MathExtremist
Joined: Aug 31, 2010
• Threads: 88
• Posts: 6526
October 4th, 2010 at 8:50:08 PM permalink
Hi,

I'm working on updating stats for my dominoes game and I'm dealing with the issue of pushes and how they should be incorporated into my house advantage / EV statistics. For the record, I have a question pending with the NGCB on this issue, but the right answer may still be to just report both numbers - pushes included (aka EV) and pushes excluded (aka HA, or so I've been told).

But I'd also like to do a comparative analysis with other popular games, including blackjack and pai gow poker. That raises several interesting questions, such as:
1) The common house edge on BJ is somewhere between 0.5% and 1%. The percentage of pushes for most games, under basic strategy, is in the vicinity of 8% (IIRC). Is that house edge figure of between 0.5% and 1% inclusive or exclusive of pushes?

2) More generally, are there tables of outcomes out there for blackjack that list probabilities of player/dealer final outcomes under basic strategy, given a set of rules and # decks? That would let you analyze the actual effects of pushing on the game at a more in-depth level. For example,
player busts, dealer busts, p() = 6%
player busts, dealer 21, p() = 1.2%
...
player 19, dealer busts, p() = 2.5%
...
player 13, dealer 17, p() = 0.6%

Then you could add up all the ones that have player == dealer and get the push percentage, as well as other interesting statistics like "if I have a 15, what are the chances of the dealer busting so I still win?"

Do these tables exist on the Internet somewhere or do I have to create them? I'm just trying to avoid running the rather large number of permutations of # decks * rule variations. I know about the files with basic starting-hand EVs (e.g. how much is 9-6 worth if you stand, hit, double, etc), but I don't think I've seen end-of-hand tables anywhere...

3) While we're at it, the house edge in Pai Gow poker on WoO is listed as 2.73%. That number counts pushes; by discounting pushes, which happen quite a lot, the house edge goes up to 4.67%. Should this larger figure be cited as the house edge and 2.73% be cited as "EV" instead? I just want to compare apples-to-apples.

Thanks!
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
Wizard
Administrator
Joined: Oct 14, 2009
• Threads: 1353
• Posts: 22448
October 4th, 2010 at 9:16:22 PM permalink
I'm strongly in the camp that the house edge should count pushes. The reason for the house edge is for the player, and casino, to estimate expected loses over time easily. If you ignore pushes, then you have to know the probability of a push to get an expected win/loss per unit of hour. So, to answer your questions.

1. Every legitimate blackjack, and video poker, writer counts pushes as a bet. So the house edge figures are INCLUSIVE of pushes.
2. I don't know of any such tables, or any value to them. I could be talked into making them, but it would help to know why they would help anybody.
3. EVERY house edge figure on my site is the ratio of the expected player loss to the initial wager. This is how the casinos define the house edge, this is how all legitimate blackjack and video poker writers define it.

In closing I invite you to join the camp that count pushes in the house edge, but don't count subsequent wagers in the denominator. You will be in very good company. I challenge anyone to name one gambling writer, who does his own math, other than John Scarne, who does not define the house edge that way.
It's not whether you win or lose; it's whether or not you had a good bet.

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