Discount Gambling

Wild Six Card Draw Poker

Posted in collusion by stephenhow on June 14, 2013

While I was playing Six Card Poker at my local Viejas Casino, another player told be about the Wild Six Card Draw that he plays in Colorado. It’s a poker game with two wild Jokers in the 54-card deck, and the player gets 5 cards plus one free replacement card vs the dealer’s 6 card hand. I ran a Monte Carlo analysis to see if ideal 6-way collusion would yield any edge (you never know, the game has two Jokers after all). But even with 6-way collusion, you can’t get the house edge below 2.2%. I guess that makes sense, since it’s probably rare where you’d chose a weird draw over the more obvious discard. Anyway, it’s really easy to check these things, and you never know what you’ll find.

High Card Flush

Posted in +EV, collusion by stephenhow on April 22, 2013

GGsite_HCF_ProdBanner_041713A couple of readers have asked about Galaxy Gaming’s new High Card Flush game, which has a few placements now, and may be picking up some steam. The game is pretty simple, where each player and the dealer receive 7 cards. Each hand is measured by its highest flush, where a flush is first ranked by its length (number of cards of same suit), then by its card values. Each player must Ante before the hand, then wagers a 1x-3x Play bet (depending on flush size), or folds. The dealer qualifies with a three-card, 9-high flush. If the dealer doesn’t qualify, the Play bets push, and the remaining Antes are paid even-money. If the dealer qualifies, the Ante and Play bets receive even-money action against the dealer hand.

As you would expect, collusion helps in this game. A Monte Carlo analysis shows that with 6 confederates, perfect knowledge of the dealt cards gives each spot at least a +7.3% edge over the house. But practically, you’d be lucky if you could even communicate the suit counts (number of cards of each suit) dealt. If you figure out a non-suspicious way of doing this, then the following simple strategy yields a +3.1% edge over the house:

High Card Flush Advanced Strategy (6 Player Hands Seen)
Flush Size Play Bet
1 or 2 cards 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Jack-high or lower 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Queen-high 1x if lowest suit count is 9 or higher,
else fold.
3 card, King-high or better 1x if lowest suit count is 8 or higher, else fold.
4 cards 1x
5 cards 2x
6 or 7 cards 3x

where the suit counts 4-tuple is the sorted number of cards of each suit.

Phil’Em Up Poker

Posted in collusion by stephenhow on April 19, 2013

When I playing Mississippi Stud in Vegas last week, I overheard someone mention a game called Phil’Em Up Poker. I looked at the game, to see if collusion would yield an edge. The rules are pretty simple. The game is played with a 52-card deck plus a Joker which may be used for Aces, straights, and flushes. Each player bets an Ante, and receives two hole cards. Two community cards are dealt face up. Each player may either make an additional 1x bet (i.e., “double-up” his action), or check, before the 3rd community card is exposed. If a player makes a pair of Tens or better, he wins according to a paytable. There is no dealer hand. The house edge is a reasonable 3.3%.

Optimal Outcomes for Phil’Em Up Poker
Hand Bet Combinations Probability Payout Return
FIVE_ACES 2 5 0.00000035 1000 0.000697
natural ROYAL_FLUSH 2 20 0.00000139 250 0.000697
wild ROYAL_FLUSH 2 100 0.00000697 100 0.001394
natural STRAIGHT_FLUSH 2 180 0.00001254 50 0.001254
wild STRAIGHT_FLUSH 2 720 0.00005018 25 0.002509
FOUR_OF_A_KIND 2 4,140 0.00028853 20 0.011541
FULL_HOUSE 2 21,840 0.00152212 15 0.045664
FLUSH 2 39,020 0.00271946 9 0.048950
STRAIGHT 2 77,460 0.00539850 7 0.075579
THREE_OF_A_KIND 2 211,200 0.01471939 3 0.088316
TWO_PAIRS 2 365,640 0.02548294 2 0.101932
High Pair 2 1,562,112 0.10886993 1 0.217740
Low Pair 2 75,648 0.00527222 -1 -0.010544
HIGH_CARD 2 339,708 0.02367563 -1 -0.047351
STRAIGHT 1 25,200 0.00175629 7 0.012294
THREE_OF_A_KIND 1 105,600 0.00735969 3 0.022079
TWO_PAIRS 1 327,360 0.02281505 2 0.045630
High Pair 1 922,608 0.06430030 1 0.064300
Low Pair 1 3,514,752 0.24495734 -1 -0.244957
HIGH_CARD 1 6,755,112 0.47079118 -1 -0.470791
total 14,348,425 -0.033067
expected 14,348,425

Collusion doesn’t help. That’s because only 3.8% of hands are bet on a draw only. Collusion will change few decisions, and result in little gain. With 7-player collusion, perfect play will only reduce the house edge to 3.2%.

