,400

0.000056

13

0.000728

Win 3x Play w/ FULL_HOUSE vs. non-qualified dealer

87,580,605,564

0.001318

6

0.007909

Win 3x Play w/ FLUSH vs. non-qualified dealer

65,138,805,780

0.000980

4.5

0.004412

Win 3x Play w/ STRAIGHT vs. non-qualified dealer

62,396,153,160

0.000939

4

0.003756

Win 3x Play w/ THREE_OF_A_KIND vs. non-qualified dealer

292,929,250,596

0.004409

3

0.013226

Win 3x Play w/ TWO_PAIRS vs. non-qualified dealer

508,732,581,216

0.007657

3

0.022970

Win 3x Play w/ ONE_PAIR vs. non-qualified dealer

1,045,415,883,192

0.015734

3

0.047201

Win 3x Play w/ HIGH_CARD vs. non-qualified dealer

71,345,648,256

0.001074

3

0.003221

Lose 3x Play vs. non-qualified dealer

22,949,828,544

0.000345

-4

-0.001382

Tie 3x Play vs. non-qualified dealer

1,115,495,892

0.000017

0

0.000000

Win 2x Play w/ ROYAL_FLUSH vs. qualified dealer

902,421,300

0.000014

503

0.006832

Win 2x Play w/ STRAIGHT_FLUSH vs. qualified dealer

13,145,193,656

0.000198

103

0.020377

Win 2x Play w/ FOUR_OF_A_KIND vs. qualified dealer

48,452,947,496

0.000729

13

0.009480

Win 2x Play w/ FULL_HOUSE vs. qualified dealer

891,034,206,768

0.013410

6

0.080462

Win 2x Play w/ FLUSH vs. qualified dealer

1,208,401,617,832

0.018187

4.5

0.081840

Win 2x Play w/ STRAIGHT vs. qualified dealer

1,833,221,994,724

0.027590

4

0.110362

Win 2x Play w/ THREE_OF_A_KIND vs. qualified dealer

1,548,010,040,012

0.023298

3

0.069894

Win 2x Play w/ TWO_PAIRS vs. qualified dealer

5,325,367,541,112

0.080148

3

0.240444

Win 2x Play w/ ONE_PAIR vs. qualified dealer

2,347,190,713,200

0.035326

3

0.105977

Lose 2x Play vs. qualified dealer

5,884,001,322,956

0.088556

-4

-0.354223

Tie 2x Play vs. qualified dealer

356,306,998,604

0.005363

0

0.000000

Win 2x Play w/ ROYAL_FLUSH vs. non-qualified dealer

164,326,140

0.000002

502

0.001242

Win 2x Play w/ STRAIGHT_FLUSH vs. non-qualified dealer

2,738,580,240

0.000041

102

0.004204

Win 2x Play w/ FOUR_OF_A_KIND vs. non-qualified dealer

1,966,734,000

0.000030

12

0.000355

Win 2x Play w/ FULL_HOUSE vs. non-qualified dealer

49,091,947,236

0.000739

5

0.003694

Win 2x Play w/ FLUSH vs. non-qualified dealer

339,829,796,400

0.005115

3.5

0.017901

Win 2x Play w/ STRAIGHT vs. non-qualified dealer

618,260,337,144

0.009305

3

0.027915

Win 2x Play w/ THREE_OF_A_KIND vs. non-qualified dealer

144,904,859,028

0.002181

2

0.004362

Win 2x Play w/ TWO_PAIRS vs. non-qualified dealer

1,898,618,140,944

0.028575

2

0.057149

Win 2x Play w/ ONE_PAIR vs. non-qualified dealer

2,363,506,427,088

0.035571

2

0.071143

Win 1x Play w/ FOUR_OF_A_KIND vs. qualified dealer

349,955,952

0.000005

12

0.000063

Win 1x Play w/ STRAIGHT vs. qualified dealer

6,321,867,792

0.000095

3

0.000285

Win 1x Play w/ THREE_OF_A_KIND vs. qualified dealer

55,773,134,484

0.000839

2

0.001679

Win 1x Play w/ TWO_PAIRS vs. qualified dealer

677,790,208,140

0.010201

2

0.020402

Win 1x Play w/ ONE_PAIR vs. qualified dealer

1,790,737,887,912

0.026951

2

0.053902

Lose 1x Play vs. qualified dealer

10,014,543,966,612

0.150721

-3

-0.452164

Tie 1x Play vs. qualified dealer

195,881,895,144

0.002948

0

0.000000

Win 1x Play w/ STRAIGHT vs. non-qualified dealer

1,944,063,612

0.000029

2

0.000059

Win 1x Play w/ THREE_OF_A_KIND vs. non-qualified dealer

521,670,744

0.000008

1

0.000008

Win 1x Play w/ TWO_PAIRS vs. non-qualified dealer

57,782,013,408

0.000870

1

0.000870

Win 1x Play w/ ONE_PAIR vs. non-qualified dealer

2,716,947,685,944

0.040891

1

0.040891

Win 1x Play w/ HIGH_CARD vs. non-qualified dealer

218,296,927,728

0.003285

1

0.003285

Lose 1x Play vs. non-qualified dealer

142,599,511,272

0.002146

-2

-0.004292

Tie 1x Play vs. non-qualified dealer

21,633,803,136

0.000326

0

0.000000

Fold

15,265,300,263,840

0.229747

-2

-0.459493

total

66,444,101,724,000

1.000000

-0.020583

expected

66,444,101,724,000

 

Chase The Flush

Posted in Uncategorized by stephenhow on May 21, 2018

Chase-the-Flush-thumb

Ok, I’ll just come out and say it.  AGS’s (relatively) new Chase-the-Flush game “Makes Gambling Great Again” 🙂  I actually did the game development math on this a few years ago, when I wasn’t gambling much.  But this weekend I discovered a table at my local Hollywood Jamul Casino, so I worked out the Basic Strategy.  And guess what?  It’s a fun and elegant game!  It has a similar structure to Ultimate Texas Hold’Em, with an equal Ante and “Blind” (called the X-Tra Bonus), and 3x/2x/1x Play decisions.  But, the game is much more fun, because of the novel flush decisions, and because it’s a lot less frustrating and dread-inducing than the loved/hated UTH.  (Hint: the X-Tra pays off more frequently than the Blind, and a 3x pre-flop raise doesn’t miss the board as often as a 4x UTH bet.)

The Rules

The layout below shows the betting spots and payouts of the main game, and the pay table of the optional, independent Same Suit Bonus.  The game is played with a standard 52-card deck, where each player and the dealer receives three hole cards and shares a four-card community board to make their best-of-seven flush hand.

