House Edge Calculation

Hi everyone, I've got a question that seems like it should be very simple (and maybe it is, I'm just tripping over my own shoelaces?)

To calculate the house edge of a game using a monte carlo simulation, you can simply divide your total profits/losses by the total amount bet. For example, if your average bet is $10, and you lose $50 million over 1 billion hands, then the HE would be -.5%. No problem there.

But, how do you calculate your average bet when it comes to splits, doubles, and surrender?

Doubles and splits seem straightforward. I would imagine every time you put more money on the table, that gets added to your total wager. Surrender is less obvious though. Are you halving your wager and losing 100%? Or are you simply losing 50%? The first option would yield a more extreme HE than the second, because profit would stay the same while total amount bet decreases.

Or maybe I'm wrong about all of this and the only thing that matters is your initial wager? Does someone have guidance on which way is correct?

Quote: Chuckles

…For example, if your average bet is $10, and you lose $50 million over 1 billion hands, then the HE would be -.5%. No problem there.

But, how do you calculate your average bet when it comes to splits, doubles, and surrender? …
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The .5% in your example is NOT the house edge. It’s merely the results of your session.


The Wizard has an article & video where he creates basic strategy on a blank Excel document, which also provides the house edge.

Note that the article is actually a transcript of the video. It’s tricky to follow, but useful after you watch the video.

Also, the YouTube page has a link to the completed Excel document.

https://wizardofodds.com/video/blackjack-basic-strategy/

Note that this math is for an infinite deck. But it gives you the basics.

I’ve adapted the spreadsheet to calculate the basic strategy and house edge for a radical / interesting modification to the game. Sorry. No details. NDA and all…

Thanks for your reply,

I should clarify, my goal here isn't actually to calculate the house edge for a game, but rather to validate my simulation by comparing the results to the HE given by CVCX/WOO. That's why I'm not trying to use combinatorics. But I'll check out the video and see if there is something that could indicate how to do this better.

I think I got it. Michael is multiplying the probability of each hand by the EV of each hand, and the EV is represented as a proportion of the initial bet only. That is to say, EV for surrender is -0.5, and EV for a double is the sum EV of standing on each outcome, multiplied by 2. Since the ratio of initial wager to total amount wagered is already included in the EV calculation, my sim should converge on the HE if I only divide the profit by the initial bet.

After testing this with 160 million rounds, my calculation is much closer than it was. (I'll run it longer to make sure it converges, but it'll take some time) Inputting the same rules into my simulation, CVCX, and WOO house edge calculator, these are the results:

WOO: -0.55051%
CVCX (400 million rounds): -0.556%
my sim (160 million rounds): -0.56595%

The CVData software breaks it down into IBA%, which is relative to your initial bet, and TBA%, relative to all the money you have bet.

Which is correct? Both. Which should you use? Either one you want. In the case of blackjack they are not different enough to make a significant difference in how you play your game.

Quote: Chuckles

WOO: -0.55051%
CVCX (400 million rounds): -0.556%
my sim (160 million rounds): -0.56595%
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This is fantastic! It took about five years for me to understand the concept of EV, but it seems you figured all these out in an hour or so.

Oh, good to know, I was wondering what those numbers meant. I'm guessing for other games the difference will be larger, such as Spanish with redoubles and all that.

Quote: aceside

Quote: Chuckles

WOO: -0.55051%
CVCX (400 million rounds): -0.556%
my sim (160 million rounds): -0.56595%
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This is fantastic! It took about five years for me to understand the concept of EV, but it seems you figured all these out in an hour or so.
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Haha, I've definitely been working on my sim for longer than an hour, but thanks!

House Edge is calculated in reference to your initial unit bet. It is intended to help you with your first decision which is: should I play this game? If I wager $100 at the start of the hand, what is my mathematical expectation at the end of the hand?

Many games, including BJ and Miss Stud and UTH and 3 Card Poker, give you decisions and opportunities to wager more during the course of the gameplay - either when the cards have become favorable or as a preferred alternative to folding. But I think of these follow-on decisions about wagering as being driven by optimal game strategy as opposed to "do I want to play this game?" Of course, the outcomes of those optimal decisions are reflected in House Edge, but in units of your initial wager.

Quote: Chuckles

<snip>After testing this with 160 million rounds, my calculation is much closer than it was. (I'll run it longer to make sure it converges, but it'll take some time) Inputting the same rules into my simulation, CVCX, and WOO house edge calculator, these are the results:

WOO: -0.55051%
CVCX (400 million rounds): -0.556%
my sim (160 million rounds): -0.56595%
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Chuckles,

Some of the small differences in your EV values may be due to the "Cut Card Effect," which the Wiz explains at this link:

https://wizardofodds.com/games/blackjack/cut-card-effect/

Hope this helps!

Dog Hand