Lunar Poker @ Pechanga Casino

Posted in +EV, collusion, lunar poker by stephenhow on August 24, 2012

Well, someone is finally bringing the infamous Lunar Poker (aka Russian Poker) to the US, starting at my nearby Pechanga Casino. The game is a very interesting version of the old Caribbean Stud Poker, with a lot more options like drawing cards, buying an extra card, buying insurance, and forcing the dealer to draw (all for a price).

The game has been infamous, because the many player options result in an incalculable number of possible hand combinations (6.27x 10^20 according to the WoOs), and because of the absence of a published strategy. It sounds like people have played this game by the seat of their pants for years in Europe and Asia. But a lot of us won’t play a game without first knowing the basic strategy and house edge. So I grinded out the analysis, just in case you run across this game.

Rules

The rules follow the basic structure of Caribbean Stud Poker. You place an Ante before the hand starts, and the players and dealer each receive five cards. The dealer exposes one of his cards. You eventually decide to either Raise 2x, or fold your Ante. The dealer turns up his hand, and needs Ace-King or better to qualify. If the dealer doesn’t qualify, then the remaining Antes are paid even-money, and the Raise bets push. If the dealer qualifies, then the Antes push, and the Raise bets are paid according to a paytable.

So far, these rules are just like Caribbean Stud, except here, the Ante only pays when the dealer doesn’t qualify.

Now, Lunar Poker offers the following player options before the Player makes his 2x Raise decision:

  • The player may either receive an extra (6th) card, or may replace 2-5 of his cards, for the cost of 1x the Ante.
  • With three-of-a-kind or better, the player may take even-money insurance against the Dealer not qualifying (up to 1/2 the amount of the winning payout).

The players make their 2x Raise or Fold decision, then the dealer turns up his hand. If the dealer doesn’t qualify, the Antes and Insurance pay even money. If the dealer qualifies, then the player must beat the dealer to win his Raise bet and push his Ante. Else, the player loses his Ante and Raise. Insurance loses if the dealer qualifies and the player wins. If the dealer qualifies and the player loses, Insurance pushes. (Note: Pechanga lets you can take Insurance on up to the full amount your potential win.)

Finally, if the dealer doesn’t qualify, the player has an option to:

  • Pay 1x Ante to force the dealer to replace his highest card with a draw from the deck.

If the dealer qualifies after the draw, then the player’s Ante and Raise resolve as before. If the dealer doesn’t qualify, then the Ante and Raise push. Note: if you decide to Force the dealer to draw, then you forfeit the pay on the Ante you would normally receive. (It is expensive to Force the dealer; you forfeit your win on the Ante, AND you have to pay 1x!)

Paytable

For winning hands against a qualified dealer hand, the Raise bet pays according to the following paytable. More importantly, you are paid on a second hand from the paytable, when the second hand uses at least one different card from your first payout hand. (Note: “hands” do not include kickers; e.g., a three-of-a-kind hand contains only 3 cards for purposes of the paytable.) I’m not going to provide examples of the second payout, as this is described elsewhere.

Hand Payout
Royal Flush 100-to-1
Straight Flush 50-to-1
Four-of-a-Kind 20-to-1
Full House 7-to-1
Flush 5-to-1
Straight 4-to-1
Three-of-a-Kind 3-to-1
Two Pairs 2-to-1
One Pair 1-to-1
AK 1-to-1

Basic Strategy

I worked out a simple strategy for the game that simulates at a 1.43% house edge. That’s not bad as far as carnival games go, but it looks like their claim of “House Advantage Under 1%!” is false.

Draw Decision

The first decision on what to hold and draw is presented in the table below.

Draw Decision for Lunar Poker
5-Card Hand Decision
Royal Flush
Straight Flush
Flush
Straight
Always buy 6th card.
Four-of-a-Kind Stand.
Full House Buy 6th card unless dealer upcard copies you.
Three-of-a-Kind Stand if 4-of-a-kind not possible,
else hold trips and exchange 2 cards.
Two Pairs Stand.
One Pair w/ AK Discard 2′s or 3′s (hold AK and exchange 3) against higher upcard, Queen or lower,
else stand.
One Pair Buy 6th card for open-ended, flush draw, or gutshot.
Hold pair and exchange 3 if pair below upcard,
else stand.
AK Buy 6th card for open-ended, flush draw, else
Buy 6th card for perfect gutshot to 6-card straight, else
Buy 6th card for gutshot straight draw against A or K upcard, else
Hold AKs and royal cards higher than dealer upcard, else
Hold AK and exchange 3
Nothing Buy 6th card for open-ended or flush draw, else
Buy 6th card for perfect gutshot to 6-card straight, else
Hold AKs and any Royal cards, else
Hold two or more Royal cards higher than the dealer upcard, else
Hold three straight flush cards higher than the dealer upcard, else
Hold A against K upcard or lower, else
Hold K against J upcard or lower, else
Hold Q against copied J upcard or lower, else
Hold Q against 5 upcard or lower,
Else fold.

where open-ended straight draws include double-gutshot straight draws.