Chase-the-Flush-layout

The Player wagers an Ante and equal X-Tra Bonus bet before the hand begins.  Each Player and the Dealer receive three hole cards.  A Player may wager a 3x All-In bet based on his hole cards, or he may check and see the two-card flop.  The flop community cards are dealt, and a previously checking Player may now wager a 2x All-In bet, or check again.  The final two community board cards are dealt, and a previously checking Player must now wager a 1x All-In bet, or else fold his hand.

Each Player and the Dealer forms the highest flush made from their three hole cards and the four card community board.   The Dealer qualifies with a 3-card Nine-high flush or better.  If the Dealer doesn’t qualify, the remaining Antes are pushed back to the Player.  The qualified Antes and the All-In bet then receive even-money action against the Dealer hand.  The Player must beat the Dealer to receive the X-Tra Bonus payout listed in the pay table.  If the Dealer’s hand beats the Player’s hand, the X-Tra Bonus loses.  All bets push on a tie.

The House Edge

The house advantage for the main Chase-the-Flush game is only 2.65% of an Ante.  That’s very reasonable, and is comparable to the UTH house edge.  The Basic Strategy yields a practical -2.99% return to the player.  The Same Suit Bonus for the pay table in the above layout has a reasonable house edge of 5.67%.

Basic Strategy

I crafted out as simple a Basic Strategy as possible, in terms of how people intuitively view their hands during the game.  The following strategy shows the betting conditions for each of the 3x (pre-flop), 2x (flop), and 1x (river) decision points.  Check your hand for any of the betting requirements listed per decision point.  If your hand doesn’t match any of the listed conditions for the decision point, then you shouldn’t bet it.

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

Decision Betting Requirements
3x

(“pre-flop”)

Pair of Aces with Four+ kicker, or K-K-K
Three suited cards
Suited Ace
Suited King w/ Six+ singleton
Suited Queen w/ Queen+ singleton
Suited Jack w/ King+ singleton
Suited Ten w/ Ace singleton
A-Q-T or higher ranks
Check all others
2x

(“flop”)

4-card flush or better
3-card flush w/ 2-card flush
3-card flush vs offsuit board
3-card flush using suited board with Six+ kicker (hole card)
Nut 2-card flush using hole A or K plus another 2-card flush
Two 2-card flushes using board w/ Jack+ average hole cards
Check all others
1x

(“river”)

4-card flush or better
3-card flush vs rainbow board
3-card flush w/ less than 15 one-card beats
2-card flush w/ less than 10 one-card beats
Fold all others
Chase-the-Flush Basic Strategy

For optimal play, you’ll 3x raise about 23.8% of your hands, bet 2x on the flop about 24.9% of the time, 1x call on the river about 35.2% of the time, and otherwise fold about 16.1% of the time.

3x Pre-flop Examples

You should 3x Play any suited Ace.  For example, Ac-2c-2d has a EV(3x) of +68.6% and an EV(check pre-flop) of +59.9%.  So, it’s still worth +8.7% to 3x raise the hand instead of checking it.

You should 3x any pair of Aces with a Four or better kicker.  However, the hand is only +EV for A-A-6 or higher.

You should 3x a suited King with a Six or higher singleton (i.e., the offsuit card).  For example, Kd-9d-6c has a EV(3x) of +45.2% and an EV(check pre-flop) of +43.3%, showing it’s slightly better to 3x raise the hand than check it.

You should 3x raise a rainbow hand if each of the ordered ranks are higher than, or equal to, A-Q-T.  This means you should 3x A-Q-T, A-K-T, A-A-T, A-Q-J, etc.  You should check A-J-T, A-K-6, A-J-J, etc.

2x Flop Examples

The Basic Strategy bets almost all 3-card flushes on the flop.  The only exception is when the board is suited, and you’re using a hole card less than a Six to make the flush, AND you don’t have another 2-card flush.  Otherwise, you’re betting all other 3-card flushes (or better).  For example, say you’re holding Kh-7d-5c and the flop is Ac-2c.  You shouldn’t bet your 3-card flush, because your 5c kicker is less than a Six.  Notice however, if you were instead holding Kh-7h-5c, you’d 2x Play your 3-card club flush with Five kicker, because you also have a two card Kh-7h flush.

You can 2x bet a 2-card “nut” flush when you have any another 2-card flush.  For example, if you’re holding Ac-6d-5h and the flop is 5c 7d, you have the 2-card “nut” (i.e., highest possible) flush in clubs, along with another 2-card flush (7d-6d).  You should 2x Play this hand, because one of your hole cards makes the “nut” 2-card flush with a board card, and your hand makes another 2-card flush.  Note you shouldn’t bet your Ac-7d-6d hand with a board of 8c-Ah, because many single dealer heart cards (Nine or higher) beat your 2-card flush.

You can also bet two 2-card flushes that use both offsuit board cards with two hole cards averaging a Jack or higher.  For example, you can 2x Play your Kh-Qs-2d when the board flops a heart and a spade.

1x River Examples

The Basic Strategy bets almost all 3-card flushes on the river.  The only exception is when you’re playing a single small hole card to make your hand, and the board is double-suited.  In most of these cases, there are 15 or more single dealer cards that’ll beat your hand.

Otherwise, if the board is rainbow, you’ll always 1x Play any 3-card flush.

If the board has only two cards of one suit, and you have any 3-card flush, there are always less than 15 single dealer cards that’ll beat your hand, so you’ll always 1x play any 3-card flush.

If the board has a 3-card flush on board, you’ll 1x Play the board since Basic Strategy says to always call when there are less than 15 single dealer cards that’ll beat your hand (there are only 10 remaining cards of the flush suit).  However, you can get a little fancy, and fold if the board singleton is higher than the 3-card flush AND you don’t hold any cards of the singleton suit.

You can play a very high 2-card flush against a rainbow board if there are less than 10 single dealer cards that’ll beat your hand.  This usually means you can play a very high 2-card flush using the highest board card if it’s not paired on board.  For example, if the board is 9s-7h-6d-5c, you can 1x Play a Kh in the hole, since the only single dealer cards that will beat your Kh-7h is an As, Ks, Ah, Ad, Ac (5 of them).

Same Suit Bonus

While straight flushes don’t have any meaning in the main game, they are included in the pay table (along with 4+ card regular flushes) in the optional Same Suit Bonus bet.  The resulting payouts are very attractive, and add a nice dimension to the game.  The breakdown of the 7-card hand outcomes is listed in the table below, and show a total house edge of 5.67% (good as far as bonuses go).

Outcome Combinations Frequency Payout Return
6-or-7 Card Straight Flush 1,624 0.000012 2000 0.024278
5 Card Straight Flush 39,312 0.000294 100 0.029385
4 Card Straight Flush 636,272 0.004756 20 0.095119
7 Card Flush 6,664 0.000050 300 0.014899
6 Card Flush 256,620 0.001918 50 0.095908
5 Card Flush 3,550,872 0.026542 10 0.265417
4 Card Flush 25,735,424 0.192365 1 0.192365
Nothing 103,557,792 0.774064 -1 -0.774064
Total 133,784,560  1.000000 -0.056694

Optimal Play Statistics

The following table breaks down the total outcomes for the main Chase-the-Flush game, over all possible starting hands, using optimal decisions.  The total return in the lower right corner shows a house edge of 2.65% of the Ante.

