Discount Gambling

Arizona Stud @ Red Wind Casino, WA

Posted in Uncategorized by stephenhow on July 30, 2014

Screen Shot 2014-07-30 at 9.21.14 PMArizona 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.

Optimal Outcomes for Arizona Stud
Outcome Combinations Frequency Net Return
Win 4x Play w/ ROYAL_FLUSH against qualified dealer 59,240,916 0.000001 5 0.000005
Win 4x Play w/ FULL_HOUSE against qualified dealer 89,284,476,240 0.001605 5 0.008025
Win 4x Play w/ FLUSH against qualified dealer 10,295,059,284 0.000185 5 0.000925
Win 4x Play w/ STRAIGHT against qualified dealer 13,761,723,420 0.000247 5 0.001237
Win 4x Play w/ THREE_OF_A_KIND against qualified dealer 674,048,087,712 0.012117 5 0.060586
Win 4x Play w/ TWO_PAIRS against qualified dealer 1,233,004,030,272 0.022165 5 0.110827
Win 4x Play w/ ONE_PAIR against qualified dealer 3,533,244,131,304 0.063516 5 0.317580
Win 4x Play w/ HIGH_CARD against qualified dealer 44,095,696,596 0.000793 5 0.003963
Lose 4x Play against qualified dealer 5,323,636,585,296 0.095701 -5 -0.478507
Push 4x Play against qualified dealer 90,869,346,720 0.001634 0 0.000000
Win 4x Play w/ ROYAL_FLUSH against unqualified dealer 20,545,164 0.000000 4 0.000001
Win 4x Play w/ FLUSH against unqualified dealer 4,736,232,972 0.000085 4 0.000341
Win 4x Play w/ STRAIGHT against unqualified dealer 5,959,832,148 0.000107 4 0.000429
Win 4x Play w/ THREE_OF_A_KIND against unqualified dealer 350,836,147,584 0.006307 4 0.025227
Win 4x Play w/ TWO_PAIRS against unqualified dealer 28,557,204,480 0.000513 4 0.002053
Win 4x Play w/ ONE_PAIR against unqualified dealer 2,530,675,447,344 0.045493 4 0.181973
Win 4x Play w/ HIGH_CARD against unqualified dealer 1,146,771,919,728 0.020615 4 0.082461
Lose 4x Play against unqualified dealer 25,248,339,684 0.000454 -4 -0.001816
Push 4x Play against unqualified dealer 25,026,495,696 0.000450 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against qualified dealer 94,841,496 0.000002 2 0.000003
Win 1x Play w/ STRAIGHT_FLUSH against qualified dealer 541,732,152 0.000010 2 0.000019
Win 1x Play w/ FOUR_OF_A_KIND against qualified dealer 6,309,658,080 0.000113 2 0.000227
Win 1x Play w/ FULL_HOUSE against qualified dealer 45,729,841,680 0.000822 2 0.001644
Win 1x Play w/ FLUSH against qualified dealer 51,704,956,552 0.000929 2 0.001859
Win 1x Play w/ STRAIGHT against qualified dealer 90,255,233,808 0.001622 2 0.003245
Win 1x Play w/ THREE_OF_A_KIND against qualified dealer 591,850,723,248 0.010640 2 0.021279
Win 1x Play w/ TWO_PAIRS against qualified dealer 1,068,681,540,840 0.019211 2 0.038423
Win 1x Play w/ ONE_PAIR against qualified dealer 4,443,972,518,832 0.079888 2 0.159776
Win 1x Play w/ HIGH_CARD against qualified dealer 188,023,085,280 0.003380 2 0.006760
Lose 1x Play against qualified dealer 10,848,202,319,420 0.195015 -2 -0.390029
Push 1x Play against qualified dealer 161,798,077,992 0.002909 0 0.000000
Win 1x Play w/ ROYAL_FLUSH against unqualified dealer 14,941,584 0.000000 1 0.000000
Win 1x Play w/ STRAIGHT_FLUSH against unqualified dealer 220,863,744 0.000004 1 0.000004
Win 1x Play w/ FLUSH against unqualified dealer 23,880,526,848 0.000429 1 0.000429
Win 1x Play w/ STRAIGHT against unqualified dealer 48,873,031,380 0.000879 1 0.000879
Win 1x Play w/ THREE_OF_A_KIND against unqualified dealer 34,291,273,536 0.000616 1 0.000616
Win 1x Play w/ TWO_PAIRS against unqualified dealer 301,557,935,124 0.005421 1 0.005421
Win 1x Play w/ ONE_PAIR against unqualified dealer 3,348,961,937,952 0.060203 1 0.060203
Win 1x Play w/ HIGH_CARD against unqualified dealer 2,842,212,690,936 0.051094 1 0.051094
Lose 1x Play against unqualified dealer 1,090,180,312,308 0.019598 -1 -0.019598
Push 1x Play against unqualified dealer 100,426,510,008 0.001805 0 0.000000
folds 15,186,972,477,600 0.273011 -1 -0.273011
total 55,627,620,048,000 1.000000 -0.013402
expected 55,627,620,048,000