Insurance

It’s only correct to take insurance in a few cases. Never insure your hand against an Ace or King upcard. Otherwise, take insurance when you copy the dealer upcard 2 or more times. If you only copy the dealer upcard once, then take insurance when you also hold 2 or more Aces or Kings in your hand.

2x Raise / Fold

You should 2x Raise any pair or better. Fold any non-qualifying hand. Otherwise, play AK according to the table below.

2x Raise Decision
Hand Decision
Pair or better Raise 2x.
AK Call with any copies of the dealer upcard, Q or lower, else
Call with AKJ83 or better with any copies of the dealer upcard (including A, K), else
Fold all others.
non-qualifying Fold.

Force Dealer Bet

Your potential Raise payout and the possible dealer outs determine when you should try to force the dealer to draw. The table below tells you when to pay 1x to replace the highest dealer card with one from the deck. Remember, you’re forfeiting your instant Ante win by Forcing the dealer to draw. Plus, you’re paying 1x for the Force, so you need at least a 4:1 payout to make it profitable (i.e., don’t Force trips-only hands).

Force Draw Strategy
Potential Payout Conditions
3-to-1
or lower
Never force.
4-to-1 Don’t force dealer flush or open-ended draws that beat you unless all dealer pair outs are available, else
Don’t force if you hold 2 or more of the dealer’s pair outs,
else force.
5-to-1 Force unless you hold 4 or more of the dealer’s pair outs.
6-to-1
or higher
Always force.

Simple Two Player Collusion

If you’re friendly with your table-neighbor, you can slightly modify basic strategy to get a +EV return of +0.43% on the Ante. The drawing decision is modified accordingly:

Buy/Exchange Decision for Two Player Collusion
5-Card Hand Decision
Three-of-a-Kind Stand pat if your neighbor holds your quad out, else
hold trips and exchange 2 cards.
One Pair
w/o AK
Buy 6th card for open-ended or flush draw, else
Buy 6th card with over-pair (above dealer upcard) and gutshot if all straight outs remain, else
Buy 6th card with under-pair (below dealer upcard) and gutshot if any straight outs remain, else
Stand pat against dead upcard (3 copies) Q or lower, else
Hold under-pair (below dealer upcard) and draw 3 if all outs remain, else
Stand pat for all others.
AK Buy 6th card for open-ended or flush draw, else
Buy 6th card with 2+ outs to perfect gutshot (6-card straight), else
Buy 6th card with 3+ outs to gutshot against A/K upcard, else
Stand pat against dead upcard (3 copies), Q or lower, else
Hold two or more royal cards, exchange rest, else
Buy 6th card with at least 2 gutshot draws to AKQJT, else
Hold AK and exchange 3 cards.
Nothing Buy 6th card for open-ended or flush draw, else
Buy 6th card with 2+ outs to perfect gutshot (6-card straight), else
Stand pat against dead upcard (3 copies), Q or lower, else
Hold two or more royal cards, exchange rest, else
Hold your highest card, 9 or better, higher than the upcard and not copied by your neighbor, else
Hold 3 straight flush cards higher than the upcard, else
Fold all others.

Only take insurance when you and your neighbor hold 3 total copies of the upcard, Queen or lower. Never insure against an Ace or King upcard.

Finally, modify the 2x Raise decision:

  • Call any 2:1 pay or better, else
  • Fold pair deuces against uncopied upcard 3 thru Q, else
  • Call any other pair, else
  • Call any hand when you and your neighbor hold all 3 copies of the dealer upcard Queen or lower, else
  • Call AKJ83 or better when you and your neighbor hold any copies of the upcard, else
  • Call AK when you and your neighbor hold 2 copies of the dealer upcard Queen or lower, else
  • Fold all others.

Six Card Poker @ Venetian, Las Vegas

Posted in +EV, collusion, six card poker by stephenhow on May 11, 2012

On my trip to Vegas last month, I saw this new game at the Venetian, and all I could think of was collusion. I figured it had to be beatable, since the dealer shows half his hand (3 upcards), which should exploitable given confederate card information. Well, I finally got around to looking at it, and of course, its not as exploitable as I hoped.

