Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

Ok, I posted earlier about doing a more complete mathematical analysis of a questionable play. I did a quick and dirty version of the analysis. First, I ignored the fact that there would be more than one caller. This will lead to an inflated EV number in my analysis (although not to dramatic, b/c the addition of another hand, even if it is AA-KK, does not eliminate the equity of 99, instead almost halfing the EV of a showdown). Also, I ignored other options (opportunity cost, sorry I'm an Econ grad student).

ANALYSIS

Situation: 110 players remain, 40 places pay. I have 3200 TCs, average is 4400. Blinds 100/200. I am sitting in MP with 5 players yet to act behind me (including the blinds).
Everyone is comfortably stacked, ranging from 2600-5500. Play has been fairly tight, but not overly so.

Calculations
Two possible outcomes, I get called, or I win the blinds
I give my opponents the calling range of AA-77, AK-AJ


% of time called =
Total number of hand combinations remaining after removing 99 = c(50,2)=1225

The number of hand combinations that my opponents will call with = (7*6)[PP other than 99] + (16*3)[AK-AJ] + 1[99] = 91

This next part is not exact, and may very well be dead wrong

1225 total hand possibilities, 5 random hands = c(1225,5)=
22 trillion and change

1134 total hand possibilities that will not call (1225-91), 5 random hands = c(1134,5) = 15 trillion and change

So c(1134,5) / c(1225,5) = .6794

67.94% chance of stealing the blinds

So....

Steal the Blinds 300
% of time 0.679359427
subtotal 203.8078281

Showdown pot 6550
equity vs range 0.463
EV of showdown -167.35
% of time 0.320640573
subtotal -53.65919987

TOTAL EV 150.1486283

Now assuming that CEV = $EV, does anyone see a problem with my calculations? I am a little worried about how I handled the % of times that I will steal the blinds. For instance, someone will fold A7, but that will affect the number of Aces available for hands like AK that will call.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

shameless bump

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

[ QUOTE ]
For instance, someone will fold A7, but that will affect the number of Aces available for hands like AK that will call.

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No it doesn't. Other players folded hands are treated as if they were part of the original card pool. If they weren't, poker math would get ridiculously complex very quickly. And since doing it that way is really not far off, don't worry about it.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

I actually created a spreadsheet that I use to run analysis like this. I plugged in your numbers and got a slightly EV: +$167.

One difference is how I calculate the chances of winning the blinds. I look at each individual player's chances of picking up a calling hand (evaluate hand combos) and take into account the number of players there are: ((1-7.92%)^5) = 66.2%)

I think your method might inflate this number slightly because it doesn't isolate each opponent's chances of calling. But even still, you number is higher here which should increase your EV, yet my EV for you is still greater.

The other input that might be different is the value we are getting from PokerStove. I'm getting a winning % for 99 of 46.35%. If this number were slightly changed, your EV could move quite a bit (around 46%, the effect is roughly $22 of EV for every 1% change in winning %).

Once last thing I noticed is that your pot size is 6,550. How did you get that number? Your chips are 3,200 and the blinds are 300 so (3,200 + 3,200 + 300 = 6,700).

Did you take an average of the 5 stacks (with larger ones only counting as 3,200) and add that number to 3,200 and 300? I think this is actually a good idea but I didn't have that info so I just plugged in 3,200.

Luke

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

[ QUOTE ]
I actually created a spreadsheet

[/ QUOTE ]
Same here, PM me and I'll email you mine.

[ QUOTE ]
((1-7.92%)^5) = 66.2%)

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I think my method may be slightly more accurate here, but Im not sure. Mine represents al possible 2 hand combos dealt to five players, and shows the % of these 5 player sets that do not contain a hand in the calling range.



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I'm getting a winning % for 99 of 46.35%

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I used 46.3%

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your pot size is 6,550. How did you get that number?

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I just doubled my stack, and took 1/2 of the blinds. I did this to approximate the change in pot size if the caller was one of the blinds, vs the button, etc. There are much more exact ways to do it, probably pretty easy too, I will add that in my next run.

As for the changing pot size for smaller stacks calling, I dont think it is really necesary. If the goal of this analysis is to decide wether to push, EV will always be higher if you are only going to be called by a small stack.

The % of times stealing blinds shouldnt change, but the negative showdown equity will be less. This obviously wont matter when you are a favorite, b/c you want as much money in as possible.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

I think your call range is too large. AA-TT, AK, AQ seems more reasonable. You're pushing more than 10x the BB.. getting called by 77 seems a little far-fetched.

You're also at a spot in the tournament where $EV starts to deviate significantly from chip EV, so that assumption seems poor. I'd reccomend just using the ICM (independant chip model) here.

BTW, I don't think this is a questionable play, I think it is a bad play. Even if it turns out to be slightly +EV it is a bad play, as other plays will be more +EV.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

[ QUOTE ]
I think your call range is too large. AA-TT, AK, AQ seems more reasonable.

