Some math that's supposed to help me figure out SNGs

[pergesu]

Okay so Bones and I have been talking about that 77 ITM hand I posted a little while ago. Here is a hypothetical situation that we're using to try and figure it out.

You: 2000
SB: 2000
Button: 4000

Blinds 100/200


SB shows AKo and pushes. You look down and see 77. ICM says a call here is correct (+1.3%).

The main reason the call is correct because of the payout weights.

So here's what we were thinking. In a $22 SNG, your profit for finishing 3rd will be $18. Winning this hand will give you a 50% chance of winning the tourney, ignoring difference in skill level. So when you go heads up, you have a profit EV of $58 ((78 + 38) * .5).

45% of the time you make $18, and 55% of the time you make $58.

58/18 = 3.22, so you need to finish 3rd place a little more than 3 times to equal the profit you gain from getting heads up once.

So that means to me that you only need to be 1-3.22 or better to win the hand, because getting heads up gives you the same expected profit as three third place finishes. So you ought to only need a hand that wins ~24% of the time to make the call.

That would be a massive range, but in fact you need 22+,AKs to make this call profitable. That's a range that wins 57% of the time, only 6% of hands.

I have to be looking at this all wrong, simply because SNGPT is slapping me around. I'd really appreciate it if someone could tell me what I'm missing, cause I think whatever it is may be the next big step towards me understanding SNGs.

[Freudian]

All three of you already have $40. What you are playing for now is a prize structure of $60/$20/$0.

The reason you don't take a 24% call now is that you can no matter what you have push the next hand for even more value. And if you call with a 24% hand, you will miss out on that opportunity 76% of the times. So instead of just having two outcomes (calling and winning/calling and losing) there is a third, folding, which means you get to play on.

Might add that I don't use ICM or SNGPT. So I don't think in terms of +1.4% etc. Mostly because I think we are often so poor at guessing the other guys range that the numbers is at best slightly inaccurate.

[pergesu]

No, the correct play according to ICM is to call here. So there are two outcomes - winning or losing.

You're talking about folding against ICM because we can have a bigger edge later on. But our edge increase with 77 vs AKo is 1.1%, which is just far too big to pass up.

[Freudian]

[ QUOTE ]
No, the correct play according to ICM is to call here. So there are two outcomes - winning or losing.

You're talking about folding against ICM because we can have a bigger edge later on. But our edge increase with 77 vs AKo is 1.1%, which is just far too big to pass up.

[/ QUOTE ]

With the 77, calling is correct.

But once you introduce hands that have 24% against his range, surely folding is an option.

[pergesu]

[ QUOTE ]
With the 77, calling is correct.

But once you introduce hands that have 24% against his range, surely folding is an option.

[/ QUOTE ]
According to ICM, it's not that folding is an option - it's that calling ISN'T an option.

Which is what I'm trying to figure out. If you need 3 3rd place finishes to make up for one heads up finish, you'd be willing to call with a hand that's a 3-1 dog, because you're getting an overlay.

[Freudian]

[ QUOTE ]

According to ICM, it's not that folding is an option - it's that calling ISN'T an option.

Which is what I'm trying to figure out. If you need 3 3rd place finishes to make up for one heads up finish, you'd be willing to call with a hand that's a 3-1 dog, because you're getting an overlay.

[/ QUOTE ]

But why would you call with a hand thats 3-1 dog when you the next hand, no matter what cards you have, will have a more +EV situation?

Are you working against a time limit here?

Are you really aiming for a finish distribution where you have twice as many 3rds as 1st and 2nds?

I might add that I have gotten much more patient when ITM and HU lately. I have learned to fold hands I might have called with/pushed in the past. And my 1sts have gone up. Of course I still play aggressive but not uncontrolled.

[kamrann]

[ QUOTE ]
If you need 3 3rd place finishes to make up for one heads up finish, you'd be willing to call with a hand that's a 3-1 dog, because you're getting an overlay.