The game is pretty simple, where both dealer and player get 6 cards to make a 5-card poker hand. There’s only an Ante, and a 1x Play bet. The dealer shows 3 upcards, and you decide to either 1x Play or fold your hand. If the dealer doesn’t qualify with Ace-King, then the Ante pushes regardless of the player hand. The 1x Play bet always receives even-money action against the dealer hand. The Wizard of Odds provides a basic strategy, and lists the house edge at 1.27%.

I figured 6-player collusion would help you know when to play Ace-high, and maybe help you fold a pair when a lot of dealer outs remain that beat you. But first, I simulated a bunch of hands finding the optimal decision given confederate card info. This gave me a very close approximation to the ideal edge obtained by perfect collusion. This 6-player edge amounted to only +1.17%. This isn’t much, especially since any actual collusion strategy approaching this limit would be impractically complex.

At this point, I only made a half-hearted attempt at finding a practical collusion strategy. There’s so many cards involved, its difficult to come up with a workable signalling system. Also, I looked over the collusion decision points, and it wasn’t simple to identify the conditions for making a counter decision to basic strategy. For what it’s worth, I came up with the following “simple” 6-player collusion strategy that simulates at +0.15%:

  • Call two pairs or better, else
  • Call one pair unless there are 7 or more dealer one-card outs remaining that beat you, else
  • Call Ace-high when 2 or more Aces and Kings seen with 9 upcard copies, else
  • Call Ace-high with 4 or more Aces and Kings seen with 8 upcard copies, else
  • Call Ace-high with 6 or more Aces and Kings seen with 7 upcard copies,
  • else fold

Update: I worked out an improved 6-way collusion strategy that yields a +0.43% return with only a couple simple rules.

WPT3X All-In Collusion (3 Players)

Posted in collusion, wpt3x by stephenhow on July 27, 2011

I’m headed out to Vegas next week with two friends, so I re-worked a 3-player collusion strategy for WPT-3X All-In. They probably won’t want to play the game, because one friend loves fast-paced, hi-limit blackjack, and the other friend doesn’t like gambling. Well, I like working out these collusion strategies, whether I use them or not. So, the following 3-player collusion strategy reduces the basic strategy house edge from 0.74% to a mere 0.05% (effectively zero). Of course, you usually raise 3x in this game, so the variance is high relative to the Ante. But, with the casinos increasing the minimum bets, and if you play for hours on end, a zero edge game is much cheaper over the course of a weekend.

In fine-tuning the strategy, I found that the low straight cards are important for weak hands like 5-2, and 3-2. For example, if you have 3-2, and are not copied, it’s also necessary that your friends don’t hold any 4,5, or 6′s. I’ve included this requirement in the strategy table, where the asterisks indicate the maximum straight cards seen.

Below is the 3-way collusion strategy table for WPT-3X All-In poker. The first table is for offsuit cards, and the second table is for suited cards. You should use the suited table ONLY IF your friends hold at most one of your suits. Otherwise, use the offsuit table. The yellow squares indicate basic strategy folding hands. The numbers in the boxes indicate the maximum number of copies allowed in order to raise the hand. The asterisks indicate the maximum number of straight cards seen, in combination with the max copies. One asterisk allows at most 1 straight card when max copied. Two asterisks requires no straight cards seen when max copied.

3 Player Collusion Strategy for WPT-3X All-In.

Examples

You have 32o. The chart says 0**. Your two friends don’t have any deuces or treys. You should 3x raise if none of your straight cards are seen (4, 5, 6). Otherwise fold.

You have 72o. The chart says 0*. You should raise if your friends don’t copy your hand, AND there’s at most one (3, 4, 5, or 6). Otherwise fold.

You have 52s. The chart says 1**. You should raise if your friends don’t copy your hand, regardless of straight cards. However, if you’re copied once, you should raise only if your friends hold no straight cards (3, 4, 6).

You have 54s. The chart says 2**. You should raise if your friends hold 0 or 1 copies of your card. If you’re copied twice, then you should only 3x raise if your friends hold no straight cards (2, 3, 6, 7).

You have T5o. The chart says you should raise if you’re copied up to two times. Otherwise, fold if you’re copied 3 or 4 times.

You have A2o. You should raise even if you’re copied 4 times (i.e., both your friends also hold A2, or they hold 22 and AA).