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Very possible. This was based on my table image. I had been working my way back from a short stack for a few orbits and had open pushed a few times.

[ QUOTE ]
You're also at a spot in the tournament where $EV starts to deviate significantly from chip EV, so that assumption seems poor. I'd reccomend just using the ICM (independant chip model) here.

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Can you explain why you think CEV and $EV deviate so much here? Also, there is a ICM for 110 players? I was under the impression this was a final table tool.

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Even if it turns out to be slightly +EV it is a bad play, as other plays will be more +EV.

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I addressed that in my post. I was ignoring other possibilities and seeing if this play was + or - EV.
Also, Im not sure I am crazy about making a standard raise and playing 99 out of position either.

I appreciate criticismm, so please expand on your points so I can understand your point of view.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

ICM is used mostly for SNGs / final tables. But there's no fundamental reason for this. There isn't a tool that I know of that lets you input arbitrary numbers of people and arbitrary prize structures, so I guess if you actually want to use ICM you'd have to write it.

When you're very far from the money chip EV is close to $EV. At the beginning of a tournament 10% of the people will get paid, and most of that goes to the top 5 spots. If you're in a tournament that only pays one spot, chip EV always = $EV. Very far from the money it's fairly obvious that the assumption that the two are equal is a good one. Conversely, this assumption breaks down the most late in the tournament when you're close to the money (on the bubble eg) or close to a pay jump.

Your tournament is like a tournament that pays 36% of the seats. This is more than gets paid in a SNG. If you want to just get a really rough idea, doubling up on the first hand of a SNG only gets you to 1.84x your original $EV. If you assumed you were getting to 2x you'd be making a significant error - one big enough that you wouldn't want to trust your results it was at all close.

This is super-rough, as it doesn't take into account people "starting" the tournament with different stack sizes or prize structure, or anything, but it's an illustration of how far wrong the CEV=$EV assumption could be.

As to table image, I think good players tend to overestimate their opponents' adjustments. Unless you've been pushing A TON so that it's completely obvious that you're often pushing on trash, people will be overly cautious in dealing with you. Since your stack is pretty decent now, someone with 77 just can't be too excited about gambling. Even if you're raising light they're probably not that much of a favorite.


If you make a standard raise here, most of the time you'll also steal the blinds, and most of the time you fail you'll be heads-up with the big blind. Playing out of position shouldn't be too big of a concern, imo.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

it is positive EV.
but that doesn't mean it is the best course of action.
I would much prefer doing a standard raise here.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

I know what the ICm is, and the logic behind it. I think the leap you made is probably wrong, or at least needs to be explained.

You can not treat a MTT with 110 people left 40 paying the same as an with 11 people and 4 paying.
Firstly, b/c the '4 places' pay at least 14 different payouts in a MTT. Also b/c if it works for 110 people, where do you draw the line and why?

This is not a bubble situation. Please explain why you think CEV varies from $EV here. I can not say it doesnt, but I am leaning that way until someone can explain why it doesnt.


Again, very possible that I overestimated range, I agree.

Question for you:

If I will 'most likely' be called by the big blind with a standard raise, creating a 6.5xBB pot. 50% of my stack, and I am giving him 3 cards to beat my hand. Deeper, this is how I play it, I just dont know if I love it with these stacks

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

[ QUOTE ]
it is positive EV.
but that doesn't mean it is the best course of action.
I would much prefer doing a standard raise here.

[/ QUOTE ]

How about a little color?

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

well you call your post a mathematical hand analysis.
you only did it on push or fold.
if it is an analysis on the hand, you should do the other scenario, raise.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

I fully realize that. I thought I made this clear in my post, but I forgot that this was kind of done in two posts.

What I was trying to do, was work through a simple problem. (push or fold), to see if my math was correct for the simple example. I was hoping for feedback as to the accuracy of my analysis. I 100% fully realize that the anlysis is incomplete, but I did not want to move on unless I was sure the 1st step was correct.

I only disagreed with the "you should have raised" posters, b/c I am not sure they are correct (they may be), I was hoping they would provide some insight into why they thought what they did so that we could work through the problem that way.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

The problem with "other plays will be more EV" is that you will lose 1.5BB each orbit 80% of the time if you do nothing, which not creates a smaller stack, but also

- affects the hands that others call with, making it more likely that you will be called, and be busted out
- if you are called and doubled up, then you are 3BB down from where you would have been and 6BB on the next double up

So, my question is how to best account for these 3 effects in a mathmatical model - smaller absolute stack, more chances of being called on future all ins, smaller stack even when doubled up

Possibly the best approach would be to write a simple software simulator which would play out these scenarios over several orbits.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

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You can not treat a MTT with 110 people left 40 paying the same as an with 11 people and 4 paying.