[/ QUOTE ]
This is where you're going wrong, you don't need 3 3rd places to make up for one heads up. What you did wrong is you used expected profit in your calculations rather than expected payout. The $22 you invested is no longer yours, it's part of the prize pool. Using the payouts, you get E(3rd) = $40, E(HU) = ($100 + $60) * 0.5 = $80. So E(HU) = 2 * E(3rd). So 2 3rd places equates to 1 Heads Up, therefore you need to be even odds to make a call a break even play, as you'd expect.

In reality, due to the overlay of the blinds, it should be correct, by ICM, to call here as slightly less than 50%. I'm not sure why this doesn't quite fit with your calculations.

[pergesu]

According to what you said, you need to finish 3rd twice to make up for a single first place finish, so you can be at worst a 2-1 dog to make this break even.

[pergesu]

Of course I'm not aiming for more thirds than firsts.

What you're saying is to pass on a hand where you're a big dog because you can get a big edge later on. That makes perfect sense. Simple gambling concept to maximize EV.

What I'm saying is you'd be WILLING to take a gamble where you're 3-1 against winning. Sure, you can have a bigger edge later on, but taking this gamble still isn't bad.

But what I'm also saying is that's wrong. The reason I'm saying it's wrong is because ICM is telling me it's wrong. I'm trying to understand WHY it's wrong.

[kamrann]

Yeah, I messed up. It's a bit confusing. But basically it's to do with what Freudian said about the remaining payouts. You're all guaranteed 3rd place money, so you're playing for 60/20/0. As such it doesn't make sense to apply any sort of valuation to finishing third, because it currently has zero value. I've just woke up and my head isn't working too well. I'll have a think about it...

[Freudian]

[ QUOTE ]

But what I'm also saying is that's wrong. The reason I'm saying it's wrong is because ICM is telling me it's wrong. I'm trying to understand WHY it's wrong.

[/ QUOTE ]

ICM is telling you it is wrong because once you are ITM, getting into 3rd is worth $0. So if you take plays where you end up in 3rd a vast majority of the time, you are giving a lot of EV to the other players at the table.

[kamrann]

Yep, it's pretty simple really. The decisions you make at this point are based on making money from your current situation, which means finishing 2nd or 1st. If it was winner take all, then forgetting the overlay of the blinds you would call with 50%+ favourite hands, since if you were 50% to double up/bust in the case your equity doesn't change. 0.5 * E(3rd) + 0.5 * E(HU) = 1/4 of the prizepool, since E(3rd) = 0 and E(HU) = 1/2 the prizepool. And this is the same equity you already have with 1/4 of the chips.

What I forgot before was the payput structure affects this. With a 60/20/0, doubling up does not quite double your equity, so you need more than just 50% to make a call correct.

Go here (http://www.bol.ucla.edu/~sharnett/ICM/ICM.html) and play around with the stacks with the payouts adjusted to 0.75, 0.25, 0. That should help you understand the situation better.

[pergesu]

I have a few problems with that.

First, that suggests that no matter how many times you get third, you can never make up for a single first place finish. Each time you finish third, you make $0. Each time you finish first, you make $58. So if I played a million tourneys and finished third, I wouldn't have made a penny. In fact, you'd have made more than me if you finished first one time. Which obviously isn't true.

Second, I think that idea would lead to "one more spot" thinking, where you basically are just trying to move up one more spot. When you're on the bubble, you're just trying to make it ITM. When you're ITM, you're just trying to make it to second. And when you're heads up, you're just trying to win. But we always say "play for first" when you're on the bubble and ITM. Because that maximizes EV. Why? Because of the payout structure. I'm just not going to take "because of the payout structure" as an acceptable answer - I want to see it quantified.

For what it's worth, I'm not trying to be argumentative or stubborn at all. I just had an idea, and it doesn't match up with the common (and correct, I believe) ideas that we use to help make decisions. I just want to know why.

[Freudian]

[ QUOTE ]

First, that suggests that no matter how many times you get third, you can never make up for a single first place finish. Each time you finish third, you make $0. Each time you finish first, you make $58. So if I played a million tourneys and finished third, I wouldn't have made a penny. In fact, you'd have made more than me if you finished first one time. Which obviously isn't true.