Collusion Analysis For Wild 52 @ Las Vegas Flamingo

Posted in collusion, wild52 by stephenhow on June 18, 2011

I got pretty excited last week about possibly exploiting Joker information for the new Wild 52 game at the Las Vegas Flamingo. Its a 7-card poker game with a Joker, where up to 6 players hold 5 cards each, and play against a dealer hand. There are 2 community cards, an Ante, and two 2x betting rounds. I figured it was a lock that sharing Joker Busy status with confederates, combined with an optimized strategy both, would yield at least a 5% player edge. So I worked it all out, and was shocked to find only a ~1% improvement from around a 2% house edge to a 1% house edge.

Honestly, I was thinking “Vegas trip”, and betting $25 or $100 Antes with my friends, winning thousands each until they shut us down. I thought it’d end up being obvious to everyone that knowing where the Joker would yield a huge player advantage. I figured they’d set up the game with a small house edge, and they didn’t foresee players sharing Joker information. Well, whether they looked into it or not, sharing Joker info didn’t help out much to change the overall odds :(

Here’s how the game is played:

  1. The game is played with a 52 card deck plus one completely wild Joker.
  2. Each player Antes before the hand begins.
  3. Each player and the dealer receives 5 cards, dealt face down.
  4. The player looks at his hand, and decides to either Play it by 2x raising his Ante, or folding his hand and losing his Ante.
  5. The dealer then turns up the first community card.
  6. Based on his 6-card hand, each player either checks, or makes the Option bet (2x the Ante).
  7. The dealer then turns up the 2nd community card.
  8. Action is complete, and the dealer then turns up his 5 cards.
  9. The dealer’s 7 card hand (5 hole + 2 community) qualifies with a pair of 5′s or better.
  10. If the dealer does not qualify, the 2x Play and Option bets push, and the player automatically wins the Ante bet.
  11. If the dealer qualifies, then the Ante, Play, and Option bet all play even-money against the dealer hand.

Here’s a simple basic strategy that yields a 2.4% house edge:
For the 2x Play bet (5 card hand):

  • Play any pair or better.
  • Play any flush draw or straight draw (including gutshots).
  • Play A-high, if 2nd card is at least a Queen, and 4th card is at least an Eight.
  • Fold all others.

For the 2x Option bet (6 card hand):

  • If the community card is a Joker, only bet trip-8′s or better.
  • Bet any hand with a non-community Joker.
  • Bet two pairs, if the community card is below your top pair, or if your top pair are 8′s or better, or if you also have a straight or flush draw.
  • Only bet a pair of Kings or better if you also have any straight or flush draw.
  • Check all others.

Effect of Collusion

I looked into the advantage obtained if 6 players colluded to share “Joker busy” information. This knowledge changed the 5th and 6th street strategies, but only at the margins, which don’t happen frequently enough to significantly change the overall EV :(

The differences on 5th street are that:

  • If the Joker is busy, you can play any Ace-high, or 4 cards higher than a Six.
  • Else, if the Joker is hiding, you can only play a pair or better.

The differences on 6th street are that:

  • If the Joker is busy, you can bet a pair of Jacks or better (and Ten’s if the community card is below a Ten).
  • Else, if the Joker is hiding, you can only bet two pairs or better (Jacks up or better, or if the community card is under your top pair).

These differences only add up to a 1.2% improvement, and the house edge is still 1.2%.

This was pretty disappointing. I called my friends back to tell them we *weren’t* headed to Vegas that weekend :(

Joker/Ace Collusion Analysis For Pai-Gow Poker

Posted in collusion, pai-gow poker by stephenhow on June 5, 2011

Experienced Pai-Gow players (i.e., those who play every day) often tell each other where the Joker and Aces are. At a full table, 6 players hold 42 cards, so if no one says “joker busy”, then the dealer probably has it (63.6%). Also, if the players don’t hold many Aces, then the dealer is going to have a stronger-than-usual hand. I’ve often wondered how practical and powerful a Pai-Gow collusion strategy would be, so I worked out an analysis of the problem.

Immediately, I saw that Ace/Joker info would help mostly for deciding whether to play “pair-pair” (small pair in front, big pair in back) or “two pair behind” (kickers in front, two pair in back). The two pair hands (or other “break/keep” decisions) occur about 20% of the time, frequently enough for a collusion strategy to have some promise. For these two pair decisions, I thought it’d be important to see how the Ace/Joker count affects the dealer front pair %, and the dealer two pair or better behind %. This seemed natural, because I figured if the players held 3 Aces, then the chance of a single dealer Ace was pretty high, which he’d probably play up front (the “3 Ace Effect”).

I ran the analysis (CA rules where the Joker is completely wild), and was amazed to see the following results!

