Discount Gambling

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.

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.

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.

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

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

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

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

Basic Strategy

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Collusion Analysis

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

Bonus Side Bet

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

Detailed Stats

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

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.

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.

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.

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.

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.

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%.

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.

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.

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.

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.

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.

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%.

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.)

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

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

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

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

Basic Strategy

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

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

Advantage Play

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

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

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

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

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

Under-The-Gun 31 Game

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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