The basic strategy for the game is listed in the table below, which returns a 1.70% house edge. The player should 4x his hand about 27% of the time, 1x call about 46% of the time, and fold the remaining 27% of the time.

The game looks like fun. The strategy is actually pretty simple, but you get to make the occasional decision. I’ll actually be in Seattle next week (my first time), so I’ll try to check out the game. Maybe I could hit a nice bad beat for once.

Arizona Stud Basic Strategy
Decision Strategy
discard Hold pair, else
hold two highest cards.
Advanced exception: hold highest and lowest cards when suited, AND
highest two cards aren’t suited, AND highest card is Eight or better, AND
middle card is Six or lower, AND lowest card is only one rank below middle
card.
4x / check 4x raise any pair, else
4x raise suited Ace and Nine or better, else
4x raise Ace and Ten or better, else
check.
1x / fold 1x call three-of-a-kind, else
1x call any pair beating the dealer by more than a kicker, else
1x call same pair as dealer plus Ten or better kicker, else
1x call open-ended straight flush draw, else
1x call flush draw or any straight draw when beating the dealer, else
1x call flush draw or open-ended straight draw when dealer has no pair, else
1x call with higher hand (Jack or better kicker), else
fold all others.

The optional 2 Pair Plus Bonus bet pays for the final hand made by the player. The house edges for the various offered paytables are listed below.

2 Pair Plus Paytables
Player Hand Paytable A Paytable B Paytable C Paytable D
Royal Flush 500-to-1 500-to-1 500-to-1 500-to-1
Straight Flush 200-to-1 200-to-1 200-to-1 200-to-1
Four-of-a-Kind 100-to-1 100-to-1 100-to-1 80-to-1
Full House 50-to-1 50-to-1 40-to-1 40-to-1
Flush 30-to-1 25-to-1 30-to-1 30-to-1
Straight 20-to-1 20-to-1 20-to-1 20-to-1
Three-of-a-Kind 6-to-1 6-to-1 6-to-1 6-to-1
Two Pairs 4-to-1 4-to-1 4-to-1 4-to-1
others lose lose lose lose
House Edge 3.03% 5.12% 5.53% 6.58%

The Player Bad Beat Bonus bet pays when a player’s Jacks-or-Better hand is beat by the Dealer. The following table shows the optimal outcomes for the strategy maximizing the Bad Beat Bonus return. The house edge for the optimal Bad Beat Bonus strategy is 8.00%.

Player Bad Beat Bonus
Player Beat Hand Combinations Frequency Payout Return
Straight Flush 142,560 0.000000 1000 0.000008
Full House 1,324,642,176 0.000071 500 0.035719
Flush 2,719,437,696 0.000147 300 0.043998
Staight 1,597,456,728 0.000086 200 0.017230
Three-of-a-Kind 49,285,841,520 0.002658 30 0.079740
Two Pairs 257,968,615,536 0.013912 20 0.278245
Jacks-or-Better 923,384,598,264 0.049798 8 0.398385
other 17,306,259,281,520 0.933327 -1 -0.9333273
total 18,542,540,016,000 1.000000 -0.080002

Screen Shot 2014-07-30 at 9.51.37 PM

1 Bet Threat @ Casino Pauma

Posted in Uncategorized by stephenhow on June 15, 2014

1bet_smI 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.