Probabilities of dealer hands as a function of known Ace/Jack locations (6 players).

Sure enough, the expected results pop right out of the graphs. On the left side are hands where the Joker is “busy”, i.e., one of the players holds it, so the dealer cannot have it. On the right the Joker is not busy, and is probably in the dealer’s hand. The top graphs represent the strength of the dealer front hand vs. the known Ace count, and the bottom graphs show the strength of the dealer back hand vs. the Ace count.

As expected, the dealer hand is strong when the Joker is not busy. Also, the dealer front hand is weakest when the players hold 3 Aces. As I pointed out earlier, the “3 Ace effect” results from the high probability that the dealer has the remaining Ace, and will play it up front. This is the “sweet spot”. So, we see the weakest dealer hand happens when the Joker is Busy and the players hold exactly 3 Aces (12.5% front pair or better) and the strongest dealer hand happens when the Joker is not busy, and the players hold no Aces (80% front pair or better). This is a huge, huge difference, and suggests a big difference in the right way to play two pairs for these two cases.

So, by sharing Ace/Joker information at a full table, you learn when the dealer has either a very weak hand (Joker Busy, 3 Aces seen; most common case, 34.3% of the time), or a very strong hand (Joker not Busy, and 2 or less Aces seen; rare 3% of the time). You should alter your 2 pair decisions to take advantage of this information, as shown below.

Examples

The following examples show cases where Joker/Ace info would save you around 20% of your bet, on average.

Strong Dealer Hand

Let’s say you have two pairs, Jacks and 5′s, and no kickers: Jd Jh 5s 5d 8s 7h 3c. Normally, you’d play this pair-pair, with a pair of 5′s up front, and a pair of Jacks behind. No one in the world would play 8-high in front, and two pairs (Jacks and 5′s behind). But, in the rare (3%) case where the Joker is hiding, and the players have 0, 1, or 2 Aces, then you’re actually better off playing two pair behind!

Jacks and 5′s, Joker Hiding, 8-high Kicker
Aces Seen EV(Pair-Pair) EV(2 Pair Behind) Decision
0 -0.72 -0.52 play 8-high front, J’s and 5′s behind
1 -0.57 -0.36 play 8-high front, J’s and 5′s behind
2 -0.34 -0.25 play 8-high front, J’s and 5′s behind
3 -0.06 -0.24 play 5′s in front, J’s behind
4 -0.11 -0.26 play 5′s in front, J’s behind

Weakest Dealer Hand

Now, say you had the J’s and 5′s again, but this time you have an Ace kicker. This is a decision point that people may think about. Most house-way strategies will play two pair behind. You should definitely play two pair behind when the Joker is hiding. However, when the dealer hand is extremely weak because the Joker is busy and exactly 3 Aces are seen (“3 Ace Effect”), ignore your Ace, and play pair-pair. This is normally considered aggressive, but against the weakest dealer hand (the most common case, 34.3% of the time), you should go for it. The “3 Ace Effect” creates the “sweet spot” that minimizes the expected dealer front hand strength.

Jacks and 5′s, Joker Busy, Ace Kicker
Aces Seen EV(Pair-Pair) EV(2 Pair Behind) Decision
1 -0.12 +0.17 play Ace front, J’s and 5′s behind
2 +0.18 +0.30 play Ace front, J’s and 5′s behind
3 +0.52 +0.32 play 5′s in front, J’s behind
4 +0.41 +0.55 play Ace front, J’s and 5′s behind

How To Exploit Ace/Joker Info

It’s kind of pointless to work out the extreme details of the optimal Pai-Gow strategy given Joker/Ace info. Most people won’t remember the details, or even the broad strokes, given the gory details. However, it’s pretty easy to boil it all down to a simple collusion strategy. The players should first find out if the Joker is “busy” or not. If players have not seen the Joker, then they should play conservatively and favor two pair behind (like house way). If the Joker is hiding, check if there are only 0 or 1 Aces out. If so, the dealer has a very strong hand, so play extremely conservatively. For example, I would play no front and Kings and 6′s behind against the very strongest dealer hand. But the strongest dealer hand is rare 0.4%), so it’s not worth checking for, unless you have a huge bet out there. The Joker is busy 80% of the time. When the Joker is busy, and you have a possible two pair decision, find out if 3 Aces are seen, and thus the dealer has the weakest possible hand (the most common case, 34.3% of the time). If so, play two pair aggressively according to the below table.