[/ QUOTE ]

Thats an argument for getting ITM, not for getting into 3rd while ITM. If your starting point is when three players remain, getting into 3rd often sucks. If your starting point is when ten players remain, getting a lot of 3rds is ok.

[Moonsugar]

You already have a lock on 3rd. If you win this hand do you automatically win 1st?

Also, as Freudian stated your analysis of prize structure is suspect.

[pergesu]

[ QUOTE ]
You already have a lock on 3rd. If you win this hand do you automatically win 1st?

Also, as Freudian stated your analysis of prize structure is suspect.

[/ QUOTE ]
No, you don't automatically win first. Which is why I said when you get heads up, your expected profit is $58, or 40% of the prize pool.

Here's the best way to think about my analsis of the prizepool. Let's say that Freudian and I are both exceptionally good players. So exceptional, in fact, that we both finish ITM 100% of the time. The difference is that Freudian always finishes in 1st, and I always take 3rd. If he plays 100 tourneys, he'll make $7800. But if I play 100 tourneys, I'll make $1800. If I want to make as much as he does in 100 tourneys, I need to play 433 of my own.

That's what I was saying I didn't like about his idea that you make nothing by finishing third. It's simply not true. I can make the same amount of money by playing 4.3x as many tourneys as he does.

[Freudian]

[ QUOTE ]

That's what I was saying I didn't like about his idea that you make nothing by finishing third. It's simply not true. I can make the same amount of money by playing 4.3x as many tourneys as he does.

[/ QUOTE ]

The $40 you get for finishing third isn't created when you bust in 3rd, but when you avoid busting while players 4-10 bust.

So I don't see how it is important once you get ITM. The only money up for grabs is the extra money you get for winning or coming second.

[pergesu]

[ QUOTE ]
[ QUOTE ]

That's what I was saying I didn't like about his idea that you make nothing by finishing third. It's simply not true. I can make the same amount of money by playing 4.3x as many tourneys as he does.

[/ QUOTE ]

The $40 you get for finishing third isn't created when you bust in 3rd, but when you avoid busting while players 4-10 bust.

So I don't see how it is important once you get ITM. The only money up for grabs is the extra money you get for winning or coming second.

[/ QUOTE ]
I just can't make sense of that, cause if I play 4.3x as many tourneys as you, then you and I have the same bankroll. Yeah, you're way more efficient than me, but we earn the same amount of money.

Anyway, that's not even the point of all this. The point is WHY is the 77 call so clearly correct? I was playing around with some math and fell flat on my face. People are telling me it's wrong, but as I said in my first post and nearly every response since, I already know that. But nobody has shown me why it's wrong yet, which is what I want to learn.

[Costanza]

I don't have an ICM calculator handy but I think I can tell you how to set up this problem. Let me make sure I have the situation correct:

Button: 4000
SB: 2000
BB (you): 2000
Blinds 100/200
Button folds, SB flips up AK and pushes, you hold 77

Because of the stack sizes you have 2 options: fold or call

Calling: on average, the chip stacks will change to:
BB 4000
SB: (4000)(0.48) = 1920
BB: (4000)(0.52) = 2080

Folding: chip stacks will change to:
BB: 4000
SB: 2200
BB: 1800

So, in this specific situation calling is +chipEV.


Of course, this isn't what's really important. What you want to know is whether or not the call is +EV from a tournament equity standpoint. That's where the ICM calculations come into play. If you plug the average results into an ICM calculator it will tell you which play, on average, has the best expected EV.

As a step toward understanding the results you'll get, think of is this way:

The $22 you paid to enter the tournament is gone.

If you call, 48% of the time you win $40, and 52% of the time you'll be even with the other stack HU and therefore have a tournament equity of $80 [(0.5)(60)+(0.5)(100)]. So calling gives you an average tournament equity of $60.80.

If you fold, your tournament equity is [(chance of 3rd)(40)+(chance of 2nd)(60)+(chance of 1st)(100)]. If this works out to more than $60.80, you should fold. If it's less, then it's a call according to ICM.

Someone let me know if this makes sense because I want to know I'm thinking about this correctly.

[pergesu]

That makes sense, but it's not the issue at hand. We know that calling is +$EV according to ICM, so folding isn't even a consideration here.

