How do you calculate unique rules?

Here are some rules I've come across at 5Dimes that are making it a nightmare for me to know what the house edge is:

- Blackjack pays amounts that are not 2:1, 3:2, or 6:5
- Insurance pays amounts other than 2:1
- Player blackjack is automatic winner
- Late Surrender vs A-8
- Suited blackjacks pay more
- All ties win/lose half

And to make things more complicated, many of the rules have a "even after splits" version.

Now before you tell me that I shouldn't even bother considering a game with a certain rule, know that there are favorable rules to counter the bad ones.

Quote: SilentBob420BMFJ

- Blackjack pays amounts that are not 2:1, 3:2, or 6:5

The probability that you get a blackjack, and the dealer does not is about 4.5% (1 in 22). So, in a game, where BJ pays 3:2, it's contribution to the expected return is 1.5/22 - about 6.8%. If it was 6:5, then make it 1.2/22 = 5.45%. In other words, the difference in house edge between 3:2 and 6:5 games is 6.8-5.45 = 1.35%

You can use the same approach for other payout ratios. Basically, if black jack pays X:1, then it costs you (1.5-X)/22 in house edge compared to the normal game (negative number means a reduction to the house edge).

Quote:

- Insurance pays amounts other than 2:1

As long as it less than 2.25:1, don't take it. If more, then do.
It has no effect on the outcome or house edge of the main game. Just think of it as a side bet.
If it pays X:1, the expected value (i.e. minus house edge) is 4/13*X - 9/13 = (4*X - 9)/13
When X is more than 2.25 this number becomes positive, meaning that the house edge turns into a player edge.

Quote:

- Player blackjack is automatic winner

This only matters when both you and the dealer have black jack, once in 446 hands. So, this rule is worth X/446, if X is the blackjack payout. For example, a regular 3:2 game with this rule has its house edge reduced by 1.5/446 or about 0.34%

Quote:

- Late Surrender vs A-8

Does this mean, you can not surrender against 10 or 9?
That pretty much leaves only the ace.
The only hand you'd want to surrender against an ace is 16, and your gain in that case is about 1.7% (-50% vs. -51.7% if you hit).
To get 16 on the first to cards, you'd need either 10-6 or 9-7 (you don't surrender 8,8). The probability of that is 10/169.
So, the overall reduction in house edge (vs. a no-surrender game) is 1.7% * 10.169, or about 0.1%

Quote:

- Suited blackjacks pay more

The probability that your blackjack is suited is 1/4. If non-suited BJ pays X, and suited BJ pays Y, you can expect to get (3X+Y)/4 on average from blackjack. You can use this number as blackjack payout to calculate the house edge the way I explained in the beginning.

Very nice post, thanks. And for the "player blackjack is automatic winner", you don't mean the house edge goes down .34%, you mean take .34% of the house edge and reduce it by that, right? Because .34% is significant otherwise, but I can't imagine once is 446 hands is significant, unless the payout was like 10:1 for something.

It is one in 446 hands, but it is a 150% gain. Yeah, pretty significant. It does reduce the house edge by .34% in absolute numbers. That is, if house edge was, say, .5% without it, then with it, it'll become .16%

I usually modify my infinite-deck spreadsheet when I come across an unusual rule. That is what I did for the "no doubling" rule discussed in another thread.

Quote: weaselman

This only matters when both you and the dealer have black jack, once in 446 hands. So, this rule is worth X/446, if X is the blackjack payout. For example, a regular 3:2 game with this rule has its house edge reduced by 1.5/446 or about 0.34%

The additional player edge is 0.5/446 for a 3:2 Blackjack. As a BJ vs BJ is normally a push, you gain only half your stake.

Quote: Wizard

I usually modify my infinite-deck spreadsheet when I come across an unusual rule. That is what I did for the "no doubling" rule discussed in another thread.

Could you explain this more?

Quote: SilentBob420BMFJ

Could you explain this more?

Infinite decks games are so easy that you can calculate games in a small spreadsheet. The results are not exact, but give you a quick glance at house edge and playing strategy. What you give up on infinite decks is the removal effect of a card when drawn. If you play 4 or more decks this is a reasonable approximation.

Quote: MangoJ

The additional player edge is 0.5/446 for a 3:2 Blackjack. As a BJ vs BJ is normally a push, you gain only half your stake.

No, in a push your winnings is 0. But if the tie wins you get 1.5. 1.5 - 0 = 1.5. 1.5/446 is the edge.

Quote: weaselman

No, in a push your winnings is 0. But if the tie wins you get 1.5. 1.5 - 0 = 1.5. 1.5/446 is the edge.

Yes you are right. I always think in european odds....