Schwza's A4s post from yesterday - A very interesting ICM problem

[Indiana]

Yesterday, schwza posted this question:

http://forumserver.twoplustwo.com/showth...=10#Post6571824

and I attempted to set up a solution with unrealistic ranges for BTN and BB.

At the end of the thread, Yugo suggested the following ranges:

"35% push for btn, BB calling 40% when you fold and 7% when you push."

Well, I find this problem very interesting and my immediate reaction was
to push but then I spent some time thinking about it and decided
to just fold. Others thought calling was optimal. This is a great ICM
question and very interesting to me so I want to rework the math with these
more realistic ranges that Yugo offers and come to a good decision about
what we should be doing in this hand. This is a great opportunity for
everyone here to absorb the ICM concepts and understand why being able to do this
kind of math will allow you to explore many concepts in your own personal
laboratory like the pros do. Here goes:


Part I: Compare pushing to folding

So what's our equity if we fold? Two possible scenarios:

i) We fold and BB folds. We have resulting fraction of $EV= .1884

ii)We fold and BB calls. We have resulting fraction of $EV= .5115*(.182) +
.4885*(.2695)=.2247

So our average equity of folding is the weighted average=

(.6)*.1884 + (.4)*.2247 = .203



Now what is our equity if we push? Two scenarios:

i)We push and BB folds:

.492*(.3269)+.508(.0864)= .205

ii)We push and BB calls. This is the tricky part of the push/fold calculation
because there are a few different scenarios. Without loss of generality, here's the
scenarios that result in a nonzero EV:

a)SB wins and we beat BB:

*Using P(AB)=P(A|B)P(B), probibility of this event is = .278*.347=.096,
with a resulting equity of .1587

b)SB loses and we lose to BB:

*probability is .722*.653=.471 with equity of .2 (because we get 3rd place)

c)SB loses and we beat BB:

probability is .722*.347=.250 with equity of .372

*****So the weighted avg for this section is: .096*.1587 + .471*.2 +.250*.372=.202


So our average equity of pushing is the weighted avg=
.93(.205) + .07(.202)=.205


-----What does this tell us? This tells us that if BB is going to be on a tight 7% range then pushing is
better than folding because $EV of a fold=.203 < $EV of a push=.205. Of course both of these are very close.
If the BB loosens up a little then the decision to fold becomes better than the decision to push. This is
because our chances of folding and watching the BB take BTN out go up a lot.

Ok, now let's do the ICM for just calling and assume that BB will be
smart and it will be checked down postflop whenever he calls us. Let's assume that BB
will call with top 15% of hands when we flat call. We have to assume that he will
be looser with his action when we just flat call than when we push.

Then the EV is as follows(using same methods as above):


i)BB folds: .492*(.3269)+.508(.0864)= .205 equity

ii)BB calls along with us

a)SB wins and we beat BB:
*probability of .096
with a resulting equity of .0868

b)SB loses and we lose to BB:
*probability of .471 with equity of .2306

c)SB loses and we beat BB:
*probability of .250 and equity of .3523

** so the weighted avg for ii) is: .096*.0868 + .471*.2306 + .250*.3523 = .205

***So the full weighted avg representing the equity of calling is: .15*(.205) + .85*(.205) = .205

This is interesting. Whether or not BB calls along with us does not change our equity much.


So here's the cliff notes:

EV for calling is .205, which equals the EV for pushing, and the EV for folding is .203. So the three decisions
are going to be just about the same EV if BB is tight. If BB opens up past 7% we actually gain more EV by
folding because BB can take BTN out of the picture with higher probability.

All told, unless I've made a math error somewhere....We had an argument about nothing yesterday because
all three decisions are pretty similar if BB is at the 7% range when we push and the 15% range when we flat
call.

$10 to the first guy to check my math thoroughly and provide meaningful feedback to this problem (stars xfer),

Indy

[UMTerp]

Indy,

I only glanced at this, but I think some things are a tiny bit off - the first one that I noticed was that you have the same percentages for the A>B>C, A>C>B, etc. scenarios whether we push and BB calls (7%), or if we call and BB calls (15%). This can't be right since the BB is calling with two different ranges.

Secondly, with all this math that makes the decisions sooo close, I think it's quite an assumption that the BB would just check down a winning hand in the call/call scenario. It seems to me that there'd be some non-zero number that you'd have to work in there to be accurate. Intuitively, I'd think the chance that the BB would bet postflop would hurt your EV for calling.

And I doubt I did enough for it anyway, but I don't need the $10 - these forums are designed for people to discuss things like this - no need to bribe them!

[mosch]

[ QUOTE ]
What does this tell us? This tells us that if BB is going to be on a tight 7% range then pushing is better than folding because $EV of a fold=.203 < $EV of a push=.205. Of course both of these are very close.

