Theoretical problem about coinflips

[AleoMagus]

Maybe easy to answer, and I certainly think I know the answer to this, but I'd like to see a definitive closure to the following question:

Assuming Independent chip model is accurate, opponents are equal in ability, and blind sizes are negligible to the problem, is there any situation where a known coinflip should be taken when an opponent pushes all in before you? Also assume that no matter how many players are at the table, showdown will only involve you and the lone pusher.

Put another way, if I know for CERTAIN that my odds of beating an opponent in a showdown are exactly 50/50, and this opponent pushes all in ahead of me, are there any conditions under which I can call profitably.

So just pull out the ICM calculator and find me one example.

Some more complicated questions to answer if you are feeling really ambitious:

If there are circumstances where you can call an opponent profitably with a true coinflip, what is special about those circumstances in general that makes this so?

Conversely, if there are no circumstances under which this is the case, I'd like to see a proof.

Regards
Brad S

As a corrollary to this problem, does the answer to this make sense given what we know about the relative value of chips in different sized stacks?

Ie - common tourney knowledge often suggests that calling with a suspected coinflip is wise if we have a huge stack in comparison. This is becasue the small stack's chips are worth more than ours and we may actually have pot odds to do so when considering the 'extra' value of shorty's chips.

[tigerite]

I'm sure you know this already but yes, there certainly are situations when you should call as a 50/50, and even as a 45% dog and worse. They're rare but they do occur.

To give an example let's say you're in the BB with A8s and the SB has pushed. If you fold your EV is 0.1556 (and the SB's goes up to 0.1556 also). If you call and win your EV goes up to 0.2716, and the SB busts, thus 0. If you call and lose, your EV goes down to 0.0475, and the SB's up to 0.2430. In this case, you only have to be a 49.6% favourite, and can call as a coin flip, even assuming you want a 0.3% edge.

[AleoMagus]

show me one

[tigerite]

Just have.. was editing, sorry.

[AleoMagus]

sorry, what are the stacks in this example and how many players are at the table?

Regards
Brad S

[EverettKings]

Unless I misread that post, you make no assumptions about equal stack sizes. So I can think of two spots.

1) You have the pusher well covered. As an extreme, say you're 5 way and he has one chip. Or you have 7k and four people have t250 and one pushes. In these cases you benefit from busting a guy and really can't hurt your stack. I'm not going to sit on the ICM calculator but I think most people in the forum would make that call.

2) You're down to 2 players. Kind of a trivial example but hey, it works.

[AleoMagus]

[ QUOTE ]
1) You have the pusher well covered. As an extreme, say you're 5 way and he has one chip. Or you have 7k and four people have t250 and one pushes. In these cases you benefit from busting a guy and really can't hurt your stack. I'm not going to sit on the ICM calculator but I think most people in the forum would make that call.



[/ QUOTE ]

It's extremely close, but ICM calc disagrees if blind sizes are negligible. At least as far as I have checked. That said, I too make these calls all the time, though I think it's just because usually the blinds really matter.

[ QUOTE ]
2) You're down to 2 players. Kind of a trivial example but hey, it works.

[/ QUOTE ]

I suppose when down to 2, blind sizes can never really be negligible, so you are right

I guess I am thinking more about situations with 4+ players, (and maybe with 3, though I am less sure about that)

Regards
Brad S

[Big Limpin']

I've been waiting to say this. Its just my get though.

The *vast* majority of tournament situations dictate folding a "known" 50/50. Perhaps all? But what exceptions there may be are not ones where [ QUOTE ]
Assuming Independent chip model is accurate, opponents are equal in ability, and blind sizes are negligible to the problem,

[/ QUOTE ]

Its the boundary conditions, where these assumptions tend finally away from zero, where +$EV 50/50s would present themselves.

[stupidsucker]

but blind sizes are rarely negligible. Why worry about a situation that is basicly never going to happen. I assume there must be a reason, but I don't understand it.

[kevkev60614]

With those assumptions, I don't think there are any conditions which would make a call profitable for that hand. But when you take table image into account...

Also, I can definitely imagine a scenario where calling and busting another player would speed up the game, improving your $/hr.

