Discount Gambling

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.

I just tried out ShuffleMaster’s new Dealer Bluff Six Card Poker game at Pala Casino, just north of San Diego. It’s a new idea in poker-based table games, where the dealing machine knows the value of its hand, and bets against you. When I first saw this game last week, I was really puzzled. How would you play your hand? Would it cost a lot of money to figure out how aggressive the machine was? How was I supposed to analyze this game?

I looked into the details of the game, and its actually pretty simple how it all works. By its stated math, its not a bad game at around 0.5% house edge (element of risk). The player first bets an equal Ante and Blind bet. All players and the dealer are dealt six cards, from which to make a five card hand. The dealer acts first, and either checks (0x), or bets 1x ,2x, or 3x the Ante, based on its hand strength, and simple randomization. This simple randomization is stateless (independent from hand-to-hand), and uses a published wager matrix (“house-way”). The player may either fold (or check if no dealer bet), call, or raise (double the dealer bet). The dealer calls all player bets/raises. The hand is resolved just like their Ultimate Texas Hold’em game, where the Ante plays only if the dealer qualifies with a pair or better, the Play bet always gets even-money action, and the Blind bet pays only if the player wins, and according to a paytable. The game is easy to analyze, because of the simple, stateless, and published dealer betting strategy (“house-way”). Given house-way, a player basic strategy is determined, and the house edge is calculated.

The game is kind of fun, because the dealer acts first, and you benefit from this position. If you raise the dealer, it must always call (it can’t come over the top, and re-raise you). Sometimes you have easy folds, or easy raises. However, basic strategy requires you to call or raise a lot of hands, even when the dealer is betting 3x, since it bluffs a lot. This of course adds to your variance, and sometimes you’ll wonder why you had to call 3x “to keep the dealer honest”.

As usual in these ShuffleMaster games, the Ante bet is -EV because of the pair qualifier needed to pay, and the Blind bet is -EV because of the paytable. The Play bet is +EV, because you’re acting last, and the dealer must call your bets/raises, and cannot re-raise. However, the dealer may bluff, so you need to know (exactly) how the betting logic works. If you have this info (or can learn it), then the Play bet will always be +EV. The game is designed so that the sum of these 3 EVs yield a reasonable house edge.

This is why I have reservations about the game. Player basic strategy is completely dependent upon how the dealer bets his hand. ShuffleMaster goes out of its way to provide all the data on this process, and even provides the players with basic strategy and percentage tables in their literature. At Pala, there are ample supplies of this literature at the table. The dealer house-way is provided. However, if house-way should change for any reason, the player using an out-of-sync basic strategy may be severely penalized.

This table provides the percentages of the way the dealer will bet his hand, using a random number generator. No previous knowledge of the player’s history, bet amount, or anything else is used to determine the dealer’s bet.

Based on this house-way wager matrix, basic player strategy is optimized as follows:

If everything operates exactly per these assumptions, then the house edge is only 0.5% (element-of-risk), or about 2% of the Ante bet. This is only $.10 per $5 Ante bet, and is very reasonable. However, if the dealer strategy is not exactly as described by the stateless house-way table above, the player return could be much worse. For example, the dealer cannot sub-divide the last “Nothing” row, into various sub-percentages based on A-high, K-high, etc. It’s assumed that it treats all no-pair hands the same, and applies an equal bluff probabilities to them, regardless of high card value.

What’s worse, if the dealer behaves significantly differently from the house-way table, the effect on the player using a basic strategy optimized for it could be disastrous. As an experiment, I tweaked the wager matrix to make the dealer more honest (bluff less) in a few rows, to see the effect.

First, I made the dealer more conservative with its high pairs. I shifted the majority of these bets to 1x, thereby giving the player less opportunities to come over the top with a raise to 2x, or even 6x. I modified the following single table row to

and the results were very significant, increasing the house edge to about 5.2% (element-of-risk), or -17.8% of the Ante bet! This really surprised me. The house could (not intentionally, of course) change a few small settings, and really hurt the player relying on the published basic strategy. Unless the player could look at the stats maintained by the betting mechanism, it’s a real leap of faith to get involved with this game for any period of time.

Next, I looked at the effect of making the dealer more conservative with its nothing hand, making it check the vast majority (80%) of the time, and betting 1x the rest of the time (20%). Again, this reduces the ability of the player to just call and win with a reasonable hand, or come over the top with a big raise.