1 Bet Threat Optimal Outcomes
Outcome Combinations Frequency Net Return
Bet 2x and 1x and beat qualified dealer 884,580,718,240 0.031804 4 0.127215
Bet 2x and 1x and beat non-qualified dealer 505,981,246,728 0.018192 0.5 0.009096
Bet 2x and 1x and lose to qualified dealer 374,729,986,984 0.013473 -4 -0.053891
Bet 2x and 1x and lose to non-qualified dealer 5,856,935,220 0.000211 -1 -0.000211
Bet 2x and 1x and tie dealer 25,182,150,868 0.000905 0 0.000000
Bet 2x only and beat qualified dealer 293,907,701,760 0.010567 3 0.031701
Bet 2x only and beat non-qualified dealer 387,449,913,432 0.013930 0.5 0.006965
Bet 2x only and lose to qualified dealer 524,307,039,216 0.018851 -3 -0.056552
Bet 2x only and lose to non-qualified dealer 76,858,269,780 0.002763 -1 -0.002763
Bet 2x only and tie dealer 25,553,189,772 0.000919 0 0.000000
Bet 1x only and beat qualified dealer 2,434,367,467,360 0.087524 2 0.175047
Bet 1x only and beat non-qualified dealer 1,467,870,962,280 0.052775 0.5 0.026387
Bet 1x only and lose to qualified dealer 1,215,166,965,412 0.043689 -2 -0.087379
Bet 1x only and lose to non-qualified dealer 17,931,292,692 0.000645 -1 -0.000645
Bet 1x only and tie dealer 164,852,060,176 0.005927 0 0.000000
Bet ante only and beat qualified dealer 3,363,692,256,360 0.120936 1 0.120936
Bet ante only and beat non-qualified dealer 4,003,403,426,760 0.143936 0.5 0.071968
Bet ante only and lose to qualified dealer 9,229,633,097,868 0.331836 -1 -0.331836
Bet ante only and lose to non-qualified dealer 1,896,770,105,748 0.068195 -1 -0.068195
Bet ante only and tie dealer 915,715,237,344 0.032923 0 0.000000
Total 27,813,810,024,000 1.000000 -0.032157

I worked out the basic strategy for the game, just in case anyone wants to play the game. The strategy is actually pretty simple. Since the dealer qualifies with a pair of 6’s or better, you generally only bet the flop if there’s a qualified hand to beat. You can bet kickers and draws against a qualified flop, otherwise you should only bet a qualifying pair when there’s a board card lower than your pair, but 6 or higher.

The basic strategy below has an error rate of 4.5%, that only results in a cost of 0.23% to the player. So the practical house edge is 3.5% for the game.

1 Bet Threat Basic Strategy
Wager Player Hand Rules
2x Pairs 2x bet a pocket pair of 7’s or better, else
check pocket 2’s thru 6’s.
Suited Bet QJs, KTs, KJs, KQs, and A8s or better, else
check all others.
Offsuit Bet KQo, and ATo or better, else
check all others.
1x Straight or better Always bet.
Three-of-a-Kind Always bet, except if trips on flop and less than 2nd nut kicker.
Two Pairs Bet if flop not paired, else
bet if flop qualified (pair 6’s or better), else
bet if board has undercard to pairs, else
bet 9’s up or better, else
check all others.
One Pair
(qualified board has pair 6’s or better)
Bet nut kicker, else
bet flush draw, else
bet open-ended straight draw with both holecards > 8, else
check all others.
One Pair
(small pair on board)
Always check.
One Pair
(unpaired board)
Bet if board has any qualifying undercards to pair, else
bet pair w/ flush draw, else
bet pair 9’s or better, else
check all others.
No Pair Bet 1st or 2nd nut flush draw, else
check all others.

There’s not much opportunity for collusion in the game. Knowledge of the hole cards of all 6 players will modify some of the preflop 2x decisions, but the frequency and value of these counter-(basic)strategy decisions aren’t enough to overcome the 3.2% house edge. Trust me, I’d have worked it out if it was worthwhile.

There’s two bonus bets offered, where the Pocket Bonus pays when your hole cards make a pocket pair, and the Final Hand bonus on your final 7-card hand. The paytables offered at Casino Pauma aren’t very good.