[HesseJam]

[ QUOTE ]
but blind sizes are rarely negligible. Why worry about a situation that is basicly never going to happen. I assume there must be a reason, but I don't understand it.

[/ QUOTE ]

same here.

Also playing ability is usually not equal. If up against a good player, try to knock him out with a coinflip.

It's much more important to know how to play when abiltity is unequal and blind sizes are not negligible.

[AleoMagus]

[ QUOTE ]
but blind sizes are rarely negligible. Why worry about a situation that is basicly never going to happen. I assume there must be a reason, but I don't understand it.

[/ QUOTE ]

We never really KNOW we are in a coinflip situation either, but these kinds of things can be assumed if we are just trying to generalize.

If it really bothers you that blind sizes are negligible, assume that that blind sizes can be ignored becasue you know (or estimate) that you are a slight underdog. here is a concrete example:

Player A) t1500
Player B) t1500
Player C) t1000
Player D) t6000

You are player D. Blinds are 50/100 and you are in the BB with AQ. All fold to player C who pushes all in. For some reason (very good read? Flashed cards?) you know he has a medium pair and you know that you are a slight underdog. It really doesn't matter what you or he have, just assume that your best estimate is that you will win this hand 46% of the time if you call.

So, if you fold:

Player A) 1500, with an equity of .2148
Player B) 1500, with an equity of .2148
Player C) 1100, with an equity of .1699
Player D) 5900, with an equity of .4004

if you call, possible outcomes are

win (.46)
Player C) 0, with an equity of 0
Player D) t7000 with an equity of .4347

lose (.54)
Player C) .2387
Player D) .3704

so, your total equity is ((.54)(.3704)+(.46)(.4347))=.3999

You should fold, though I admit it is very close.

Is everyone content to fold in a situation like this?

Regards
Brad S

[ QUOTE ]

Is everyone content to fold in a situation like this?


[/ QUOTE ]

Nope, I'd rather not send the message that my BB is ripe for the taking.

[Nicholasp27]

yes, as big stack there, i'd rather fold here and then open-push a WIDE range of hands when i'm first in pot...none of them wanna end up in fourth, so they will fold most hands to u and u can pick up the t300 in blinds often (and soon it'll be t750)

with those stacks i don't want the bubble to burst just yet so i definitely wouldn't take a coinflip here, even if icm said it was +ev

[AleoMagus]

I think I am still being unclear. What I am really driving at is this:

We all seem to agree that taking a coinflip for your survival is a bad idea in a SNG and we try to avoid those sorts of confrontations. What about when you have a really big stack and it's not about your survival? Can you start to make more gambling kinds of plays if your stack is huge and you have the opportunity to bust a player?

Now, though I mentioned it in my original post, I don't buy the argument that we can do this because shorty's chips have 'extra' value. As we add them to our stack, they don't have that value anymore so that can't exactly be the reason why we would justify these kinds of plays.

Still, many tournament experts seem to advocate taking chances against small stacks where you have the opportunity to bust them.

Is our avoidance of confrontation a universal theme of good tournament play, or can we take advantage of small stacks by getting them all-in we have a huge stack. Even if we are only 50/50 or worse?

ICM says no

I guess my reason for this thread is just that when I was thinking about this myself, I suspected it would say no, but that wasn't my gut feeling about what was right. Now that may simply be because in reality the blinds will give us the proper edge we need.

Still, might it be something more than this.

While I don't like to deal in vague imprecise statements, I am reminded of a gigabet (I think) statement where he considers the extra chips in a big stack (when chipleader) somewhat useless, as he cannot double those chips on a single hand. Does this, or something like this change our opinion about putting those chips into play on even money confrontations, or even confrontations where we might be a slight underdog?

Regards
Brad S

[microbet]

I would say no, but I don't have time to think about it much now.

Tigerite, are you counting blinds in there? Aleo wants blinds to be negligible.

Aleo, are you trying to reconcile ICM with Block theory? I don't think that's gonna happen.

I don't put a lot of stock in the feeling that coinflips are good early in an MTT. I think in an MTT you are far away from the money and the equity is a lot closer to a cash game and thus a true 50/50 is not too bad, and being on the good side of a coin flip is probably good enough. In an STT we start pretty close to being on the bubble.