Outcome Combinations Frequency Net Return
Win 3x Play w/ 7-card flush against qualified dealer

20,439,619,200

0.000051

204

0.010459

Win 3x Play w/ 6-card flush against qualified dealer

534,992,418,432

0.001342

24

0.032207

Win 3x Play w/ 5-card flush against qualified dealer

4,296,578,849,136

0.010777

9

0.096997

Win 3x Play w/ 4-card flush against qualified dealer

16,130,726,914,176

0.040462

5

0.202309

Win 3x Play w/ 3-card flush against qualified dealer

16,796,416,174,704

0.042132

4

0.168527

Lose 3x Play  against qualified dealer

30,809,847,740,400

0.077283

-5

-0.386413

Push 3x Play against qualified dealer

2,751,669,318,312

0.006902

0

0.000000

Win 3x Play w/ 6-card flush against unqualified dealer

24,404,889,600

0.000061

23

0.001408

Win 3x Play w/ 5-card flush against unqualified dealer

1,075,217,004,000

0.002697

8

0.021576

Win 3x Play w/ 4-card flush against unqualified dealer

6,377,470,048,800

0.015997

4

0.063988

Win 3x Play w/ 3-card flush against unqualified dealer

12,970,988,479,440

0.032536

3

0.097608

Win 3x Play w/ 2-card flush against unqualified dealer

1,600,580,385,168

0.004015

3

0.012045

Lose 3x Play  against unqualified dealer

1,162,087,560,552

0.002915

-4

-0.011660

Push 3x Play against unqualified dealer

478,678,665,600

0.001201

0

0.000000

Win 2x Play w/ 6-card flush against qualified dealer

227,291,635,008

0.000570

23

0.013113

Win 2x Play w/ 5-card flush against qualified dealer

4,704,150,904,080

0.011800

8

0.094398

Win 2x Play w/ 4-card flush against qualified dealer

21,499,155,021,948

0.053928

4

0.215712

Win 2x Play w/ 3-card flush against qualified dealer

14,714,103,160,440

0.036908

3

0.110725

Lose 2x Play  against qualified dealer

32,751,544,964,688

0.082153

-4

-0.328613

Push 2x Play against qualified dealer

622,124,227,116

0.001561

0

0.000000

Win 2x Play w/ 5-card flush against unqualified dealer

187,837,403,616

0.000471

7

0.003298

Win 2x Play w/ 4-card flush against unqualified dealer

6,488,002,635,144

0.016274

3

0.048823

Win 2x Play w/ 3-card flush against unqualified dealer

16,304,458,158,816

0.040898

2

0.081795

Win 2x Play w/ 2-card flush against unqualified dealer

987,169,878,672

0.002476

2

0.004952

Lose 2x Play  against unqualified dealer

710,513,189,700

0.001782

-3

-0.005347

Push 2x Play against unqualified dealer

79,383,252,492

0.000199

0

0.000000

Win 1x Play w/ 5-card flush against qualified dealer

393,192,506,064

0.000986

7

0.006904

Win 1x Play w/ 4-card flush against qualified dealer

10,828,061,228,676

0.027161

3

0.081482

Win 1x Play w/ 3-card flush against qualified dealer

20,718,789,206,988

0.051970

2

0.103941

Lose 1x Play  against qualified dealer

68,485,489,408,332

0.171787

-3

-0.515362

Push 1x Play against qualified dealer

7,086,006,696,552

0.017774

0

0.000000

Win 1x Play w/ 5-card flush against unqualified dealer

5,385,180,384

0.000014

6

0.000081

Win 1x Play w/ 4-card flush against unqualified dealer

1,985,444,394,456

0.004980

2

0.009960

Win 1x Play w/ 3-card flush against unqualified dealer

26,514,857,520,000

0.066509

1

0.066509

Win 1x Play w/ 2-card flush against unqualified dealer

1,746,004,992,372

0.004380

1

0.004380

Lose 1x Play  against unqualified dealer

2,094,365,166,192

0.005253

-2

-0.010507

Push 1x Play against unqualified dealer

362,165,402,664

0.000908

0

0.000000

folds

64,139,016,142,080

0.160885

-2

-0.321769

total

398,664,610,344,000

1.000000

-0.026470

expected

398,664,610,344,000

Ultimate Casino War

Posted in Uncategorized by stephenhow on July 24, 2015

UCW_table cardI saw this new variant of Casino War at Barona Casino, where they player gets an option to swap his card and make a 1x Raise bet. Of course, the catch is the dealer gets two cards, and gets to use the highest one. I wanted to see what the strategy and house edge were, and to check if it was at all countable out of the One-2-Six CSM they use.

The rules are pretty simple. You’re dealt one card face up, and the dealer is dealt two cards face down. The dealer will use his highest card. You have the option to replace your card with the next card out of the shoe (CSM), but you must wager an additional 1x bet to do make this swap. Finally, you may wager an optional 1x Raise on your final hand.

The dealer reveals his hand, and all your bets receive action against the dealer high card. Wins win a Six or lower pay 2:1, else it pays even-money. Ties push, and there’s no “going to War”.

For a 6-deck CSM game, the house edge is a fair 2.56%.

The basic strategy is pretty simple. You should swap an Eight or lower card. You should Raise a Jack or higher final card.

I checked the countability in a CSM by assuming perfect play given 16 known cards before every hand. The EV barely changes by +/- 0.3%, and thus is never +EV.

Ultimate Casino War Optimal Outcomes (6 Decks)
Outcome Combinations Frequency Net Return
Win 3x bet with drawn A 165,477,312 0.035607 3 0.106820
Win 3x bet with drawn K 138,914,496 0.029891 3 0.089673
Win 3x bet with drawn Q 114,674,112 0.024675 3 0.074026
Win 3x bet with drawn J 92,756,160 0.019959 3 0.059877
Lose 3x bet with drawn card 163,441,152 0.035169 -3 -0.105506
Tie 3x bet with drawn card 97,187,328 0.020912 0 0.000000
Win 2x bet with drawn T 73,160,640 0.015742 2 0.031485
Win 2x bet with drawn 9 55,887,552 0.012026 2 0.024051
Win 2x bet with drawn 8 40,772,160 0.008773 2 0.017546
Win 2x bet with drawn 7 28,274,400 0.006084 2 0.012168
Win 2x bet with drawn 6 18,057,600 0.003886 4 0.015542
Win 2x bet with drawn 5 10,121,760 0.002178 4 0.008712
Win 2x bet with drawn 4 4,466,880 0.000961 4 0.003845
Win 2x bet with drawn 3 1,092,960 0.000235 4 0.000941
Lose 2x bet with drawn card 1,409,976,288 0.303394 -2 -0.606788
Tie 2x bet with drawn card 88,157,160 0.018969 0 0.000000
Win 2x bet with original A 306,488,448 0.065949 2 0.131898
Win 2x bet with original K 257,453,856 0.055398 2 0.110796
Win 2x bet with original Q 212,690,880 0.045766 2 0.091532
Win 2x bet with original J 172,199,520 0.037053 2 0.074107
Lose 2x bet with original card 301,682,880 0.064915 -2 -0.129830
Tie 2x bet with original card 179,437,536 0.038611 0 0.000000
Win 1x bet with original T 135,979,776 0.029260 1 0.029260
Win 1x bet with original 9 104,031,648 0.022385 1 0.022385
Lose 1x bet with original card 409,808,160 0.088181 -1 -0.088181
Tie 1x bet with original card 65,156,976 0.014020 0 0.000000
Total 4,647,347,640 1.000000 -0.0256406
Expected 4,647,347,640

