Chase The Flush Easy Advanced Strategy

AGS’s Hold’Em-style flush game has recently arrived at my local Viejas Casino, and now it’s all I play anymore. It’s fun and relaxing to just play symbols and colors, and there’s a lot more different outcomes than in simple Ultimate Texas Hold’Em. Structurally, it’s similar, where the player paces equal Ante and X-Tra wagers (similar to UTH’s Ante = Blind) before the hand begins. The players and dealer then each receive three hole cards, which is more interesting than UTH’s two hole cards, because of the increased ways to make a hand.
I did the analysis for AGS back in 2017 when they developed the game, and I’ve previously posted all the details about the house edge and the original bonus. I included a basic strategy for the game in that original post, but now that I’ve played it at length, I think I can present a much more concise description of a nearly optimal strategy.
To summarize the rules, the player posts an equal Ante and X-Tra bonus bet before the hand begins. Each player and the dealer each receive three hole cards. There are four community cards, exposed via a 2-card flop, then a 2-card turn/river. The player may go all-in with a 3x Play bet pre-flop, or with a 2x bet after the flop, or with a 1x bet after the turn/river. Otherwise the player folds and loses his Ante and X-Tra bets. The dealer qualifies with a 3-card Nine-high or better hand, otherwise the Ante pushes. The Play bet always receives even-money action, and the X-Tra bonus pays odds when the player beats the dealer with a 4-card flush or better. The only hand for the main game are flushes (i.e., not straight flushes, or any other rank) rated by the number of cards of the same suit in a hand, followed by their descending card ranks.
Fortunately, the near-optimal strategy works out to be very straight-forward, and plays fairly automatically. You know what to look for in advance of each step, and how to react to what cards hit the board. Basically, you preflop raise (3x) strong starting hands, generally 2x/1x bet a 3-card flush (except for a few exceptions), and occasionally 1x call a two-card flush against a rainbow board. The exceptions are few and very specific, and will make sense to you as soon as you encounter them.
You can practice this game online on AGS’ website.
Improved and Intuitive Strategy
Preflop, you should 3x the following strong hands:
Bet 3x: any three suited cards any suited Ace suited King with sufficiently high offsuit kicker: KQs.3+, KJs.4+, KTs.5+, K9s.5+, K8s.6+, K7s.7+, K6s.7+, K5s.7+, K4s.7+, K3s.8+, K2s.8+ suited Queens w/ high kickers: QJs.J+, QTs.J+, Q9s.J+, QXs.Q+ suited Jack with King+ offsuit kicker suited Ten with Ace offsuit kicker three high rainbow cards: AA.4+, KK.J+, AK.7+, AQ.T+ Else check
Then, after the flop:
Bet 4+ card flush: always Bet 3-card flush: flop: offsuit board: when have both suits of board when 3-card flush > other board suit with 9-6-x flush or better suited board: with flush hole card SIX or higher river: rainbow board: always 2-suited + 2-suited board: when < 15 single dealer cards beat you 2-suited + 2-offsuit board: any card of majority board suit any 3-card flush > 2-suited board cards any 3-card flush when 2-suited board cards < NINE any qualifier > board suits not part of your flush 3-suited + 1-offsuit board: play any 3-card flush > board play board if offsuit board card is <= 3-card flush Bet 2-card flush: flop: offsuit board: when have both board suits, and these two 2-card flushes make: nut 2-card flush 2nd-nut 2-card flush & 8th nut 2-card flush or better 3rd-nut 2-card flush & 5th nut 2-card flush or better river: when rainbow board and < 10 single dealer cards that beat you Else check/fold
Why It’s Fun
You may have noticed that UTH has become very crowded these days, and everyone knows the game well enough to take all the fun out of it. The games are usually expensive, with minimums at $10-$15-$25 Antes, because of its popularity. And I don’t have to tell you how bad that game can run, especially when the boards miss your 4x raising hands.
Chase The Flush is kind a return to the days when Ultimate was new, and nobody knew how to play. It’s kind of fun to see people learning a new game, trying out new strategies. UTH has become a bit boring, and less communal. People selfishly call out for their specific card(s) to hit the board, which would probably hurt you. In Chase The Flush, multiple people make various hands out of all kinds of boards. This is probably due to the fact that players hold three hole cards instead of two.