Lastly, what's it like living in Victoria? Is it something like heaven?

[pooh74]

[ QUOTE ]
I think I am still being unclear. What I am really driving at is this:

We all seem to agree that taking a coinflip for your survival is a bad idea in a SNG and we try to avoid those sorts of confrontations. What about when you have a really big stack and it's not about your survival? Can you start to make more gambling kinds of plays if your stack is huge and you have the opportunity to bust a player?

Now, though I mentioned it in my original post, I don't buy the argument that we can do this because shorty's chips have 'extra' value. As we add them to our stack, they don't have that value anymore so that can't exactly be the reason why we would justify these kinds of plays.

Still, many tournament experts seem to advocate taking chances against small stacks where you have the opportunity to bust them.

Is our avoidance of confrontation a universal theme of good tournament play, or can we take advantage of small stacks by getting them all-in we have a huge stack. Even if we are only 50/50 or worse?

ICM says no

I guess my reason for this thread is just that when I was thinking about this myself, I suspected it would say no, but that wasn't my gut feeling about what was right. Now that may simply be because in reality the blinds will give us the proper edge we need.

Still, might it be something more than this.

While I don't like to deal in vague imprecise statements, I am reminded of a gigabet (I think) statement where he considers the extra chips in a big stack (when chipleader) somewhat useless, as he cannot double those chips on a single hand. Does this, or something like this change our opinion about putting those chips into play on even money confrontations, or even confrontations where we might be a slight underdog?

Regards
Brad S

[/ QUOTE ]

Thats funny, because my gut instinct always said "yes", it is a bad idea. The inherent advantage of calling a short stack's all-in is usually in the fact that their range will be much wider. But I dont see why taking 50/50s (putting aside that we never "know" for sure we have a flip) is commonly regarded as a good thing.

I think the value of having a short stack present is mainly to be able to exploit the medium stack(s). This, to me, is much more valuable than knocking out a player. I would rather use those so called extra chips exploiting someone who still has hopes of making the money ( or higher payout) than knocking out someone on a flip and then having much LESS leverage against those medium stacks. ICM will not take this into account but these concepts are huge IMO.

[AleoMagus]

[ QUOTE ]
Lastly, what's it like living in Victoria? Is it something like heaven?

[/ QUOTE ]

no, but it's not too bad either. lots of rain in the winter, but one of the prettiest cities in Canada.
Why? thinking of moving here?

We are supposed to be getting a legit casino poker room pretty soon, so one major drawback of victoria will soon be gone.

I'll also say this about Victoria. People who don't live in Victoria think that living in Victoria would mean lots of trips to Vancouver and Seattle. This is a myth. Considering the time, expense and hassle involved with getting off the isalnd, it actually feels pretty isolated here sometimes. Then again, a lot of people like it that way.

Regards
Brad S

[valenzuela]

I came up with an example where it doesnt matter. I mean its not close, its the same thing!!. I repeat just in case, you lose 0,0055 of the prize pool and u win 0,0055 of the prize pool, im not considering blinds.
Im now trying to come up with a profitable example.

edit: I jus realized that its only the same thing becuase of ICM lack of more decimals.

[microbet]

[ QUOTE ]
[ QUOTE ]
Lastly, what's it like living in Victoria? Is it something like heaven?

[/ QUOTE ]

no, but it's not too bad either. lots of rain in the winter, but one of the prettiest cities in Canada.
Why? thinking of moving here?

We are supposed to be getting a legit casino poker room pretty soon, so one major drawback of victoria will soon be gone.

I'll also say this about Victoria. People who don't live in Victoria think that living in Victoria would mean lots of trips to Vancouver and Seattle. This is a myth. Considering the time, expense and hassle involved with getting off the isalnd, it actually feels pretty isolated here sometimes. Then again, a lot of people like it that way.

Regards
Brad S

[/ QUOTE ]

I definitely want to move somewhere north of LA (where I am) on the pacific coast within a few years. Victoria or Vancouver I think would be great as I could afford a much nicer house and I think they are really beautiful. I lived around Portland, OR for 5 years so I know about rain.

More likely I'll end up staying in Cali though. /images/graemlins/frown.gif

[Simplistic]

[ QUOTE ]
[ QUOTE ]
Lastly, what's it like living in Victoria? Is it something like heaven?