According to Dan Lubin, there’s a version that pays 2:1 for a win with a Six, 3:1 for a win with a Five, 5:1 for a win with a Four, and 8:1 for a win with a Trey. For a 6-deck game, these payouts reduce the house edge to 1.27%. The basic strategy remains the same. Still, the game never gets +EV with only 16 known cards.

Ultimate Casino War Optimal Outcomes (6 Decks, 2-3-5-8 Pay Table)
Outcome Combinations Frequency Net Return
Win 3x bet with drawn A 165,477,312 0.035607 3 0.106820
Win 3x bet with drawn K 138,914,496 0.029891 3 0.089673
Win 3x bet with drawn Q 114,674,112 0.024675 3 0.074026
Win 3x bet with drawn J 92,756,160 0.019959 3 0.059877
Lose 3x bet with drawn card 163,441,152 0.035169 -3 -0.105506
Tie 3x bet with drawn card 97,187,328 0.020912 0 0.000000
Win 2x bet with drawn T 73,160,640 0.015742 2 0.031485
Win 2x bet with drawn 9 55,887,552 0.012026 2 0.024051
Win 2x bet with drawn 8 40,772,160 0.008773 2 0.017546
Win 2x bet with drawn 7 28,274,400 0.006084 2 0.012168
Win 2x bet with drawn 6 18,057,600 0.003886 4 0.015542
Win 2x bet with drawn 5 10,121,760 0.002178 6 0.013068
Win 2x bet with drawn 4 4,466,880 0.000961 10 0.009612
Win 2x bet with drawn 3 1,092,960 0.000235 16 0.003763
Lose 2x bet with drawn card 1,409,976,288 0.303394 -2 -0.606788
Tie 2x bet with drawn card 88,157,160 0.018969 0 0.000000
Win 2x bet with original A 306,488,448 0.065949 2 0.131898
Win 2x bet with original K 257,453,856 0.055398 2 0.110796
Win 2x bet with original Q 212,690,880 0.045766 2 0.091532
Win 2x bet with original J 172,199,520 0.037053 2 0.074107
Lose 2x bet with original card 301,682,880 0.064915 -2 -0.129830
Tie 2x bet with original card 179,437,536 0.038611 0 0.000000
Win 1x bet with original T 135,979,776 0.029260 1 0.029260
Win 1x bet with original 9 104,031,648 0.022385 1 0.022385
Lose 1x bet with original card 409,808,160 0.088181 -1 -0.088181
Tie 1x bet with original card 65,156,976 0.014020 0 0.000000
Total 4,647,347,640 1.000000 -0.0126955
Expected 4,647,347,640

Arizona Stud @ Red Wind Casino, WA

Posted in Uncategorized by stephenhow on July 30, 2014

Screen Shot 2014-07-30 at 9.21.14 PMArizona Stud is a new poker-based table game debuting at the Red Wind Casino in Olympia, WA next week (6 Aug 2014). In this game, both the Dealer and the player each receive three hole cards. The player must discard one of his hole cards before the flop, while the Dealer must use exactly two hole cards to make a hand. After the player discards, he may wager a Play bet of 2x-4x the Ante, or check pre-flop. The two card flop is then revealed, as well as one of the Dealer’s hole cards. If the player checked pre-flop, he must then make a 1x Play bet, or fold. Finally, the community river card and all Dealer hole cards are revealed. The Dealer qualifies with a hand of AK-high or better. The Ante pushes if the Dealer doesn’t qualify. The Play bet always receives even-money action against the Dealer hand.

The set of all possible outcomes for the optimal player is listed in the table below. The total in the lower right corner shows a house edge of 1.34% of the Ante. Note that you should either 4x bet pre-flop, or check. You should never only bet 2x.