Also interesting is that the 7-card flush (pays 250:1 on both the X-Tra and the Same Suit Bonus) and the 6-card flush (pays 20:1 on the X-Tra, and 50:1 on the Same Suit Bonus) hit much more frequently than their Royal and Straight Flush counterparts on UTH. It’s perceptible. Also the Same Suit Bonus hits about 25% of the time, but at my local casino (Viejas), they’ve lowered the payouts a bit, and the house edge is substantially higher than the typical UTH Trips side bet.
At my local casino, they always seem to have a $5 minimum table, and sometimes it’s not very full, and you can enjoy having some more personal space than at a packed UTH table. Also, because of the 3x (instead of 4x) preflop raise, the game tends to feel like it’s less volatile (until you hit a nice 6-card flush or better, which generally happens every session).
How to Play Quickly
Once you’ve played a bit and understand the strategy, you’ll probably come up with some kind of similar ways of remembering your hand so you don’t have to constantly look at them to decide what to do. Normally, when I 3x raise, I just tuck the cards and forget what I have, and wait for showdown. Otherwise, if I have “rainbow” cards, I just remember the one suit I don’t have. Then, I’ll know if I flop a 3-card flush, or flop two 2-card flushes. If one of the suits I don’t have hits the flop, then chances are I’ll fold if any more hit the turn or river.
If my hole cards are two of one suit, and one of another suit, I’ll mentally note something like “spades, heart”. That way, if both hit the flop, I’ll automatically bet, and otherwise I’ll know if I flop a 3-card flush that I might bet. Similarly, remembering “spades, heart”, I’ll know if I have a 3-card flush or better or not on the river.
Same Suit Bonus (w/ 3-Card Straight Flush Payout)
Outcome | Combinations | Frequency | Payout | Return |
---|---|---|---|---|
6-or-7 Card Straight Flush | 1,624 | 0.000012 | 500 | 0.006069 |
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 | 250 | 0.012415 |
6 Card Flush | 256,620 | 0.001918 | 50 | 0.095908 |
5 Card Flush | 3,550,872 | 0.026542 | 5 | 0.132709 |
3 Card Straight Flush | 6,736,184 | 0.050351 | 3 | 0.151053 |
4 Card Flush | 22,152,936 | 0.165587 | 1 | 0.165587 |
Nothing | 100,404,096 | 0.750491 | -1 | -0.750491 |
Total | 133,784,560 | 1.000000 | -0.062246 |
Multiplier BlackJack @ Jamul Casino
You know what we all need, besides a +EV game, or some ridiculously countable side-bet? We need more fun at a blackjack table. We also sometimes need a miracle just to get even at a blackjack table. Both of these are offered by the new Multiplier Blackjack game at my local Jamul Casino, in Jamul, CA.
I did the math for the game inventors about 6 years ago. I understand the game won the Best-in-Show award at the table games conference, and is growing in casino placements around the country. It’s finally available in San Diego county, and I checked it out last night. Woo-hoo, it’s fun, and I actually won at a blackjack table for once!
The idea behind the game is simple. All the rules are the same as Blackjack, but when you win against a dealer bust, you’re paid odds based on the dealer bust card. The payouts are higher for a low bust card, and decrease to 1-to-2 for a picture card. When the dealer busts with a 10-spot card (Ten), your winning wager just pushes. Of course, the game excitement comes when you’re hoping for a dealer bust with a Six (pays 4-to-1), or any other non-10 valued card.
Dealer Bust-Card | Payout |
---|---|
Six | 4-to-1 |
Seven | 3-to-1 |
Eight | 3-to-2 |
Nine | 3-to-2 |
Jack/Queen/King | 1-to-2 |
Ten | push |
Why It’s Fun
Regular blackjack is boring for a lot of people. I almost never play it, because I just don’t care about even-money games. I walk by all the blackjack tables. I’ll look at some of the odds-paying side bets, but it’s still not worth the effort of playing the main blackjack game.