[/ QUOTE ]

no, but it's not too bad either. lots of rain in the winter, but one of the prettiest cities in Canada.
Why? thinking of moving here?

We are supposed to be getting a legit casino poker room pretty soon, so one major drawback of victoria will soon be gone.

I'll also say this about Victoria. People who don't live in Victoria think that living in Victoria would mean lots of trips to Vancouver and Seattle. This is a myth. Considering the time, expense and hassle involved with getting off the isalnd, it actually feels pretty isolated here sometimes. Then again, a lot of people like it that way.

Regards
Brad S

[/ QUOTE ]common consensus in canada is that victoria is one of the best places to live. your real estate market is running on a heater too.

i'm not the biggest fan of live poker other than home games with friends. in edmonton we have at least 4 cardrooms but profit isn't there compared to online and the typical pokerroom degenerates can be tough to deal with.

[se2schul]

[ QUOTE ]
[ QUOTE ]

Is everyone content to fold in a situation like this?


[/ QUOTE ]

Nope, I'd rather not send the message that my BB is ripe for the taking.

[/ QUOTE ]

Isn't this situation similar to the rule about calling a single all-in with any 2 cards if you are getting 2:1 for less than a third of your stack??

In this case, you are calling a single all-in, the pot is laying you roughly 1:1, and you are roughly 50% to win the hand. By the same logic as the previous example, shouldn't it be a clear call as long as it's for less than half your stack?? In this case it's for significantly less than half your stack which should make it an attractive option.

What am I missing?

[schwza]

i don't know the math behind icm very well, but i'd be shocked if there were a time to take a coinflip when there are 3+ players.

the reason that coinflips are ok HU is not that blinds can't be negligible (i think you said that) it's that cashEV = chipEV (at least according to ICM) so flips are neutral.

[Degen]

Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[Simplistic]

[ QUOTE ]
Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[/ QUOTE ]which is against ICM theory

[se2schul]

[ QUOTE ]
[ QUOTE ]
Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[/ QUOTE ]which is against ICM theory

[/ QUOTE ]
... and it will surely sometimes contradict a basic pot odds call as well, but it doesn't mean that it's wrong.

Just because it doesn't jive with the chip model that is commonly accepted within this forum, it certainly doesn't mean his play is wrong. He's just using a different chip model, and it may be better than ICM.

It's hard to argue with his success .

[Lori]

Heads up, tiny stacks, enormous blinds.

300-600
Hero (bb) 800 chips
Villain (sb) 800 chips, pushes

Hardly exciting, but I don't think there any exciting solutions.

Lori

[Simplistic]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[/ QUOTE ]which is against ICM theory

[/ QUOTE ]
... and it will surely sometimes contradict a basic pot odds call as well, but it doesn't mean that it's wrong.

Just because it doesn't jive with the chip model that is commonly accepted within this forum, it certainly doesn't mean his play is wrong. He's just using a different chip model, and it may be better than ICM.

It's hard to argue with his success .

[/ QUOTE ]

[Lori]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[/ QUOTE ]which is against ICM theory

[/ QUOTE ]
... and it will surely sometimes contradict a basic pot odds call as well, but it doesn't mean that it's wrong.

Just because it doesn't jive with the chip model that is commonly accepted within this forum, it certainly doesn't mean his play is wrong. He's just using a different chip model, and it may be better than ICM.

It's hard to argue with his success .

[/ QUOTE ]

However his success has nothing to do with this problem.

Lori

[schwza]

[ QUOTE ]
I guess I am thinking more about situations with 4+ players, (and maybe with 3, though I am less sure about that)

[/ QUOTE ]

3 will be same as 4. as long as more than one spot is paying out, you don't want to take a coinflip. 4 players is 0:2:3:5, 3 is essentially 0:1:3, so it's not fundamentally different.

[tigerite]

Correct, it never ceases to amaze me how many don't realise that ITM is a mini bubble of its own !

[schwza]

[ QUOTE ]
Correct, it never ceases to amaze me how many don't realise that ITM is a mini bubble of its own !