Optimal Outcomes for Arizona Stud
Outcome Combinations Frequency Net Return
Win 4x Play w/ ROYAL_FLUSH against qualified dealer 59,240,916 0.000001 5 0.000005
Win 4x Play w/ FULL_HOUSE against qualified dealer 89,284,476,240 0.001605 5 0.008025
Win 4x Play w/ FLUSH against qualified dealer 10,295,059,284 0.000185 5 0.000925
Win 4x Play w/ STRAIGHT against qualified dealer 13,761,723,420 0.000247 5 0.001237
Win 4x Play w/ THREE_OF_A_KIND against qualified dealer 674,048,087,712 0.012117 5 0.060586
Win 4x Play w/ TWO_PAIRS against qualified dealer 1,233,004,030,272 0.022165 5 0.110827
Win 4x Play w/ ONE_PAIR against qualified dealer 3,533,244,131,304 0.063516 5 0.317580
Win 4x Play w/ HIGH_CARD against qualified dealer 44,095,696,596 0.000793 5 0.003963
Lose 4x Play against qualified dealer 5,323,636,585,296 0.095701 -5 -0.478507
Push 4x Play against qualified dealer 90,869,346,720 0.001634 0 0.000000
Win 4x Play w/ ROYAL_FLUSH against unqualified dealer 20,545,164 0.000000 4 0.000001
Win 4x Play w/ FLUSH against unqualified dealer 4,736,232,972 0.000085 4 0.000341
Win 4x Play w/ STRAIGHT against unqualified dealer 5,959,832,148 0.000107 4 0.000429
Win 4x Play w/ THREE_OF_A_KIND against unqualified dealer 350,836,147,584 0.006307 4 0.025227
Win 4x Play w/ TWO_PAIRS against unqualified dealer 28,557,204,480 0.000513 4 0.002053
Win 4x Play w/ ONE_PAIR against unqualified dealer 2,530,675,447,344 0.045493 4 0.181973
Win 4x Play w/ HIGH_CARD against unqualified dealer 1,146,771,919,728 0.020615 4 0.082461
Lose 4x Play against unqualified dealer 25,248,339,684 0.000454 -4 -0.001816
Push 4x Play against unqualified dealer 25,026,495,696 0.000450 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against qualified dealer 94,841,496 0.000002 2 0.000003
Win 1x Play w/ STRAIGHT_FLUSH against qualified dealer 541,732,152 0.000010 2 0.000019
Win 1x Play w/ FOUR_OF_A_KIND against qualified dealer 6,309,658,080 0.000113 2 0.000227
Win 1x Play w/ FULL_HOUSE against qualified dealer 45,729,841,680 0.000822 2 0.001644
Win 1x Play w/ FLUSH against qualified dealer 51,704,956,552 0.000929 2 0.001859
Win 1x Play w/ STRAIGHT against qualified dealer 90,255,233,808 0.001622 2 0.003245
Win 1x Play w/ THREE_OF_A_KIND against qualified dealer 591,850,723,248 0.010640 2 0.021279
Win 1x Play w/ TWO_PAIRS against qualified dealer 1,068,681,540,840 0.019211 2 0.038423
Win 1x Play w/ ONE_PAIR against qualified dealer 4,443,972,518,832 0.079888 2 0.159776
Win 1x Play w/ HIGH_CARD against qualified dealer 188,023,085,280 0.003380 2 0.006760
Lose 1x Play against qualified dealer 10,848,202,319,420 0.195015 -2 -0.390029
Push 1x Play against qualified dealer 161,798,077,992 0.002909 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against unqualified dealer 14,941,584 0.000000 1 0.000000
Win 1x Play w/ STRAIGHT_FLUSH against unqualified dealer 220,863,744 0.000004 1 0.000004
Win 1x Play w/ FLUSH against unqualified dealer 23,880,526,848 0.000429 1 0.000429
Win 1x Play w/ STRAIGHT against unqualified dealer 48,873,031,380 0.000879 1 0.000879
Win 1x Play w/ THREE_OF_A_KIND against unqualified dealer 34,291,273,536 0.000616 1 0.000616
Win 1x Play w/ TWO_PAIRS against unqualified dealer 301,557,935,124 0.005421 1 0.005421
Win 1x Play w/ ONE_PAIR against unqualified dealer 3,348,961,937,952 0.060203 1 0.060203
Win 1x Play w/ HIGH_CARD against unqualified dealer 2,842,212,690,936 0.051094 1 0.051094
Lose 1x Play against unqualified dealer 1,090,180,312,308 0.019598 -1 -0.019598
Push 1x Play against unqualified dealer 100,426,510,008 0.001805 0 0.000000
folds 15,186,972,477,600 0.273011 -1 -0.273011
total 55,627,620,048,000 1.000000 -0.013402
expected 55,627,620,048,000

The basic strategy for the game is listed in the table below, which returns a 1.70% house edge. The player should 4x his hand about 27% of the time, 1x call about 46% of the time, and fold the remaining 27% of the time.

The game looks like fun. The strategy is actually pretty simple, but you get to make the occasional decision. I’ll actually be in Seattle next week (my first time), so I’ll try to check out the game. Maybe I could hit a nice bad beat for once.

Arizona Stud Basic Strategy
Decision Strategy
discard Hold pair, else
hold two highest cards.
Advanced exception: hold highest and lowest cards when suited, AND
highest two cards aren’t suited, AND highest card is Eight or better, AND
middle card is Six or lower, AND lowest card is only one rank below middle
card.
4x / check 4x raise any pair, else
4x raise suited Ace and Nine or better, else
4x raise Ace and Ten or better, else
check.
1x / fold 1x call three-of-a-kind, else
1x call any pair beating the dealer by more than a kicker, else
1x call same pair as dealer plus Ten or better kicker, else
1x call open-ended straight flush draw, else
1x call flush draw or any straight draw when beating the dealer, else
1x call flush draw or open-ended straight draw when dealer has no pair, else
1x call with higher hand (Jack or better kicker), else
fold all others.

The optional 2 Pair Plus Bonus bet pays for the final hand made by the player. The house edges for the various offered paytables are listed below.

2 Pair Plus Paytables
Player Hand Paytable A Paytable B Paytable C Paytable D
Royal Flush 500-to-1 500-to-1 500-to-1 500-to-1
Straight Flush 200-to-1 200-to-1 200-to-1 200-to-1
Four-of-a-Kind 100-to-1 100-to-1 100-to-1 80-to-1
Full House 50-to-1 50-to-1 40-to-1 40-to-1
Flush 30-to-1 25-to-1 30-to-1 30-to-1
Straight 20-to-1 20-to-1 20-to-1 20-to-1
Three-of-a-Kind 6-to-1 6-to-1 6-to-1 6-to-1
Two Pairs 4-to-1 4-to-1 4-to-1 4-to-1
others lose lose lose lose
House Edge 3.03% 5.12% 5.53% 6.58%

The Player Bad Beat Bonus bet pays when a player’s Jacks-or-Better hand is beat by the Dealer. The following table shows the optimal outcomes for the strategy maximizing the Bad Beat Bonus return. The house edge for the optimal Bad Beat Bonus strategy is 8.00%.

Player Bad Beat Bonus
Player Beat Hand Combinations Frequency Payout Return
Straight Flush 142,560 0.000000 1000 0.000008
Full House 1,324,642,176 0.000071 500 0.035719
Flush 2,719,437,696 0.000147 300 0.043998
Staight 1,597,456,728 0.000086 200 0.017230
Three-of-a-Kind 49,285,841,520 0.002658 30 0.079740
Two Pairs 257,968,615,536 0.013912 20 0.278245
Jacks-or-Better 923,384,598,264 0.049798 8 0.398385
other 17,306,259,281,520 0.933327 -1 -0.9333273
total 18,542,540,016,000 1.000000 -0.080002

Screen Shot 2014-07-30 at 9.51.37 PM

1 Bet Threat @ Casino Pauma

Posted in Uncategorized by stephenhow on June 15, 2014

1bet_smI saw a new Hold’Em type game at Casino Pauma last week, and I thought I’d work out the numbers and give it a try. The game is pretty simple. You bet an Ante before the hand begins. After seeing your two hole cards, you may bet 2x preflop, or check. After the flop, you may 1x bet or check. The turn, river, and the dealer’s hole cards are then revealed. The dealer qualifies with a pair of 6’s or better. If the dealer doesn’t qualify, all post-Ante wagers push. If the dealer beats your hand, you lose all your remaining bets. If you beat a qualified dealer hand, you win all your bets. If you beat a non-qualified dealer, you only win 1/2 your Ante.

The game is a bit calmer than Ultimate Texas Hold’Em, since you only have a single Ante, and you can check it down to showdown (in fact, this happens 69.8% of the time). Plus, players may like the fact that they can make the 2x and 1x bets only when they have an advantage. (I.e., all properly made 2x and 1x bets are +EV.) And the Ante is only a -11.4% loser, on average. The optimal player makes a 2x preflop bet 11.2% of the time, and a 1x flop bet on 25.5% of the time. The dealer qualifies 69.1% of the time. The game has relatively low variance, and I found myself increasing the Ante from the $5 minimum, to $10, and $15. (I’d never do that with UTH.)