Pocket Bonus
Outcome Combinations Frequency Payout (to-1) Return
Pocket A’s 6 0.004525 50 0.226244
Pocket J’s – K’s 18 0.013575 20 0.271493
Pocket 2’s – T’s 54 0.040724 8 0.325792
no pair 1,248 0.941176 -1 -0.941176
Total 1,326 1.000000 -0.117647
Final Hand Bonus
Outcome Combinations Frequency Payout (to-1) Return
Royal Flush 4,324 0.000032 250 0.008080
Straight Flush 37,260 0.000279 50 0.013925
Four-of-a-Kind 224,848 0.001681 15 0.025210
Full House 3,473,184 0.025961 5 0.129805
Flush 4,047,644 0.030255 4 0.121020
Straight 6,180,020 0.046194 3 0.138581
Three-of-a-Kind 6,461,620 0.048299 2 0.096597
Jacks Up 17,385,408 0.129951 1 0.129951
others 95,970,252 0.717349 -1 -0.717349
Total 133,784,560 1.000000 -0.054179

Flush Rush @ The D Casino, Las Vegas

Posted in Uncategorized by stephenhow on May 3, 2014

Screen Shot 2014-05-03 at 3.32.29 PM

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.

Ante Pay Table
Length Flush Straight Flush
7 300-to-1 1000-to-1
6 20-to-1 500-to-1
5 9-to-1 100-to-1
4 5-to-1 15-to-1

I believe The D will award the highest possible payout for a given hand. So, if you make a 5-card flush that contains a 4-card straight flush, they’ll pay you 15-to-1 (instead of 9-to-1). With this liberal rules interpretation, the house edge is 3.75%. The total possible outcomes for an optimal player are listed below.

Optimal Play Outcomes (Liberal Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
6-card Straight Flush 167,160 1.1900E-05 502 0.005974
7-card Flush 697,620 4.9662E-05 302 0.014998
5-card Straight Flush 4,127,760 0.000294 102 0.029972
6-card Flush 26,945,100 0.001918 22 0.042119
4-card Straight Flush 65,648,544 0.004673 17 0.079447
5-card Flush 372,841,560 0.026542 11 0.291959
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.037493

If the rules are interpreted strictly, and you must make a straight flush with all your cards of the same suit, then the house edge is 5.41%.

Optimal Play Outcomes (Strict Rules)
Outcome Combinations Frequency Net Return
7-card Straight Flush 3,360 2.3919E-07 1002 0.000240
7-card Flush 717,360 5.1067E-05 302 0.015422
6-card Straight Flush 147,420 1.0494E-05 502 0.005268
6-card Flush 27,960,660 0.001990 22 0.043790
5-card Straight Flush 3,112,200 0.000222 102 0.022598
5-card Flush 397,427,940 0.028292 11 0.311212
4-card Straight Flush 41,062,164 0.002923 17 0.049693
4-card Flush 2,627,978,496 0.187080 7 1.309557
Nothing 5,035,629,456 0.358475 -3 -1.075424
Fold before river 4,431,366,576 0.315459 -2 -0.630917
Fold before flop 1,481,973,168 0.105498 -1 -0.105498
Total 14,047,378,800 1.000000 -0.054059
All-Or-Nothing Side Bet
Outcome Combinations Frequency Net Return
All hole cards same suit 2,860 0.010564 30 0.316927
All hole cards different suits 28,561 0.105498 5 0.527491
Others 239,304 0.883938 -1 -0.883938
Total 270,725 1.000000 -0.039520

Of course, the only reason why I analyzed the game was to Monte Carlo the 6-way collusion edge. The return is about +4.07% for 6 players sharing perfect info under strict rule interpretation, and +5.67% under the liberal rules. That’s not much, considering it’s pretty hard to convey suit information between confederates. It’s probably not worth anyone’s trouble to attack the game. I didn’t bother working out a practical strategy.

(FYI, I’m spending a lot more time outside of the casino these days. Before, I used to practically live in the casino. About 9 months ago, I changed obsessions. You can read about my current mania on my other blog.)

Raise It Up Stud @ Pala Casino

Posted in Uncategorized by stephenhow on September 1, 2012

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.)

Raise It Up Stud Play Paytable
Hand Payout
Royal Flush 100:1
Straight Flush 20:1
Four-of-a-Kind 10:1
Full House 6:1
Flush 5:1
Straight 4:1
Three-of-a-Kind 3:1
Two Pairs 3:2
10’s or Better 1:1
Others lose
Raise It Up Stud Blind Bonus
Hand Payout
Royal Flush 1000:1
Straight Flush 200:1
Four-of-a-Kind 30:1
Full House 4:1
Flush 3:1
Straight 2:1
Three-of-a-Kind 1:1
Others push

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.