[/ QUOTE ]

i've wondered before about playing 2-table sng's with 2:4:6:8 structure. all bubbles are same size but i bet people bend over backward to avoid 5th. never played 'em though.

[valenzuela]

Im going to disagree with you here, on all the final tables Ive played on MTT( 3 of them)we are always on a bubble everyone wants to move up on the money ladder.

[mcteecho]

As a Vancouverite, Victoria seems (with respect, Aleo) a bit sleepy, but otherwise very nice. Vancouver is great but has the highest housing prices in Canada so you might want to make sure your housing $ is going to go further here before making any irrevocable decisions. My condo has gained about $300,000 in value in 2.5 years.

[schwza]

[ QUOTE ]
Im going to disagree with you here, on all the final tables Ive played on MTT( 3 of them)we are always on a bubble everyone wants to move up on the money ladder.

[/ QUOTE ]

yeah, that's how MTT's tend to play. i'm talking about 18-person sng's where the increment is always the same (2 buy-ins).

[ChrisV]

[ QUOTE ]
[ QUOTE ]
Gigabet had a great post on this awhile back...he is willing to take the worst of it at times, if the result will allow him to have a MASSIVE stack and walk over the table

[/ QUOTE ]which is against ICM theory

[/ QUOTE ]

Not against. The word you're looking for is outside.

There are certain situations where it's correct to overrule the ICM and take a stab. Typically they arise 5-6 handed and in situations where you won't be close to going allin.

[FlyWf]

Do you know what ICM theory is?
Hint: It's not normative. ICM doesn't tell you what to do, you can't go against ICM.

[Simplistic]

[ QUOTE ]
Do you know what ICM theory is?
Hint: It's not normative. ICM doesn't tell you what to do, you can't go against ICM.

[/ QUOTE ]my understanding of ICM is that it assigns tournament chips a certain value of the prize pool based on stack sizes and villain hand ranges. thus if you have XX hand and push/fold/call each will have a different value and you should choose the action which maximizes equity

[FlyWf]

ICM is just the assigning of value. It's fundamentally just an expansion of the very simple idea that holding all the chips doesn't mean you get all the money. It's useful for understanding basic concepts like don't coinflip early, but it doesn't tell you to fold Q3 where Gigabet called. To make call/fold/push decisions you need hand ranges and table context that are well beyond the scope of ICM.

[Insty]

[ QUOTE ]
if I know for CERTAIN that my odds of beating an opponent in a showdown are exactly 50/50, and this opponent pushes all in ahead of me, are there any conditions under which I can call profitably.


[/ QUOTE ]

No. The best you can do is break even.

[ QUOTE ]

I'd like to see a proof.


[/ QUOTE ]

I've managed to prove it for a winner takes all tournament. I can type this up if you think it would be useful.
It's a bit more complicated for a 5/3/2 payout structure, and I've not quite managed it yet.
Although I've simulated enough to be confident of the answer.

[ QUOTE ]

As a corrollary to this problem, does the answer to this make sense given what we know about the relative value of chips in different sized stacks?

Ie - common tourney knowledge often suggests that calling with a suspected coinflip is wise if we have a huge stack in comparison. This is becasue the small stack's chips are worth more than ours and we may actually have pot odds to do so when considering the 'extra' value of shorty's chips.


[/ QUOTE ]

In the specified scenario it's never -ev to call when you have more chips than your opponent.
I thought that was interesting.

Insty.

Here's what I know so far.

1. Assume you have the option to call (but not raise) the final bet by the
only other player still in the pot. Imagine a bet putting one of you
all-in or a bet on the river.
2. There may be dead money in the pot from blinds or earlier bets.
However, assume the bet is "fair," i.e. the pot odds equal the odds against
your winning the bet.
3. And so far, unfortunately, assume the tournament pays only two places with at least three still in it.

The ICM then implies that calling the bet has negative EV.

Sketch of the proof. Your total EV is comprised of several terms. It is
fairly easy to check that your expected value for finishing first or for
finishing second when anyone except the bettor finishes first is the same
whether or not you call the bet. The only remaining term is when the
bettor finishes first and you finish second. Here your EV decreases if you
call the bet. In a nutshell, this is because f(z)=(y+z)(x-z)/(1-y-z) is
concave down (x represents your percentage of the total chips, y the
bettor's, and z your loss, which could be negative).