What I'm saying is that you stand to profit 3.2x as much by finishing 1/2 than you do by finishing third. So you're basically getting odds of 3.1-1 on a call, thus you'd be we willing to accept a gamble as a 3-1 dog.

But it changes if we think of the payout rather than profit. 3rd gets $40, taking 1/2 gets $80. You'd still be willing to take a gamble as a 2-1 dog.

So according to my math, you should be able to profitably call with WAY more hands than just the 6% according to ICM. And I think it's wrong...I just want an explanation why.

[Ryendal]

There is at least one factor which is not taken in ICM.
The "Dynamism of the game" . I would imagine a line as this one. D=0,35 would mean there is 35% tourney where the other short stack will be broken while you fold all your hands.

An idea like that. I know it's not enough but we can't ignore that.

[microbet]

The two main problems are, you have already paid the $22 and you have already won at least 3rd place.

You have a certain chance of half of the remaining equity, which is $80, and a certain chance of none of the remaining equity. You have to weigh this against the equity you have if you fold.

If you wanted to, you could phrase the question like this:

I have won $1 million dollars playing poker in my life. If I win this hand, I will have won $1,000,058. If I lose this hand, I will have won $1,000,018.

[Moonsugar]

Just got home from church and will try to clarify:

Payouts are as follows:

1 = 30
2 = 10
3 = 0

(These are pool% numbers less the floor all players have of 3rd place money.)

Your EV right now is 10.83 (for simplicity lets say 10). Ignore blinds (for simplicity) and assume that if you win you have EV of 20. Since if you fold you have EV of 10 and if you call you have a binary payout of EV0 or EV20 (with our assumptions) then your hand has to win more than 50% v. AK for this call to be profitable.

What hands beat AK more than 50% of the time? Any pair. You call with any pair and you fold all others.

Our assumptions are not too out of line and I think the math gives the correct answer here.

Does this help?

[Moonsugar]

Now that I think about it, what you seemed to be leaving out of your reasoning was that if you fold you can still win 2nd or 1st. I think you were only thinking that you could get 3rd.

[Nottom]

The problem with all your math is that you are ignoring what your stack is worth when you fold. As you say, you are in a coinflip situation, and then will be HU with a chance at another coinflip (even stacks) to win. So you win a little more than 25% of the time and get HU a little over 50%.

If you were to fold you still have 1800 chips and should by conventional wisdom finish first just under 25% of the time and get HU a little under 60% of the time

If you are suddenly taking 2-1 the worst of it here, now you will only win about 16% and gets HU 33%. Obviously a much worse situation.

Note: You logic is pretty good if you ever end up on "Who Want's to be a Millionaire"

[PrayingMantis]

I edited this post a bit, but my original reply is that if you fold your EV is a little less than $20, and if you call and win your EV is $40, and so, assuming equally skilled players, the math is very easy.

Numbers are for a $22 SNG.

[Costanza]

OK -- I think I understand your question a little better now. So, what range of hands should you call with in this situation if your opponent flips over AKo?

As a rough guess, let's say your current equity is [(0.375)(40) + (0.375)(60) + (0.25)(100)] = $62.50

We already decided that your equity if you call and win is [(0.5)(60)+(0.5)(100)] = $80. If you call and lose it's $40. You can then figure the winning percentage you need to make this call correct: 62.50 = 80x + 40(1-x) ==> x=0.56

So if these were the real numbers, you'd need a hand which is 56% against AKo in order to increase your tournament equity. [edit: this would mean you could only correctly call with AA or KK]

But, folding is an option. Maybe not the best one, but it is an option. So the real question is what percentage of the time do you need to win for calling to be better than folding.

I can't see my last post, but I think I remember gettting something like $60.20 in tournament equity if you fold. With this number, you need a hand that is 50.5% against AKo to make calling better than folding. AKs or any pair should do.

I'll run it with some real numbers when I get home this evening but I bet these are close.

[PrayingMantis]

[ QUOTE ]
OK -- I think I understand your question a little better now. So, what range of hands should you call with in this situation if your opponent flips over AKo?

