One of the dealers at Viejas told me about a weird Hold’Em carnival game at the Hustler Casino in LA, so I decided to check it out (over the phone). As with all table games in California card rooms, you have to pay a per-hand “collection” to the house in order to pay. You play against a 3rd party bank, which is always available whenever the game is open. (The 3rd party bank is usually a separate “corporation” with an arrangement with the casino, but the house is prohibited from directly banking the game.)
Anyways, here’s how the game works. All bets receive even-money action against the dealer, including the Ante. The dealer plays two hands against the player’s one hand. There’s no qualifier to the dealer hand, and if the dealer hands don’t make a pair (including the board), the player instantly wins all wagers. Specifically, the rules are as follows:
- The player must Ante before the hand begins.
- The player receives two cards, and decides either to (A) wager an amount equal to the Ante (1x) in exchange for the right to make additional post-flop bets, or (B) to just play the hand for only the Ante bet. (There is no folding in this game.)
- On the flop, the player may either bet an amount equal to the Ante (1x), or check. (Provided he wagered the 1x preflop bet.)
- On the turn, the player may either bet an amount equal to 2x the Ante, or check. (Provided he wagered the 1x preflop bet.)
- On the river, the player may either bet an amount equal to 2x the Ante, or check. (Provided he wagered the 1x preflop bet.)
- The dealer turns up the two bank hands (two separate sets of hole cards).
- If both the dealer hands are lower than a pair, the player automatically wins even money on all wagers, including the Ante and preflop 1x bet.
- Otherwise, the player receives even money action on all his wagers (including Ante and 1x preflop bet) vs. the best dealer hand.
The game is pretty weird in that you have to play against two dealer hands. I guess this resembles more of a poker game, in that you have to beat multiple players. However, you’re not getting 2:1 odds on your money, so it’s clearly worse. The only benefit to the game is that there’s no qualifiers on the action, and you automatically win if both dealer hands are lower than a pair. Plus, your Ante never folds.
Banking The Game
As I show below, the bank has about a 6.25% edge against a near-optimal player. Needless to say, most people will probably not even come close to playing this game correctly. Even though they can find this page on the web, they won’t bother to look for it, let alone read it, or god-forbid, follow it. I can only imagine how badly people play this game. I’m guessing they under bet pre-flop, over-bet the flop, and under-bet the river. And every post-flop mistake will cost the player at least 10% of the Ante, and there are up to 3 post-flop decisions per hand. Plus the sucker bonus bet that everybody plays.
According to the house rules, you can bank twice per round. When I talked to the floor over the phone, he says that “99.9% of the time” players don’t bank the game. So you’ll share banking privileges evenly with the corporation bank (and other bankers, if present). It costs $2/hand to bank the game, so even if there are 3 players playing $5 Antes, you’ll break even if each player averages a total of 13% of EV mistakes per hand. People probably bet top pair on the flop, which is around a 20% mistake, and they probably bet 2nd pair on a paired board on the river, which is another 20% mistake.
If you live in LA, you should probably give banking a shot, especially on the weekends. It sounds like you’ll only be competing with the corporation for banking rights, and you won’t need a huge bankroll to bank $5 Antes. You might need a minimum bankroll to cover paying off a possible Royal Flush (200:1?) on the bonus bet. But the bonus bet is pretty easy money, depending on the pay table.
I worked out a relatively simple strategy that’s probably close to optimal, since there weren’t many exceptions to the following rules. The following strategy simulates at a 6.25% bank edge.
- Preflop (1x)
- See below preflop tables.
- Flop (1x)
- If trips on board, bet pocket 3′s or higher, else check.
- If the board is paired, bet trips or better, pocket 10′s or higher, or top pair if King’s or higher.
- If the board is suited, bet trips or better, or top two pair.
- If the board is not suited nor paired, bet two pair or better.
- Check everything else.
- Turn (2x)
- If quads on board, bet nut kicker, else check.
- If trips on board, bet full house except under pair, else check.
- If two pair on board, bet if you beat board by more than just kickers, else check.
- If scare flush board, bet 4th worst flush or better, else check.
- If scare straight board, bet any straight or better, else check.
- If board paired, bet trips or better, top pair or over pair if no possible flush, or top pair with 4th nut flush draw or better, else check.
- Else, bet two pairs or better, and top pair or over pair unless gutshot board or possible flush.
- Check all other hands.
- River (2x)
- If quads on board, bet 2nd nut kicker, else check.
- If trips on board, bet full house unless under pair, else check.
- If two pair on board, bet any hand that beats board by more than kickers, else check.
- If scare flush board, bet 3rd worst flush or better, else check.
- If scare straight board, bet any straight or better.
- If board paired, bet trips or better, or top or over pair (or pocket pair higher than 2nd board singleton) unless gutshot board, else check.
- If board not paired, bet two pair or better, or top pair or over pair or 2nd pair unless gutshot board, else check.
where “top pair” means you pair with the highest singleton on board, and “2nd pair” means you pair with the 2nd highest singleton on board.
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:
- The game is played with a 52 card deck plus one completely wild Joker.
- Each player Antes before the hand begins.
- Each player and the dealer receives 5 cards, dealt face down.
- 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.
- The dealer then turns up the first community card.
- Based on his 6-card hand, each player either checks, or makes the Option bet (2x the Ante).
- The dealer then turns up the 2nd community card.
- Action is complete, and the dealer then turns up his 5 cards.
- The dealer’s 7 card hand (5 hole + 2 community) qualifies with a pair of 5′s or better.
- If the dealer does not qualify, the 2x Play and Option bets push, and the player automatically wins the Ante bet.
- 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 :(
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!
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.
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!
|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.
|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
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.
where “2p” means always play two pair behind.
|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%|