This same proof shows that raising on the river if you are certain your
opponent will call is worse than calling.

[Insty]

[ QUOTE ]
Here's what I know so far.

1. Assume you have the option to call (but not raise) the final bet by the
only other player still in the pot. Imagine a bet putting one of you
all-in or a bet on the river.
2. There may be dead money in the pot from blinds or earlier bets.
However, assume the bet is "fair," i.e. the pot odds equal the odds against
your winning the bet.
3. And so far, unfortunately, assume the tournament pays only two places with at least three still in it.

The ICM then implies that calling the bet has negative EV.


[/ QUOTE ]

I disagree.
With dead money in the pot it should be possible to construct a situation where calling has a positive ev. Unless the stuff about it being "fair" means this isn't possible, in which case just simplify the problem by assuming 0 dead chips.

[ QUOTE ]
Sketch of the proof. Your total EV is comprised of several terms.

[/ QUOTE ]
Ok.

[ QUOTE ]
It is fairly easy to check that your expected value for finishing first or for
finishing second when anyone except the bettor finishes first is the same
whether or not you call the bet.


[/ QUOTE ]
I don't see how this works, can you explain this further?

[ QUOTE ]

The only remaining term is when the
bettor finishes first and you finish second. Here your EV decreases if you
call the bet. In a nutshell, this is because f(z)=(y+z)(x-z)/(1-y-z) is
concave down (x represents your percentage of the total chips, y the
bettor's, and z your loss, which could be negative).


[/ QUOTE ]
Where does this function come from?
I agree that it is -ev if you might be knocked out, but am I mistaken when I say that it is never -ev to call if you have more chips than the villain?

[ QUOTE ]

This same proof shows that raising on the river if you are certain your
opponent will call is worse than calling.

[/ QUOTE ]

I'll have to think about this.

[Insty]

[ QUOTE ]

In the specified scenario it's never -ev to call when you have more chips than your opponent.
I thought that was interesting.


[/ QUOTE ]

This may be wrong.

[fluorescenthippo]

when you have no FE and a coinflip gives you FE if you double up

[SumZero]

[ QUOTE ]
Assuming Independent chip model is accurate, opponents are equal in ability, and blind sizes are negligible to the problem, is there any situation where a known coinflip should be taken when an opponent pushes all in before you? Also assume that no matter how many players are at the table, showdown will only involve you and the lone pusher.

Put another way, if I know for CERTAIN that my odds of beating an opponent in a showdown are exactly 50/50, and this opponent pushes all in ahead of me, are there any conditions under which I can call profitably.

[/ QUOTE ]

Yes there are. But they all involve meta-game and EV past this hand. If you demonstrate that you will make calls, even coinflip and slightly the worst of it then you may prevent people from stealing/pushing in front of you.

[AleoMagus]

[ QUOTE ]
when you have no FE and a coinflip gives you FE if you double up

[/ QUOTE ]

This is outside the strict mathematical aspects of this single hand, but actually, yeah, good example.

This hits to the heart of some issues that I have with SNG mathematical methods, especially concerning the conversion of chip values to their dollar equivalent. ICM.

On the one hand, we can say 'that's not fair, you are introducing other factors which do not really fit into the model which we are assuming is correct'

Lets face it though, Chips do NOT have a dollar equivalent in tournaments, even if that is what ICM tries to approximate for us.

The reason is simple - You don't get paid until the tourney is over, and so what happens on the next hand really matters. If your chip stack grants you less mathematically describable advantages (like FE) then these things cannot be ignored.

On that same line of thinking, even a very very big stack might have slightly more value than we suppose because it may actually tighten the calling ranges of opponents we have covered. This matters in future hands, and so despite how ICM may approximate our momentary equity, we cannot ignore hands that are yet to come in which our equity will be modified for the better by tighter calling ranges.

Now, this is not original and many people have been saying these kinds of things for a while, but still, I find myself more intrigued with these issues lately. In the past, I have implied that ICM doesn't ignore future hands, because it is based on probability calculations about what those future hands will result in.

