Friends,
This is
absolutely crucial. This is a
foundational building block of economics. Forget about my magic calling range (not magic, really works), if anyone does not get this question below right, they are in serious jeopardy of making HUGE mistakes both on and off a poker table.
This is a basic exercise in game theory. This is usually in the first few pages of guess which chapter.
Below, I will use the 20.61 rake and the ranges from others in the thread, probabilities are only slightly adjusted for tiny occurrences of chops (every possible outcome must add to 100 percent).
First, we are the bettor. We will sit in the bettor’s seat and watch his betting line. When his chips cross the line we will count them, and when they come back we will count them. (We don’t trust the dealer).
Betting strategy is all-in always:
With probability 100 percent we push 100 into the middle, so
BEHIND THE BETTING LINE there is zero, and
across the line there is now 100 of our chips plus the free 10 in antes.
With probability (1326-188)/1326 the dealer gives us back the whole 100 plus the 10 so 1138/1326(110) =
94.40422
So, 94.40422 is
back in front of us, as we sit in the bettor’s seat.
With probability 188/1326 there is a call and the dealer collects
exactly 43.281 from the pot and keeps it.
This leaves precisely
166.719 in the middle of the table.
With probability (.3298) we win that pot:
(188/1326)(166.719)(.3298) =
7.79560945
And...
With probability (.0176) there is a tie, and we get back 83.3595
(188/1326)(83.3595)(.0176) =
0.20801
We get back zero otherwise.
We have watched every chip from the bettor’s chair, and now let’s count everything we won with our betting strategy:
94.40422 + 7.79560945 + 0.20801 = 102.40784
Neato! We won a net increase of 2.40784!
Next, same for caller. We count every chip.
We fold with probability (1326-188)/1326 and we keep all 100.
100*(1326-188)/1326 =
85.82202 kept behind the betting line.
With probability 188/1326 we call and push our 100 in the middle.
The dealer collects 43.281 and keeps it.
Pot is now 166.719...
With probability (188/1326)(.6526) we win and collect 166.719
(188/1326)(.6526)(166.719) =
15.4257572
With probability .0176 there is a tie:
(188/1326)(83.3595)(.0176) =
0.20801
We collect zero otherwise:
We have watched every chip from the caller’s chair, and now let’s count everything we kept or won:
85.82202 + 15.4257572 + 0.20801 = 101.4557872
Sweet! We won money too!
We started with
210 chips in play, but since this is
raked poker, it is
supposedly ok that the totals we finish with are Shover + Caller = 101.4557872 + 102.40784 =
203.8636272 = 204 Bucks.
And, if you set rake at .00001 percent,
suddenly you get back a total of 210 bucks.
I must admit, that is a pretty
slick trick. I can understand why there would be books, videos,
software etc. That would be marketed as if all of the above is true.
SO, WHY IS IT THAT:
If we keep this same sequence of actions, shover always shoves, caller calls sometimes, why is it that when there are
no antes at all our players only get back a total of 194.49 chips? (the calculation is below).
You can assign ANY ranges you want, shover has aces, caller has ATC, both players have ATC, whatever. You can NEVER FIND
ALL the missing chips.
The house is only charging the same percentage price on the 200 pot as it did on the 210 pot.
Somehow, money has disappeared??!?
So. Where did the money go?
If no one on the worlds greatest poker forum will post the solution to the missing chips, then I will post the answer in 24 hours.
Thank you for reading.
Bettor:
Caller folds with probability (1138/1326) and we get back (1138/1326)(100) = 85.82202
The dealer collects EXACTLY 41.22 and keeps it.
Pot is now precisely 158.78.
(188/1326)(.3446) we win and collect 158.78
(188/1326)(.3446)*158.78 = 7.75756
With probability .0176 there is a tie:
(188/1326)(79.39)(.0176) = 0.19810
Add up every chip:
85.82202 + 7.75756 + 0.19810 = 93.77768
Now the caller:
100*(1326-188)/1326 = 85.82202
(188/1326)(.6526)(158.78) = 14.69120
(188/1326)(79.39)(.0176) = 0.19810
Adde ‘em up:
85.82202 + 14.69120 + 0.19810 = 100.71132
Total money back to players from 200 in play: 93.77768 + 100.71132 = 194.489 = missing 5 bucks!