The total outcomes for the optimal player strategy are listed in the table below, and show a house edge of 3.2% of the Ante.

1 Bet Threat Optimal Outcomes
Outcome Combinations Frequency Net Return
Bet 2x and 1x and beat qualified dealer 884,580,718,240 0.031804 4 0.127215
Bet 2x and 1x and beat non-qualified dealer 505,981,246,728 0.018192 0.5 0.009096
Bet 2x and 1x and lose to qualified dealer 374,729,986,984 0.013473 -4 -0.053891
Bet 2x and 1x and lose to non-qualified dealer 5,856,935,220 0.000211 -1 -0.000211
Bet 2x and 1x and tie dealer 25,182,150,868 0.000905 0 0.000000
Bet 2x only and beat qualified dealer 293,907,701,760 0.010567 3 0.031701
Bet 2x only and beat non-qualified dealer 387,449,913,432 0.013930 0.5 0.006965
Bet 2x only and lose to qualified dealer 524,307,039,216 0.018851 -3 -0.056552
Bet 2x only and lose to non-qualified dealer 76,858,269,780 0.002763 -1 -0.002763
Bet 2x only and tie dealer 25,553,189,772 0.000919 0 0.000000
Bet 1x only and beat qualified dealer 2,434,367,467,360 0.087524 2 0.175047
Bet 1x only and beat non-qualified dealer 1,467,870,962,280 0.052775 0.5 0.026387
Bet 1x only and lose to qualified dealer 1,215,166,965,412 0.043689 -2 -0.087379
Bet 1x only and lose to non-qualified dealer 17,931,292,692 0.000645 -1 -0.000645
Bet 1x only and tie dealer 164,852,060,176 0.005927 0 0.000000
Bet ante only and beat qualified dealer 3,363,692,256,360 0.120936 1 0.120936
Bet ante only and beat non-qualified dealer 4,003,403,426,760 0.143936 0.5 0.071968
Bet ante only and lose to qualified dealer 9,229,633,097,868 0.331836 -1 -0.331836
Bet ante only and lose to non-qualified dealer 1,896,770,105,748 0.068195 -1 -0.068195
Bet ante only and tie dealer 915,715,237,344 0.032923 0 0.000000
Total 27,813,810,024,000 1.000000 -0.032157

I worked out the basic strategy for the game, just in case anyone wants to play the game. The strategy is actually pretty simple. Since the dealer qualifies with a pair of 6’s or better, you generally only bet the flop if there’s a qualified hand to beat. You can bet kickers and draws against a qualified flop, otherwise you should only bet a qualifying pair when there’s a board card lower than your pair, but 6 or higher.

The basic strategy below has an error rate of 4.5%, that only results in a cost of 0.23% to the player. So the practical house edge is 3.5% for the game.

1 Bet Threat Basic Strategy
Wager Player Hand Rules
2x Pairs 2x bet a pocket pair of 7’s or better, else
check pocket 2’s thru 6’s.
Suited Bet QJs, KTs, KJs, KQs, and A8s or better, else
check all others.
Offsuit Bet KQo, and ATo or better, else
check all others.
1x Straight or better Always bet.
Three-of-a-Kind Always bet, except if trips on flop and less than 2nd nut kicker.
Two Pairs Bet if flop not paired, else
bet if flop qualified (pair 6’s or better), else
bet if board has undercard to pairs, else
bet 9’s up or better, else
check all others.
One Pair
(qualified board has pair 6’s or better)
Bet nut kicker, else
bet flush draw, else
bet open-ended straight draw with both holecards > 8, else
check all others.
One Pair
(small pair on board)
Always check.
One Pair
(unpaired board)
Bet if board has any qualifying undercards to pair, else
bet pair w/ flush draw, else
bet pair 9’s or better, else
check all others.
No Pair Bet 1st or 2nd nut flush draw, else
check all others.

There’s not much opportunity for collusion in the game. Knowledge of the hole cards of all 6 players will modify some of the preflop 2x decisions, but the frequency and value of these counter-(basic)strategy decisions aren’t enough to overcome the 3.2% house edge. Trust me, I’d have worked it out if it was worthwhile.

There’s two bonus bets offered, where the Pocket Bonus pays when your hole cards make a pocket pair, and the Final Hand bonus on your final 7-card hand. The paytables offered at Casino Pauma aren’t very good.

Pocket Bonus
Outcome Combinations Frequency Payout (to-1) Return
Pocket A’s 6 0.004525 50 0.226244
Pocket J’s – K’s 18 0.013575 20 0.271493
Pocket 2’s – T’s 54 0.040724 8 0.325792
no pair 1,248 0.941176 -1 -0.941176
Total 1,326 1.000000 -0.117647
Final Hand Bonus
Outcome Combinations Frequency Payout (to-1) Return
Royal Flush 4,324 0.000032 250 0.008080
Straight Flush 37,260 0.000279 50 0.013925
Four-of-a-Kind 224,848 0.001681 15 0.025210
Full House 3,473,184 0.025961 5 0.129805
Flush 4,047,644 0.030255 4 0.121020
Straight 6,180,020 0.046194 3 0.138581
Three-of-a-Kind 6,461,620 0.048299 2 0.096597
Jacks Up 17,385,408 0.129951 1 0.129951
others 95,970,252 0.717349 -1 -0.717349
Total 133,784,560 1.000000 -0.054179

Flush Rush @ The D Casino, Las Vegas

Posted in Uncategorized by stephenhow on May 3, 2014

Screen Shot 2014-05-03 at 3.32.29 PM

A reader told me about a new ShuffleMaster game at The D Casino, where you try to make a 4- to 7- card flush, starting with 4 hole cards, and paying to see 5th/6th and 7th Streets on a community board. The betting structure is similar to Mississippi Stud, where you post an Ante, then make a 1x Play decision to see 6th Street, and a final 1x Play decision to see 7th Street. The game pays odds on the Ante if you make a 4-card flush or better, and even-money on the 1x Play bets. Otherwise, if you fold or don’t make a hand, you lose your bets.

Ante Pay Table
Length Flush Straight Flush
7 300-to-1 1000-to-1
6 20-to-1 500-to-1
5 9-to-1 100-to-1
4 5-to-1 15-to-1

I believe The D will award the highest possible payout for a given hand. So, if you make a 5-card flush that contains a 4-card straight flush, they’ll pay you 15-to-1 (instead of 9-to-1). With this liberal rules interpretation, the house edge is 3.75%. The total possible outcomes for an optimal player are listed below.

