Discount Gambling

Six Card Poker @ Venetian, Las Vegas

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

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

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

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

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

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

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

Lucky Lucky Blackjack Sidebet (+EV)

Posted in +EV, blackjack sidebets by stephenhow on May 3, 2012

Well, here’s another massively countable side bet that some people might be interested in (advantage players, casino floor supervisors, and the game publisher), but that I’ll never play. I think after this one, designers will know to check their games for vulnerabilities, especially when there’s oversized items in the paytable. And we’ll remember, “It’s not a sucker bet if the count is good.”

Again, Eliot Jacobson pointed this one out to me. (But, if Barona had this side bet, I’d have already looked at it.)

The Lucky Lucky blackjack side bet is played with your first two dealt cards, and the dealer upcard. On these three cards, you get paid for various ways to make 21, and for any 20 and 19 total. The most countable version of this side bet is for the double-deck version with the paytable below. The game is also countable for the 6 deck shoe game, but it’s only 60% as profitable.

Lucky Lucky Side Bet(Double Deck)
Hand Frequency Probability Payout Return
suited 678 32 1.757238E-4 100 0.01757238
777 56 3.075166E-4 50 0.01537583
other 678 480 0.00263586 30 0.07907569
suited 21 936 0.00513992 15 0.07709880
other 21 14904 0.08184334 3 0.24553003
any 20 13792 0.07573694 2 0.15147388
any 19 13344 0.07327680 2 0.14655362
others 138560 0.76088389 -1 0.76088389
total 182,104 1.00000000 -0.02820366

As usual, I program a function that tells me the EV for any given shoe composition. Then I simulate millions of hands, calculating the ideal EV of the side bet at the beginning of each hand. I sum up the times when the side bet is +EV, and find the average +EV bet and +EV frequency. For the double-deck Lucky Lucky, I got

double-deck, cut card @ 75th card
ideal +EV frequency: 0.2769, ideal EV/bet: +0.0591

which is not a practical counting scheme, but the theoretical limit if you used a computer that took into account all info (suits, etc.).

Then I calculated the Effect-of-Removal (EORs) of a given card on the EV, in order to make counting tags. (Outside the gambling world, people would call “EORs” sensitivities, and “tags” coefficients.)

EORs and Tags for Double-Deck Lucky Lucky
Card EOR Tag
Deuce +0.007853 +1
Trey +0.006066 +1
Four +0.004099 +1
Five +0.003171 0
Six -0.010422 -2
Seven -0.017270 -2
Eight -0.012616 -2
Nine +0.002515 0
Ten/Face +0.006270 +1
Ace -0.008477 -1

So, setting the trueCount threshold to 2.4 (bet Lucky Lucky when the trueCount is >= 2.4), you get the practical results in double deck:

practical frequency: 0.2640, average EV/bet: +0.0561

6 Deck Shoe Version

The 6 deck shoe paytable is better than the double deck version, as it pays 200:1 for a suited 777. The EORs are similar, and I came up with the same count tags as the double deck game. Using a trueCount threshold of 2.1, the practical counting scheme yields:

ideal +EV frequency: 0.2311, ideal average EV/bet: +0.0432
practical frequency: 0.2217, practical average EV/bet: +0.0409

which is only 61% of the profit rate as the double-deck game.

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