Raise It Up Stud Basic Strategy
Street Play Bet Betting Hands
3rd Street 3x Any pair,
3 pay cards,
2 pay cards 1-gapped or less,
suited cards 2-gapped or less,
suited cards with 2 pays
4th Street 2x Any pair,
any straight or flush draw,
3 pay cards with 3 suited
5th Street 1x Any pair,
any flush draw,
open-ended draw,
gutshot draw with pay card

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

Posted in Uncategorized by stephenhow on March 28, 2011

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

Double Baccarat @ Sycuan Casino

Posted in Uncategorized by stephenhow on November 4, 2010

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”.

House Way
Hand Rule
Two Pairs Set large pair behind, small pair in front.
(Pair-Pair; never break pair).
Pair Aces Pair Aces behind (never split Aces.)
One Pair Pair behind if 5 or higher front, else
Split pair if can form (8,8), (7,9), (8,9) or (9,9), else
Pair behind.
No Pair Set (5,9) if possible, else
Set hand with highest front + back total, with minimum back – front gap.

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):

Heads-Up Player Strategy
Hand Rule
Two Pairs Set large pair behind, small pair in front.
(Pair-Pair; never break pair).
Pair Aces Pair Aces behind (never split Aces.)
One Pair Pair behind if 5 or higher front, else
Split pair if can form (8,8), (7,9), (8,9) or (9,9), else
Pair behind.
(6,9), (7,9), (8,9), (9,9)
front + back = 14 (7,7), (6,8)
(5,9)
front + back = 10 (5,5)
(9,1)
(4,6), (3,7), (2,8)
front + back = 9 (0,9)
(1,8)
(4,5), (2,7)
front + back = 8 (0,8)
(1,7)
(3,5), (2,6)
front + back = 7 (3,4)
(0,7)
(2,5), (1,6)
front + back = 6 (2,4)
(0,6)
(1,5)
Set hand with highest front + back total, with minimum back – front gap.

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.

ShuffleMaster Ultimate Draw Poker Machine @ Viejas

Posted in Uncategorized by stephenhow on November 13, 2009

Table-Master_cutout_3There’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

draw min kicker to hold
2345 Ten
3456 Ten
4567 Jack
5678 Queen
6789 King
789T Ace
89TJ Ace
9TJQ

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.

Hand Win
Royal Flush 1000
Straight Flush 150
Four Of A Kind 25
Full House 8
Flush 7
Straight 5
Three of A Kind 3
Two Pairs 1
all others -1

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.

Welcome Wizard of Odds Readers!

Posted in Uncategorized by stephenhow on November 2, 2009

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!

PlayCraps Example Session with Counting

Posted in Uncategorized by stephenhow on August 27, 2009

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.

Point ΔCount
4 +4 if both cards ≤ 3, -4 if both cards ≥ 4, else 0
5 +2 if no Fives or Sixes, -1 if one Five or Six, -4 if two Fives/Sixes
6 +1 if no Sixes, -2 if one Six, -4 if boxcars
8 +1 if no Aces, -2 if one Ace, -4 if snake-eyes
9 +2 if no Ace or Deuce, -1 if one Ace or Deuce, -4 if two Aces/Deuces
10 +4 if both cards ≥ 4, -4 if both cards ≤ 3, else 0

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…)

Triple-Down BJ @ Texas Station, Las Vegas

Posted in Uncategorized by stephenhow on August 8, 2009

I’m finally getting to a reader’s request about a triple-down blackjack game at the Texas Station casino in North Las Vegas. Here are the relevant rules:

  • triple-down on first two cards totals of 9, 10, 11 (including soft totals)
  • blackjack pays even money (1:1)
  • blackjack is an automatic winner, if you stay
  • normal double-down rules
  • no triple after split
  • double-deck

The overall house edge for this game is 0.83%. The triple-down rule gives the player a +1.62% boost, but it’s not enough to overcome the even money blackjacks. The basic strategy is the same as double-deck, except that doubles on hard 9, 10, 11 are replaced by triples. Also, 9 vs a 7 upcard, and A-9 vs 4/5/6 are triples.

Interestingly, tripling a blackjack against a dealer 6 upcard isn’t too bad of an option. The EV is .998, instead of the 1.0 automatic winner for staying. It’s worth a gamble, if you feel like it. Tripling a blackjack against a dealer 5 upcard isn’t as good, as the EV here is only .946. That’s giving up 5.4% of your original bet, on average.

Someone should petition Texas Station to allow triple-down on any two cards. This still leaves the house edge at 0.59%, which is about equal to a liberal shoe game.

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