Advantage Play: Calculating Bonus Streak

So, I saved bonus streak for “last”.  I have last in quotes because there are a few more variable state poker games I’d like to take a look at, but I’ve either  never seen them in the wild or they are so far away from me that I’m not going to get enough play to worry about them, so I have not bothered.   Maybe I’ll get to them one day, but probably not anytime soon.

The reason I saved bonus streak for last is that the game is slightly more complex than most variable state poker machines because of the stacks of multipliers and because you are playing 10 coins. You have a chance of creating new multipliers and changing existing multipliers to 12X.  This is a big deal for a vulture.

Let’s take a look at the last part first.  Here are the frequencies for a 9/6 jacks machine.

Jacks Table.PNG

Here are the return frequencies for a NSUD machine:

NSUD Table.PNG

I know both of these pay tables don’t exist on bonus streak, but what I’m trying to drive home is that the chance of converting a stack of multipliers to all 12X is fairly substantial (11.07% for the job game and 11.82% for NSUD).  I could do this for any pay table, but let’s call it in the neighborhood of a 10% chance of converting a second level multiplier to 12X. 

What about a 3rd level multiplier?  Well, you have a 10% chance of converting it to a 12X on the first hand and a 10% chance of converting it to a 12X on the second hand played.  That’s an addition problem…  There is a 20% chance you will convert a 3rd level multiplier to a 12X.

So if you see this on a hand:

Next Hand 12X

Next Hand 10X

Next Hand 8X

Next Hand 4X

Next Hand 2X

You should really see this:

Next Hand 12X

Next Hand 10X *.7 + Next Hand 12X *.3 = 10.6X

Next Hand 8X *.8+ Next Hand 12X * .2 = 8.8X

Next Hand 4X *.9 + Next Hand 12X *.1 = 4.8X

Next Hand 2X

 

So, deeper stacks of multipliers add value.  In the case above, the multipliers add an extra 2.2X over the next 4 games. 

Now I could do this same work for the exact pay table and assume there is a 10% chance that even a 1X will grow, but this game has so many moving parts, that is getting too far away from “easy” for me to embrace that level of detail.  What I do is just lean towards being more aggressive with this game, which seems to be the exact opposite of what most vultures I’ve observed do with it…  And being fair to them, it is a high variance game.  Said another way, if you only have $100 in your pocket, a quarter 10 – play could send you home really quick…  The vultures in my market don’t appear to be trust fund babies…  Bankroll is always a valid concern with taking on a game and I’ll try to remember to write more about that sometime soon.

Next problem is that the deeper stacks force you to account for more than the next game you play.  Let’s say I have the big ass full house stack above on 3 of 10 hands and the other 7 hands are 1x.

I’ll institute a new notation for this game.  BAS = Big Ass Stack (shown above)

So you come upon this:

BAS        BAS        BAS

1x           1x           1x

1x           1x           1x

1x

Clearly you don’t want to play the first hand, 13X for the price of 20 bets of 5 coins each.  Do you want to play the whole stack?  That’s 5 hands at 100 coins per hand, or 500 coins...  100 bets of 5 coins each. 

How many X’s do I get for that price?

Each BAS is worth 38.2X.  There are three of them, so that's 114.6X (you see where this is going right?) plus there are a bunch of 1X to add in.  All in this 5 hand series is worth 149.6X for the price of 100 bets of 5 coins each.

I assume a 95% base game, so using the UX formula:  149.6 * .95 / 100 = 142.12% RTP.

Whew…  I need a drink after this post.