Optimal Play Outcomes (Liberal Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
6-card Straight Flush 167,160 1.1900E-05 502 0.005974
7-card Flush 697,620 4.9662E-05 302 0.014998
5-card Straight Flush 4,127,760 0.000294 102 0.029972
6-card Flush 26,945,100 0.001918 22 0.042119
4-card Straight Flush 65,648,544 0.004673 17 0.079447
5-card Flush 372,841,560 0.026542 11 0.291959
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.037493

If the rules are interpreted strictly, and you must make a straight flush with all your cards of the same suit, then the house edge is 5.41%.

Optimal Play Outcomes (Strict Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
7-card Flush 717,360 5.1067E-05 302 0.015422
6-card Straight Flush 147,420 1.0494E-05 502 0.005268
6-card Flush 27,960,660 0.001990 22 0.043790
5-card Straight Flush 3,112,200 0.000222 102 0.022598
5-card Flush 397,427,940 0.028292 11 0.311212
4-card Straight Flush 41,062,164 0.002923 17 0.049693
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.054059
All-Or-Nothing Side Bet
Outcome Combinations Frequency Net Return
All hole cards same suit 2,860 0.010564 30 0.316927
All hole cards different suits 28,561 0.105498 5 0.527491
Others 239,304 0.883938 -1 -0.883938
Total 270,725 1.000000 -0.039520

Of course, the only reason why I analyzed the game was to Monte Carlo the 6-way collusion edge. The return is about +4.07% for 6 players sharing perfect info under strict rule interpretation, and +5.67% under the liberal rules. That’s not much, considering it’s pretty hard to convey suit information between confederates. It’s probably not worth anyone’s trouble to attack the game. I didn’t bother working out a practical strategy.

(FYI, I’m spending a lot more time outside of the casino these days. Before, I used to practically live in the casino. About 9 months ago, I changed obsessions. You can read about my current mania on my other blog.)

Raise It Up Stud @ Pala Casino

Posted in Uncategorized by stephenhow on September 1, 2012

While visiting Pala Casino to check out House Money yesterday, I ran across the new ShuffleMaster game Raise It Up Stud. It has the familiar ShuffleMaster Ante, Blind, and 1x-3x Play bets, and there’s a 3-card community board. There’s no dealer hand; you’re just playing against a Paytable. You’re dealt 3 cards at the start of the hand, and you can bet 3x on your first 3 cards, or check. The dealer then turns up the first community board card, and you can now 2x bet your hand, or check. The dealer then turns up the 2nd community card, and you must either 1x bet to see the river, else fold. If you make a pair of Ten’s or better, you win even money on your ante, and odds on your Play bet. If you make trips or better, you win odds on your Blind bet. If you don’t make Ten’s or better, or if you fold, you lose all your bets.

I’d say the game plays like a more forgiving (easier) version of Mississippi Stud. You can raise a winning hand as soon as you make it, and you get paid odds on your raise. However, you can only make one bet per hand (in Mississippi Stud, you can bet a winner on all streets). But you can check until you make a hand, or have to call a draw. The Play and Blind paytables are listed below. Combining your three hold cards with the three community cards, you make your best 5 card hand.

This game is probably the long-awaited replacement for Let It Ride, which the dealers call “Let It Die”. They all hate the game, because they either stand dead at an empty table, or they just push back bets until someone occasionally wins on a 1x bet on the River. At Viejas, dealers keep their own tokes, so they hope the floor supervisor closes the game as early as possible, so they can go deal a game where they can make money. Hence, “Let It Die”.

Everyone was having a great time at Raise It Up last night, and the dealers were making lots of tokes. (Tokes are especially +EV on the Ante/Play bets; a nice little angle.) You make a lot more hands with 6 cards (compared to 5 in Let It Ride). Plus, you’re supposed to bet a lot more hands in this game than Let It Ride (small pairs, gut shot straight draws, 3 pay cards on 3rd St, etc.)

Raise It Up Stud Play Paytable
Hand Payout
Royal Flush 100:1
Straight Flush 20:1
Four-of-a-Kind 10:1
Full House 6:1
Flush 5:1
Straight 4:1
Three-of-a-Kind 3:1
Two Pairs 3:2
10’s or Better 1:1
Others lose
Raise It Up Stud Blind Bonus
Hand Payout
Royal Flush 1000:1
Straight Flush 200:1
Four-of-a-Kind 30:1
Full House 4:1
Flush 3:1
Straight 2:1
Three-of-a-Kind 1:1
Others push

Basic Strategy

The theoretical house edge for this game is 3.5022%. Below is a simple, intuitive strategy that simulates at -3.70%. The decisions on 4th and 5th Streets are fairly obvious, and you can easily learn the 3rd Street strategy.

Raise It Up Stud Basic Strategy
Street Play Bet Betting Hands
3rd Street 3x Any pair,
3 pay cards,
2 pay cards 1-gapped or less,
suited cards 2-gapped or less,
suited cards with 2 pays
4th Street 2x Any pair,
any straight or flush draw,
3 pay cards with 3 suited
5th Street 1x Any pair,
any flush draw,
open-ended draw,
gutshot draw with pay card

where “gap” is the sum of the distance between all cards (e.g., 456 is 0-gapped, 457 is 1-gapped, JT87 is 1-gapped, JT76 is 2-gapped, etc.).

Advantage Play

Even with ideal (computer) 6-way collusion, you can’t get the house edge below 0.93%.

Eliot Jacobson has published a simple hole-carding strategy that yields from +7.6% to +62.7% depending on which board card you see. Pala procedure places the bottom board card on 4th St, so I guess it’s only worth +7.6% when you see it.

Under-The-Gun 31

Posted in Uncategorized by stephenhow on March 28, 2011

Under-The-Gun 31 is a game developed and marketed by a pair of brothers who work at my local San Diego casinos. The game was on the floor at Viejas for a year, and it had a test placement at Pala too. The game is something of a cross between Blackjack and Three Card Poker. They designed the Ante bet with a small house advantage, while they pay good odds for the optional Bonus bet. The idea of the game is to make a hand total as high as possible, where only suited cards add together. Aces are always 11, and face cards have a 10 value. Since you can only add cards of the same suit, the maximum hand value is 31. The A-K-Q suited hand is a mini-Royal. The Ante pays a built-in bonus for a straight flush, a 31, or a mini-Royal.

To begin, the player makes an Ante bet. The Bonus bet is optional. The player and dealer both receive 3 cards. The looks at his hand, and decides to either fold, or to play the hand by betting an additional amount equal to the Ante. If the player stays, he also has the option to discard and draw one card. Once the action is complete, the dealer turns up his 3 cards. The dealer automatically takes a hit, and makes a hand from his 3 best cards. The player’s 31 Bonus and Stay-n-Play Bonus pay regardless of the dealer hand. The player’s Ante and Stay-n-Play bet pay even money against the dealer’s hand.