[/ QUOTE ]

Remember your high school science experiments and signficant digits.

I know you mention that at the end, but basically the EV is .2 on both but saying it's .205/.203 pretends that your estimates are far more accurate than they are.

[The Yugoslavian]

Wow they all ended up being way closer than I expected.

Imma still push, [img]/images/graemlins/wink.gif[/img].

But this probably means we're all wasting our time as it doesn't much matter what we do, lol.

I say someone pm gramps and ask him what the correct answer is.

Yugoslav

[Indiana]

[ QUOTE ]
[ QUOTE ]
What does this tell us? This tells us that if BB is going to be on a tight 7% range then pushing is better than folding because $EV of a fold=.203 < $EV of a push=.205. Of course both of these are very close.

[/ QUOTE ]

Remember your high school science experiments and signficant digits.

I know you mention that at the end, but basically the EV is .2 on both but saying it's .205/.203 pretends that your estimates are far more accurate than they are.

[/ QUOTE ]

agreed 100% but at the end of the day a more accurate estimate here isn't going to sway the decision dramatically.

Indy

[Indiana]

[ QUOTE ]
Indy,

I only glanced at this, but I think some things are a tiny bit off - the first one that I noticed was that you have the same percentages for the A>B>C, A>C>B, etc. scenarios whether we push and BB calls (7%), or if we call and BB calls (15%). This can't be right since the BB is calling with two different ranges.

Secondly, with all this math that makes the decisions sooo close, I think it's quite an assumption that the BB would just check down a winning hand in the call/call scenario. It seems to me that there'd be some non-zero number that you'd have to work in there to be accurate. Intuitively, I'd think the chance that the BB would bet postflop would hurt your EV for calling.

And I doubt I did enough for it anyway, but I don't need the $10 - these forums are designed for people to discuss things like this - no need to bribe them!

[/ QUOTE ]

UMTERP, u are correct. I made a mistake copying and pasting. I did this in between meetings in a hurry. Let me fix this error and I'll give the updated #s. U will get the $10. I'll repost in a minute.

**Of course this will only change the "calling" #s. Pushing still equals folding roughly.

Indy

[The Yugoslavian]

Terp - that $10 better go into the WEDDING FUND!!! bwhahahahhaa.....

So how is that quest going?

I think 10% of your proceeds should go to the UMTerpImmaFuckTonsOfHookersFund as a one night blowout extravaganza b4 your wedding. Or once a month..whatever, [img]/images/graemlins/wink.gif[/img].

Yugoslav

[UMTerp]

[ QUOTE ]
So how is that quest going?

[/ QUOTE ]

Kinda of slacking in the discipline part, but I half-expected that. I'm about a third of the way to the overall goal, so I guess it could be worse...

[Indiana]

Seems UMTerp pointed out a small error in my multiplication. Pushing and folding don't change, but calling does go up just a tad. Still, even when corrected all 3 decisions are still very close:

REPOST OF CALLING #s and CONCLUSION:



Ok, now let's do the ICM for just calling and assume that BB will be
smart and it will be checked down postflop whenever he calls us. Let's assume that BB
will call with top 15% of hands when we flat call. We have to assume that he will
be looser with his action when we just flat call than when we push.

Then the EV is as follows(using same methods as above):


i)BB folds: .492*(.3269)+.508(.0864)= .205 equity

ii)BB calls along with us

a)SB wins and we beat BB:
*probability of .305*.433=.132
with a resulting equity of .0868

b)SB loses and we lose to BB:
*probability of .695*.567=.394 with equity of .2306

c)SB loses and we beat BB:
*probability of .695*.433=.30 and equity of .3523

** so the weighted avg for ii) is: .132*.0868 + .394*.2306 + .3*.3523 = .208

***So the full weighted avg representing the equity of calling is: .85*(.205) + .15*(.208) = .206



This is interesting. Whether or not BB calls along with us does not change our equity much.


So here's the cliff notes:

EV for calling is .206, EV for pushing is .205, and the EV for folding is .203. So the three decisions
are going to be just about the same EV if BB is tight. If BB opens up past 7% we actually gain more EV by
folding because BB can take BTN out of the picture with higher probability.

All told, unless I've made a math error somewhere....We had an argument about nothing yesterday because
all three decisions are pretty similar if BB is at the 7% range when we push and the 15% range when we flat
call.


Indy

[Indiana]

UMTerp what's ur stars ID? Would you mind looking at my corrected #s? Its really important to me that I get these right. I am shocked that the 3 decisions are just about the same in EV. Given what u said about possibly not having this hand checked down and the possibility that BB will call more I'd say its probably best to fold and keep this hand simple.

I'd need at least A7s to get involved here, and I'd likely push.

Indy