As a rough guess, let's say your current equity is [(0.375)(40) + (0.375)(60) + (0.25)(100)] = $62.50

We already decided that your equity if you call and win is [(0.5)(60)+(0.5)(100)] = $80. If you call and lose it's $40. You can then figure the winning percentage you need to make this call correct: 62.50 = 80x + 40(1-x) ==> x=0.56

So if these were the real numbers, you'd need a hand which is 56% against AKo in order to increase your tournament equity.

But, folding is an option. Maybe not the best one, but it is an option. So the real question is what percentage of the time do you need to win for calling to be better than folding.

I can't see my last post, but I think I remember gettting something like $60.20 in tournament equity if you fold. With this number, you need a hand that is 50.5% against AKo to make calling better than folding. AKs or any pair should do.

I'll run it with some real numbers when I get home this evening but I bet these are close.

[/ QUOTE ]

If you are aware of doing it and being consistent with it, it doesn't matter much, but usually when you talk about EV, it's easier and makes more sense to ignore the money already won (here, for instance, the share of the prize pool that was already distributed to all 3 ITM players, is not part of your *expected* value). And you're not the only one doing it here. I think it's a source for a lot of confusion.

[Costanza]

Now I think I'm confused with semantics. I agree there is some confusion here, and purposefully didn't talk in terms of EV in my last post for just that reason. It's a lot easier for me to think of these situations in terms of what I call "tournament equity" which is the amount of the prize pool I can expect based on my chip stack and the stacks of the remaining players.

If there was a problem with that in my last post please let me know. I think this is a pretty fundamental aspect of understanding tournament play, I think I finally am starting to understand it, and I want to know if I'm getting things mixed up.

If we're going to talk about EV in this situation, I think we need to be careful to explain if were talking about a play being +EV in terms of adding chips to your stack or being +EV in terms of increasing your, well, what I would call "tournament equity." Because the two are definitely not always the same thing.

[microbet]

You essentially have it right. The first calc where you got .56 isn't right because keeping 2000 chips isn't an option. The second equation is right. The things to equate are equity if you fold vs. expected equity if you call.

In the previous example I don't see where you worked out the equity if you fold. Your procedure (guessing how likely you are to come into each place, multiplied by the prize amount) is reasonable, but hard to justify the numbers you pick. (say, when stacks are 4000,2200,1800) ICM is just a way of estimating the value of those complicated stack arrangements.

It's not just a coincidence that you come up with the right answer (IMHO), which is basically call if you are the favorite.

This is different than a cash game, where you should call as a slight dog because of the dead money in the pot.

[eastbay]

[ QUOTE ]
Okay so Bones and I have been talking about that 77 ITM hand I posted a little while ago. Here is a hypothetical situation that we're using to try and figure it out.

You: 2000
SB: 2000
Button: 4000

Blinds 100/200


SB shows AKo and pushes. You look down and see 77. ICM says a call here is correct (+1.3%).

The main reason the call is correct because of the payout weights.

So here's what we were thinking. In a $22 SNG, your profit for finishing 3rd will be $18. Winning this hand will give you a 50% chance of winning the tourney, ignoring difference in skill level. So when you go heads up, you have a profit EV of $58 ((78 + 38) * .5).

45% of the time you make $18, and 55% of the time you make $58.

58/18 = 3.22, so you need to finish 3rd place a little more than 3 times to equal the profit you gain from getting heads up once.


[/ QUOTE ]

You were fine until this last sentence I think, and then you took a left turn into nonsenseland.

What you are trying to do is to estimate the equity of each of your possible moves. The two moves you're considering are calling and folding.

For folding, you get this from ICM'ing the chip stacks if you fold out (and assume that it's folded around for simplicity.)

For calling, you need to use the numbers you just quoted to get the overall EV. You do this by weighting each outcome by its probability. (.45)(18) +(.55)(58) = X.

The key concept is that the overall EV of some decision is the average of the possible outcomes weighted by the probability of each. Your ratio 58/18 isn't relevant. What you want is the overall EV of calling, so that you can compare it to folding (or the EV of any other possible move, if you have some way of estimating it.)

Then you compare X to whatever you got for folding, and this helps you make your decision.

eastbay