I know the game inventors, and wrote a playable Flash demo for them. They’d love to hear your feedback. Please try it out, and leave a comment about its playabilty, appeal, etc.. They’re working hard to get it out on the floor again. Click on the screenshot below to play:

Under-The-Gun 31 Game

Double Baccarat @ Sycuan Casino

Posted in Uncategorized by stephenhow on November 4, 2010

My local Sycuan Casino offers a unique game that’s a simplification of Pai-Gow tiles. Like the tile game, the players and the bank are dealt 4 cards each. Each hand is set into a front hand of 2 cards, and a back hand of two cards. The back hand must be greater than the front hand. Hand values are ranked by poker pairs, followed by Baccarat totals (0 thru 9). After the player hands are set, the bank turns over its hand and sets it according to “House Way”.

House Way
Hand Rule
Two Pairs Set large pair behind, small pair in front.
(Pair-Pair; never break pair).
Pair Aces Pair Aces behind (never split Aces.)
One Pair Pair behind if 5 or higher front, else
Split pair if can form (8,8), (7,9), (8,9) or (9,9), else
Pair behind.
No Pair Set (5,9) if possible, else
Set hand with highest front + back total, with minimum back – front gap.

Once all hands are set, the player or banker wins the wager if hands win/win, tie/win, or win/tie. In the case of tie/tie, the bank wins the wager. All other hands push the wager.

The player posts a minimum 1% collection before each hand. If all players push their bets, all collections are returned (“free collection”). This means in a heads-up game against the house, the player only pays the collection for a win or a loss, and gets free collection on a push.

I worked at optimizing a heads-up player strategy against the a house way bank, out of curiosity at what the house edge was. Of course, its an uphill battle against the collection (even when free for pushes), and worst, losing tie/tie. Using exhaustive combinatorics, I came up with the following near-optimal strategy (I only looked at the no-pair cases):

Heads-Up Player Strategy
Hand Rule
Two Pairs Set large pair behind, small pair in front.
(Pair-Pair; never break pair).
Pair Aces Pair Aces behind (never split Aces.)
One Pair Pair behind if 5 or higher front, else
Split pair if can form (8,8), (7,9), (8,9) or (9,9), else
Pair behind.
(6,9), (7,9), (8,9), (9,9)
front + back = 14 (7,7), (6,8)
(5,9)
front + back = 10 (5,5)
(9,1)
(4,6), (3,7), (2,8)
front + back = 9 (0,9)
(1,8)
(4,5), (2,7)
front + back = 8 (0,8)
(1,7)
(3,5), (2,6)
front + back = 7 (3,4)
(0,7)
(2,5), (1,6)
front + back = 6 (2,4)
(0,6)
(1,5)
Set hand with highest front + back total, with minimum back – front gap.

This strategy simulates at -1.46% heads up against house way, when minimizing collection to 1% of the bet amount. The frequency of ties simulates at 1.12%. So even if they eliminated the bank wins tie/tie rule, you’d still lose because of collection. As an additional note, if a heads-up player also plays the same House Way as the bank, the house edge increases to 2.0%.

Overall, the head’s-up game is about as good as a free-collection Pai-Gow game. (There are a few free-collection games at the card rooms town.) However, since the casinos don’t offer free-collection Pai-Gow, the head’s-up Double Baccarat game has better odds than the Pai-Gow game, for what it’s worth.

ShuffleMaster Ultimate Draw Poker Machine @ Viejas

Posted in Uncategorized by stephenhow on November 13, 2009

Table-Master_cutout_3There’s a new multi-player video “table” game at Viejas from ShuffleMaster, called Ultimate Draw Poker. (This game is different from the cards and table version of the game, which uses community draw cards.) The new Ultimate Draw machine seats up to five players, who play against a dealer hand. The game is “virtual single deck”, meaning that as far as any one player is concerned, you’re playing heads up against the dealer using a single deck. I’ll explain how they do this below.

The minimum bet (Ante) for this game is $3, and the maximum is $100. The video table is very nice, a single horizontal display for all player and the dealer hands, with nice visual effects (card animations, etc.). A vertical display is used to show a life-size dealer from the waist up, which is close enough to soft-core pornography to make you feel slightly uncomfortable. The dealer is dealt five cards face down, and also 5 replacement cards (not shown) from which she may draw. The remaining 42-card deck is then cloned for each seated player. Each player is dealt a five card hand of out a shuffled, 42-card cloned deck. The player decides what to discard, then draws from his cloned deck.

Once all players have discarded and drawn to their final hand, the dealer turns up her hand. The dealer applies a simple house-way discard policy:

  1. hold a pair or better, ELSE
  2. hold a four-card flush draw, ELSE
  3. hold an open-ended straight draw, ELSE
  4. hold all high cards (>= Jack), ELSE
  5. discard everything.

The dealer needs to make a pair or better to qualify. If she doesn’t qualify, you win 70% of your Ante bet. If she qualifies, then your Ante bet plays for even money against her hand.

Fortunately, “house-way” is a little weak, and a better player strategy exists (0.32% better than “house-way” vs. “house-way”):

  1. hold a pair of 3’s or better, ELSE
  2. hold a pair of 2’s unless flush draw w/ Jack or better, or unless kicker is King or better, ELSE
  3. hold a four-card flush draw (unless offsuit kicker better*), ELSE
  4. hold an open-ended straight (unless kicker better**), ELSE
  5. hold two highest cards >= Jack, ELSE
  6. hold JTs, ELSE
  7. hold highest card >= Ten, ELSE
  8. discard everything.

where:
*Ace is better than four-card flush draw, unless draw contains Queen or bettter
*King is better than four-card flush draw, unless draw contains Jack or better
**the following table shows kickers better than open-ended straight draws

draw min kicker to hold
2345 Ten
3456 Ten
4567 Jack
5678 Queen
6789 King
789T Ace
89TJ Ace
9TJQ

The house edge is very small for this game, only 0.61% for the above player strategy. However, the bonus bet is really bad, since it pays something like a Jacks-or-better video poker game, but you’re playing a strategy to beat the dealer hand, not to win a bonus. For the following table, and above player strategy, the bonus bet has about a 14% house edge. If you want to play the bonus bet, go find a video poker machine, it’s faster and pays more.

Hand Win
Royal Flush 1000
Straight Flush 150
Four Of A Kind 25
Full House 8
Flush 7
Straight 5
Three of A Kind 3
Two Pairs 1
all others -1

There’s a small “collusion” opportunity in this game. Because the game is played with cloned decks, and each player acts in turn, a player acting last gets to see a lot of the 42-card cloned deck. For example, if you look at all the dealt player hands, you can see what’s available in the cloned deck (any card you see is in the cloned deck). And, when you see what’s drawn, you get more info of what’s available. There’s a few cases where this info would help you make a borderline discard decision. There’s probably aren’t enough situations like this to make it worthwhile, but I could be wrong.