Here’s a summary of the practical 6-player collusion strategy:

  • If Joker is Busy
    • If exactly 3 Aces seen, the dealer has the weakest possible hand (most common), so play aggressively.
    • Else play normal.
  • Else, Joker is Hiding
    • If 0 or 1 Aces seen (extremely rare), dealer has the strongest possible hand, play super conservatively
    • Else, play conservatively
Minimum Front for Two Pair Behind (Aggressive, only use against the weakest dealer hand.)
22 33 44 55 66 77 88 99 TT JJ QQ KK
AA p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
KK p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
QQ p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
JJ AK AK pp AK AK AK p/p p/p p/p
TT AQ AQ AQ AK AK AK AK AK
99 AJ AJ AQ AQ AQ AQ AK
88 AT AT AJ AJ AJ AQ
77 A4 A4 A8 AT AJ
66 A4 A5 A5 A7
55 A4 A7 A8
44 KJ KQ
33 K9

where “p/p” means always play pair-pair. Note that this table more aggressively plays pair-pair, because of the stronger-than-normal minimum front hand required to play two pair behind. Most tables require only an Ace for the larger pairs, and only a Jack or Queen for the lower pairs. However, the above table requires not only an Ace, but often AK, AQ, or AJ. Even the smallest pairs require K9 to play two pair behind. This is an aggressive table to play against a weak dealer hand that likely (87.5%) will not have a front pair.

Compare the above aggressive table to the more conservative strategy below, where the dealer hand is moderately strong (Joker is Hiding, and all 4 Aces are held by the players). Notice front hand requirements are much lower than the aggressive strategy. This means you end up playing “two pair behind” much more often. This is as you would expect against a stronger dealer hand. I don’t provide two pair tables for all Joker/Ace combinations, but provide these two to show the effect of expected dealer hand strength on how you play two pairs.

Minimum Front for Two Pair Behind (Conservative, used when Joker Hiding and 4 Aces Seen)
22 33 44 55 66 77 88 99 TT JJ QQ KK
AA p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p p/p
KK Ax Ax p/p p/p p/p p/p p/p p/p p/p p/p p/p
QQ Kx Kx Ax Ax Ax p/p p/p p/p p/p p/p
JJ 2p Qx Qx Kx Kx Ax Ax Ax Ax
TT 2p 2p 2p Qx Kx Kx Kx Ax
99 2p 2p 2p 2p Jx Qx Kx
88 2p 2p 2p 2p Jx Jx
77 2p 2p 2p 2p 2p
66 2p 2p 2p 2p
55 2p 2p 2p
44 2p 2p
33 2p

where “2p” means always play two pair behind.

Summary

Frequencies of Dealer Hand Strengths
Description Strength Frequency
Joker Hiding, 0 Aces Seen Strongest 0.02%
Joker Hiding, 1 Aces Seen 0.4%
Joker Hiding, 2 Aces Seen 3.0%
Joker Hiding, 4 Aces Seen 8.6%
Joker Hiding, 3 Aces Seen 8.8%
Joker Busy, 0 Aces Seen 0.1%
Joker Busy, 1 Aces Seen 2.0%
Joker Busy, 2 Aces Seen 13.2%
Joker Busy, 4 Aces Seen 29.6%
Joker Busy, 3 Aces Seen Weakest 34.3%
Total 100%

Simplified +EV Collusion For WPT-3x All-In (4 Players)

Posted in +EV, collusion, wpt3x by stephenhow on May 30, 2011

I cleaned up my +EV collusion strategy for the World Poker Tour 3x All-In Hold’Em casino table game, because my old strategy was basically unplayable. I’ve simplified the strategy to focus on copied cards, and to ignore the high cards that only slightly lower the probability of the dealer qualifying. I optimized the strategy for four players, since I wanted a +EV worth playing for.

I always see the game at the Bellagio, when I walk through it on my way to the Forum Shops at Caesar’s. It’s also dealt at my local Sycuan Casino. I always tell my friends we should play it, but no one has any interest in +EV play, or carnival games. I figure someone out there sees the value in sharing hole card info for this game, since it starts with a mere 0.74% house edge. The following simplified collusion strategy simulates at a player advantage of about 0.31%.

The game is really simple, and other than the bad bonus bets, is not very exciting. Each player posts an Ante to start the game. The player then receives 2 down cards, which combined with the 5 card board, makes a Hold’Em poker hand. The dealer also receives two down cards, for his Hold’Em hand. You look at your 2 down cards and decide to either 3x raise “all-in”, or fold and lose the Ante. Once the players action is complete, the dealer turns up his hole cards. The dealer hand qualifies if it’s a pair, or has a blackjack value of 11 or greater. If the dealer doesn’t qualify, the remaining players win the Ante bet, and the 3x Raise bet pushes. If the dealer qualifies, then he deals the flop, turn, and river. The dealer’s Hold’Em hand is compared to each player’s Hold’Em hand. If the player has the higher hand, he wins even money on the Ante and the 3x Raise. If the dealer has the higher hand, the player loses both bets. If the hands are equal, the bets push.