The potential 4-to-1 payout of Multiplier Blackjack makes the game fun. I really don’t care if I sometimes win 1-to-2 on a dealer bust with a picture, because it’s still a win. It’s all part of the build-up towards a 4-to-1, or 3-to-1 win. And, if that win comes on a multiply split hand with a double or two, that’s really exciting. (In fact, that happened during my first short session, where I re-split Deuces to three hands, had a double-after-split, and the dealer busted with an Eight.)
I also like the way the table layout has betting spots for both regular blackjack, and Multiplier Blackjack. That way, the player has the option of either standard blackjack, or the multiplier version. This makes it easy for the casino to introduce the game to the player, pretty much at no floor-space cost. This layout itself is a new innovation, where the player can choose between different blackjack versions on-the-fly, or weight wagers differently for the same hand.
As I mentioned before, there are sometimes during a session when an odds payout becomes attractive (or necessary).
Basic Strategy
The house edge for a six-deck shoe game with Late Surrender (LS), double after split (DAS), re-split up to 4 hands (SP4) including Aces (SPA4) is 1.92%. So, pretty much for the cost of a 6-to-5 blackjack game, you get a lot of fun, and up to a 4-to-1 payout on wins.
I generated a basic strategy table for the Multiplier Blackjack rules, a six-deck shoe, and the above pay table. There are a few differences from standard blackjack, notably the hits with hard-12 against a dealer 2-4 upcard. There are a few more doubles with hard 9 and even hard 8 against a dealer 6 upcard. Overall, the play is a little more aggressive, including splitting 4-4 against a dealer 5 or 6 upcard.
Hand | Dealer Upcard | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | A | |
Soft Totals | ||||||||||
A-10 | S | S | S | S | S | S | S | S | S | S |
A-9 | S | S | S | S | S | S | S | S | S | S |
A-8 | S | S | S | S | D | S | S | S | S | S |
A-7 | S | S | D | D | D | S | S | H | H | H |
A-6 | H | H | D | D | D | H | H | H | H | H |
A-5 | H | H | H | D | D | H | H | H | H | H |
A-4 | H | H | H | D | D | H | H | H | H | H |
A-3 | H | H | H | D | D | H | H | H | H | H |
A-2 | H | H | H | H | D | H | H | H | H | H |
Hard Totals | ||||||||||
20 | S | S | S | S | S | S | S | S | S | S |
19 | S | S | S | S | S | S | S | S | S | S |
18 | S | S | S | S | S | S | S | S | S | S |
17 | S | S | S | S | S | S | S | S | S | Rs |
16 | S | S | S | S | S | H | H | R | R | R |
15 | S | S | S | S | S | H | H | H | R | R |
14 | S | S | S | S | S | H | H | H | H | H |
13 | H | S | S | S | S | H | H | H | H | H |
12 | H | H | H | S | S | H | H | H | H | H |
11 | D | D | D | D | D | D | D | D | D | H |
10 | D | D | D | D | D | D | D | D | H | H |
9 | H | H | D | D | D | H | H | H | H | H |
8 | H | H | H | H | D | H | H | H | H | H |
7 | H | H | H | H | H | H | H | H | H | H |
6 | H | H | H | H | H | H | H | H | H | H |
5 | H | H | H | H | H | H | H | H | H | H |
Pairs | ||||||||||
A-A | P | P | P | P | P | P | P | P | P | P |
T-T | S | S | S | S | S | S | S | S | S | S |
9-9 | S | P | P | P | P | S | P | P | S | S |
8-8 | P | P | P | P | P | P | P | P | P | Rp |
7-7 | P | P | P | P | P | P | H | H | H | H |
6-6 | H | P | P | P | P | H | H | H | H | H |
5-5 | D | D | D | D | D | D | D | D | H | H |
4-4 | H | H | H | P | P | H | H | H | H | H |
3-3 | H | H | P | P | P | P | H | H | H | H |
2-2 | H | H | P | P | P | P | H | H | H | H |
Countability
I checked the sensitivity of the house edge to the removal of a single card from the 6-deck shoe, to see if the game was somehow especially vulnerable to card-counting, because of the odds payout. The sensitivity numbers (the effect-of-removal, or EORs) were of pretty standard magnitude. And, because of the high initial house edge, it takes a much more distorted shoe to attain any +EV opportunities. So, it looks like the game is fairly uncountable.