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I didn't. I said it is somewhat similar, as an illustration.

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Also b/c if it works for 110 people, where do you draw the line and why?

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There is no line. There is only one situation in a tournament where CEV = $EV. That is when there are two people left. (And a tournament where only first place gets a prize.) At ANY OTHER TIME, CEV doesn't = $EV. The only question is how much they differ by. ICM is a decent way of guessing that - there is no exact way.

Intuitively, CEV gets closer to $EV the further away you are from the money. In your example you are close to the money, so CEV will be significantly different from $EV. How much exactly? Can't say, nobody can. But ICM gets you a rough estimate.

[ QUOTE ]
I am giving him 3 cards to beat my hand.

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No, he paid for those 3 cards. You didn't give him anything. If you feel you're really bad at playing in these spots, maybe pushing in these spots is the right course of action. But that's a seperate thread and somehting you should be working on improving in your game.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

Alwayswrong, I appreciate your comments. My arguements are only half rooted in my opinions. Mostly I just dont know, you have some interesting points, I just want to understand your underlying reasoning.

You keep alluding to using ICM. This is a quote from Fnord_too's post on CEV ~ EV. He looked into the ICM code.

[ QUOTE ]

Just read their code. It looks like what it is doing is the following:

Everyone has a their chips/total chips chance of winning.

For each winning person, it calculates the probability of each other person coming in second by removing the winning players chips from the pool and using to above formula to see who wins amongst the remaining players. You multiply the prob of the assumed winner winning times the prob of the each player winning (in the now reduced field) to get the prob that that 1/2 scenario occurs and for each player sum the the probabilities of all 1/2 combinations where they are two.

Then basically repeat for each 1/2/3 combination.


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I'll admit, I dont fully follow his description, but it looks like CEV = EV when you are not in the money. I am not quoting this to say that your assertations are wrong, only to say that I dont think the ICM agrees.

What I think would be very good to discuss is how you have come to believe that CEV does not closely approximate EV in the middle-late stages of a MTT, when you still have 70 players remaining before payouts start. Im not saying its not true, I just dont like believing things b/c someone says so.

2nd. I really dont know if pushing 99 in this spot is max EV. maybe, maybe not. I was just playing devils advocate to your post. SO, you believe a standard raise is the best play. Could you please explain in detail why? I think most people who have made it to the bottom of this geeky ass thread are beyond needing "this way is the best" posts without an explanation of your thought process.

thanks

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

[ QUOTE ]
Everyone has a their chips/total chips chance of winning.

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At an arbitrary point in the tournament, in or out of the money. You just have a list of eg 110 people's chip stacks.

[ QUOTE ]
For each winning person

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What you're doing here is the traditional fraction of total chips in play = probability of coming in first. So if there are 110 people and you have an exactly average stack size, you have a 1/110 chance of winning. So what we're going to do is have 110 scenarios, 1 for each person winning. At the end we're going to multiply the result from each person winning by the probability that they will win. That's what he means by "for each". Obviously there cannot actually be 110 winners, there can only be one. But we consider each possible case and see how likely it is.

[ QUOTE ]
it calculates the probability of each other person coming in second by removing the winning players chips from the pool and using to above formula to see who wins amongst the remaining players.

[/ QUOTE ]

So in each of those 110 scenarios, it removes the winner from the tournament and pretends everyone is playing for second. Your chance of second = the fraction of the chips you have in play. So in our scenario if you had an average stack you'd now have a 1/109 chance of winning.

[ QUOTE ]
You multiply the prob of the assumed winner winning times the prob of the each player winning (in the now reduced field) to get the prob that that 1/2 scenario occurs and for each player sum the the probabilities of all 1/2 combinations where they are two.

[/ QUOTE ]

So 1/2 here means 1st place / 2nd place. For 110 people there are 110*109=11990 possible combinations. We're going to assign a probability to each of those combinations, and then sum all the results.

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Then basically repeat for each 1/2/3 combination.

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1/2/3 because SNGs pay 3 places. In a tournament that pays 40 places, you'd have to iterate it for each spot.

Hope that's fairly clear. Trust me on the fact that I know what I'm talking about and ICM does apply to an arbitrarily sized tournament, at any time during that tournament.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

your math is correct.
i have a model created that does the math for me. the win% when called on mine isn't perfect, but close enough.
mine says the ev is 141, which is close to yours.
yours is correct.

Re: Mathematical Hand Analysis (the EV of pushing 99 from MP)
 

there is no need to worry about this, because if nobody folded an A, then the chance somebody else has one goes up.
they completely cancel each other out.

as per the more than one caller, it really happens so incredibly little that you don't need to worry about it.


[ QUOTE ]
For instance, someone will fold A7, but that will affect the number of Aces available for hands like AK that will call.

[/ QUOTE ]