The effect of this row change was less significant, increasing the house edge to about 2.0% (element-of-risk), or -6.6% of the Ante bet. If both these row changes are applied together, the effects are additive. Of course, if the player knows about these changes, then he can modify his basic strategy to again be +EV on the Play bet. Again, since the dealer acts first, must call all player raises and cannot come over the top, the Play bet will always be +EV, given knowledge of how the dealer behaves. The above house edge effects for the wager matrix changes are based on a player using the published basic strategy.

I stopped after looking at these two simple, but significant effects. That’s all I needed to see.

Usually, I’m very naive, and always assume the casino is operating according to fair rules, and everything is on the up-and-up. However, with the advent of this new type of game, where the shuffler actually reads all hands it deals (it must, as the dealer hand is the last out of the shuffler, and the operator hits the ‘stop’ button to tell the betting logic use the last hand), I’m getting a little nervous. Also, the nature of a heads-up poker game requires knowledge of your opponent. Although ShuffleMaster provides a complete model of this opponent, its always possible that something gets lost in the process, a transcription error occurs, and/or something changes, and basic strategy gets out of sync with the actual dealer behaviour.

I think the game could be fun and safe, provided that ShuffleMaster provides a few assurances to the players. First, they need to make it extremely clear, and verify, that there’s only one version of the wager matrix. Furthermore, this can never change. Additionally, they could provide assurances to the players that the betting logic is audited by built-in stats collection in the shuffler, and periodically checked against the published, unchanging house-way. If they can provide these assurances, you can feel safe walking up to the game for the first time, and betting following basic strategy. Otherwise, while possible, it’s a pretty big task for people to figure out the dealer betting strategy, and to devise the appropriate counter-strategy. I’m just there to have fun, not to play Big Blue (that can see your cards).

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.

I got a mention on the universally-known “last word on gambling” Wizard of Odds site. Needless to say, its nice to suddenly get much more page hits in a day then I used to get in a month. So welcome, especially if you’re here for the +EV angles I’m working to find. Well, if you’re in San Diego, there are two +EV games right now, Mississippi Stud @ Barona, and CSM (card) craps at Viejas. These games have been spread for a while, and the casino isn’t afraid of any advantage players. They’re making plenty of money with these games. No one is going to get rich from these small edges. Its a lot like card-counting in blackjack, but without all the hard work. I just re-wrote the page on the Viejas craps game, to make it easier to see the edge, and how to play it. Hopefully someone will actually try it. Welcome!

I thought I’d post a thorough description of how to play the PlayCraps game at Viejas, including how to account for the shuffle, using a fair-weighted counts for all the points.

First, here’s the value of each roll, and how it contributes to the counts for each point.

Ideally, you’ll keep a count for each point. Of course, this is hard to do. I just wait for the point to come out, then I try to guess if I saw any key cards lately, and make up an initial count for the point. Then I adjust the count for each roll as in the above table. It’s very easy while the muck accumulates. Then, when the dealer shuffles the muck into the CSM, I keep counting as normal, for about 5 rolls (approx. buffer depth). After these five rolls, I start the count again, based on a guess of what’s in the muck.

simulated session with annotations after the jump! (more…)

There’s a new craps game at Viejas Casino, that I’ve been playing lately. It’s definitely my new favorite game, since it’s dealt to players seated at a table, and it provides the best odds in the casino. In fact, it’s a winning game for a player using basic strategy (yes, I know, sounds impossible, but read my full analysis). Although the game is beatable, you can only make a killing the normal way, by gambling and getting lucky. However, it’s always good to know the odds are in your favor.

PlayCraps™

If you haven’t read my full analysis yet, you’ll first need to know the player edge is on the don’t pass / don’t come side, and the house’s edge on the pass odds are higher than usual. So don’t rush out here and start betting the pass line and taking odds like you normally do. Switch over to the Dark Side first, if you’re not already over there.

If you like taking a shot a craps, come on out to Viejas, and have some fun. The layout of the game is really enjoyable, because it promotes fast action, and you get to sit while you play. It’s very relaxing, and there’s no stress of rolling the dice in front of a bunch of angry players. It’s just you watching the cards that come out, and deciding to increase/decrease your don’t pass or don’t come odds. Even for the best counts, you’re not really justified in buying any No-4′s or No-10′s, but at times, the odds overcome the vig.

If you like advantage play, and can enjoy playing the Don’ts, read up on this game, and come on out. It’s not every day that a game is beatable by basic strategy.