Single Removed Card Rank | EOR |
---|---|
Deuce | +0.000931 |
Trey | +0.001052 |
Four | +0.001308 |
Five | +0.001347 |
Six | -0.000787 |
Seven | -0.001300 |
Eight | -0.000785 |
Nine | -0.001285 |
Ten | +0.000853 |
Jack/Queen/King | -0.000082 |
Ace | -0.001042 |
Premium Hold’Em
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.
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
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.
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 -2.80% 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 Aces or Kings with minimum singleton:
AA.4+, KK.J+ |
Three suited cards | |
Suited Ace | |
Suited King with minimum singleton:
KQs.3+, KJs.4+, KTs.5+, K9s.5+, K8s.6+, K7s.7+, K6s.7+, K5s.7+, K4s.7+, K3s.8+, K2s.8+ |
|
Suited Queen with minimum singleton:
QJs.J+, QTs.J+, Q9s.J+, QXs.Q+ |
|
Suited Jack with King+ singleton | |
Suited Ten w/ Ace singleton | |
Rainbow AK.7+, AQ.T+, AJ.J+ | |
Check all others | |
2x
(“flop”) |
4-card flush or better |
3-card flush w/ 2-card flush using both board suits | |
3-card flush vs offsuit board | |
3-card qualifier using suited board with Six+ 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 using the 2 suited cards of a 3-suit board | |
3-card flush higher than suited cards of a 3-suit board | |
3-card qualifier higher than singleton of a 3-suit board | |
3-card flush against double-suited board w/ less than 15 one-card beats | |
2-card flush against rainbow board w/ less than 10 one-card beats | |
Fold all others |
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 sufficiently high 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 AK, AQ, or AJ if the 3rd card is sufficiently high. This means you should 3x AK8o, AQTo, AJJo etc. You should check AK5o, AQ9o, AJTo, 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
I 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.
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.
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 |
DJ Wild Poker
DJ Wild is a new “deuces wild” poker game against the dealer, using a standard 52-card deck plus one additional Joker. The game is pretty simple, where you wager an Ante and equal Blind bet before receiving a five card hand. You then decide to either 2x Play the hand, or fold. The Dealer also receives a five card hand, and always qualifies. The Ante and Play bets receive even money action against the Dealer hand, but the Blind only pays for a straight or better. The Blind pays nice odds for rare hands, but only pays about 6% of your hands.
The full analysis of the game shows a house edge of about 3.5% of an Ante.
When I first looked at this game, it looked like easy pickings for a table full of colluding advantage players. The confederates would silently share the number of Deuces or Jokers they held in their hands (using simple chip signaling). The whole table would know the number of outstanding Wild cards seen. Each player would 2x Play if they had better than the minimum hand needed for the given Wild count. It looked like the game was toast.
So, I quickly coded it all up to find the theoretical 6-way collusion edge. I was shocked to find that even perfect info sharing only yielded +0.5% between 6 players. You’d expect more of an edge on a perfect 2x Play decision and the always-qualifying Ante. Plus, you get the chance to “save” the Blind bet with a weak hand when the Wild count is high.
Anyways, I worked out a simple 6-way collusion strategy, just in case it turned out to be slighly +EV. The strategy just uses separate minimum calling hands for each Wild count (0 thru 5). The strategy below only decreases the house edge to 1.1%.
Wild Count | Minimum Play Hand |
---|---|
0 | Pair Jacks |
1 | Pair Nines |
2 | Pair Sevens |
3 | Pair Fours |
4 | Ace-King high |
5 | Ace high |
Well, at least we know now. No one needs to lose any sleep over this game.
Arizona Stud @ Red Wind Casino, WA
Arizona 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.
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.
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.
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 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 |
1 Bet Threat @ Casino Pauma
I 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.
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.
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.
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 |
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
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.
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.
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%.
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 |
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.)
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