The basic strategy is very simple. You’re supposed to 3x raise any pair or suited hand. The only hands you fold are 23o thru 28o, and 34o, 36o, and 37o. (That’s deuce-trey thru deuce-eight offsuit, and trey-four, trey-six, and trey-seven offsuit.)

The collusion strategy is also very simple. You 3x raise anything, unless you’re copied. Fold a weak hand (the basic strategy folding hands) if copied. Slightly stronger hands are still played if only copied once. You always play a Jack or better.

If four players share down card info, then the players have about a 0.31% edge over the house. The players need to know if their hole cards are copied by their neighbors. Here’s the modified strategy:

Hand Basic Strategy 4-Player Strategy
Offsuit
23o thru 28o Fold 3x Raise if no copies
29o, 2To 3x Raise 3x Raise if no copies
34o, 36o, 37o Fold 3x Raise if no copies
35o 3x Raise 3x Raise if no copies
38o 3x Raise 3x Raise if no copies
39o 3x Raise 3x Raise if ≤ 1 copies
3To 3x Raise 3x Raise if ≤ 2 copies
45o 3x Raise 3x Raise if ≤ 2 copies
Suited
23s thru 28s 3x Raise 3x Raise if no copies, or 1 copy and ≤ 1 suit seen
29s, 2Ts 3x Raise 3x Raise if ≤ 1 copies, or 2 copies and ≤ 1 suit seen
34s thru 37s 3x Raise 3x Raise if no copies, or 1 copy and ≤ 1 suit seen
38s 3x Raise 3x Raise if ≤ 1 copies, or 2 copies and ≤ 1 suit seen
39s 3x Raise 3x Raise if ≤ 1 copies, or ≤ 1 suit seen
3Ts 3x Raise 3x Raise if ≤ 2 copies, or ≤ 1 suit seen

Additionally, you should fold triple-copied offsuit hands T2 thru T6, 92 thru 96, 82 thru 85, 72 thru 76, 62 thru 65, and 54.

Simplified Collusion for Mississippi Stud (4 Players)

Posted in collusion, mississippi stud by stephenhow on May 22, 2011

I know it’s not always feasible to wait for 6 players in a Mississippi Stud game, nor is it easy to get 6 strangers to collude together. So while I was at it, I figured I’d work out a simple collusion strategy for 4 players, and see how well it worked. It helps a lot, improving the game from a -4.91% house edge down to about a manageable -1.4% house edge (or, an element of risk of about -0.4%). Here’s the simplified strategy for 4 colluding players:

  • 3rd Street
    • small pair: 3x bet if 0 copies, 1x bet if 1 copy, fold if 2 copies
    • 3x bet 6 high suited outs
    • if suited, 1x bet at least 3 high outs, or 2 high outs and 1 mid out, or 1 high out and 3 mid outs, or 4 mid outs
    • if offsuit, 1x bet at least 3 high outs, or 2 high and 2 mid outs, or 5 mid outs
    • fold all others
  • 4th Street
    • 3x bet 7 suited high outs
    • 3x bet suited 6 high and 1 mid outs
    • 3x bet 0-gap straight flush draw
    • 1x bet small pair
    • 1x bet suited cards
    • 1x bet 3 high and 2 mid outs
    • 1x bet 2 high and 4 mid outs
    • 1x bet 1 high and 6 mid outs
    • 1x bet 7 mid outs
    • 1x bet 6 mid outs w/ 2-gap
    • 1x bet 5 mid outs w/ 1-gap
    • 1x bet 4 mid outs w/ 0-gap
    • fold all others
  • 5th Street
    • 3x bet flush draw, or open-ended straight draw with all outs remaining
    • 1x bet low pair, or straight draw
    • 1x bet 6 high outs
    • 1x bet 5 high and 1 mid outs
    • 1x bet 4 high and 3 mid outs
    • 1x bet 3 high and 5 mid outs
    • 1x bet 2 high and 7 mid outs
    • 1x bet 1 high and 9 mid outs
    • fold all others

The strategy is very close to the simplified 6 player strategy, where you only play a little tighter — you only need one more mid out on the 4th and 5th Street decisions. So it’s really easy to remember both strategies. It’s also pretty easy to collude with 4 players, since 4 players sitting together can easily see each others cards, unless someone is deliberately hiding their hole cards.

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