Let me make clear, that I am happy with the ICM. I think it does what it is supposed to. It gives us a reasonable and functionally accurate estimate of what our equity in a tourney prize pool is, based on each player's respective chip position. nothing more.

What matters, I think, is that we get better able to recognize where future hands will modify what ICM calculations (when extended to hand analysis) are telling us, because there are factors outside the scope of the model.

Fold equity is a good example. Table image is another good example, and already mentioned in this thread. A well timed, slightly -$EV play when you are big stack can reap long term rewards (not considered in the usual ICM hand analysis) if it tightens up the range of hands that opponents will push into you with. Another example is having a shortstack at the table, and our occasional desire to keep a bubble going if the presence of the shortstack reduces the number of hands that our own all-ins will be called with.

I do not think that we need to modify ICM to account for these things. I have thought in the past about 'weighted' models which slightly overvalued/undervalued stacks, but I think this is a bad approach.

What we need to do is change the extension of our reasoning when we use the results of the model in hand analysis. For example, even if we find a situation where we can push with 74o for a slight gain in equity, this slight gain may not be worth the future ramifications of (possibly) having to show a 74o push.

To make what I mean clear, consider good old fashioned pot odds calculations. Nobody ever felt the need to alter our value of money when implied odds were brought into the picture. We simply realized that the potential for winning future bets meant that those pot odds calculations were incomplete, and that we could go against these calculations when we took this into account. We know we can call with a flush draw even if we aren't immediately getting proper odds, so long as we suspect that we have the potential to win future bets which make the play profitable.

In the same respect, I think ICM only provides us with immediate equity estimates, and can be overriden by future +$EV opportunities in certain situations. Rather than having this equity come from future bets in the hand, it now comes from future hands within the tournament. I think this was once suggested as 'implied equity'.

I am just thinking out loud here, and as I say, I'm sure this isn't news to many of you. This is especially obvious to me now that I am looking over this thread another time. Oh well...

Any other thoughts?

Regards
Brad S

[Insty]

[ QUOTE ]

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Sketch of the proof. Your total EV is comprised of several terms.

[/ QUOTE ]
Ok.

[ QUOTE ]
It is fairly easy to check that your expected value for finishing first or for
finishing second when anyone except the bettor finishes first is the same
whether or not you call the bet.


[/ QUOTE ]
I don't see how this works, can you explain this further?


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I have dusted off my algebra and managed to derive this result.
(basically the +chips and the -chips cancel each other out.)

[ QUOTE ]

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The only remaining term is when the
bettor finishes first and you finish second. Here your EV decreases if you
call the bet. In a nutshell, this is because f(z)=(y+z)(x-z)/(1-y-z) is
concave down (x represents your percentage of the total chips, y the
bettor's, and z your loss, which could be negative).


[/ QUOTE ]
Where does this function come from?
I agree that it is -ev if you might be knocked out, but am I mistaken when I say that it is never -ev to call if you have more chips than the villain?


[/ QUOTE ]

I also managed to derive a similar, although less elegant formula. Which shows the same thing.
Your mathematics is obviously much better than mine.

I assert that for tournaments that pay out more than 2 places further placings may slightly alter the curve they will not fundamentally alter it's shape.
Therefore the result above holds for all (regular) payout structures.
I don't have the mathematical ability to prove this, so I challenge anyone to disprove it. /images/graemlins/smile.gif

In summary:

I now agree with NoSelfControl and believe his proofs and mathematics to be good.

It is always -ev (or at best breakeven) to call.
Even if you have the bigger stack..

Insty
who needs to study some more mathematics.

[Insty]

Good post.

If forced to GUESS, I would agree with Insty that it will always be negative EV when more than two players remain in the tournament and only one player is in the pot. However, it is impossible to use a term-by-term approach to prove it.

On the other hand, it MAY be positive EV to call a fair bet in a multiplayer pot because your being in the pot may affect some players' probabilities of winning more than others.

Interesting problems; I'll think about them some more over the next few days.

[puzzlemoney]

[ QUOTE ]
I am just thinking out loud here, and as I say, I'm sure this isn't news to many of you. This is especially obvious to me now that I am looking over this thread another time. Oh well...

[/ QUOTE ]

Brad, allow me, as somebody who's still coming to grips with ICM in many ways, to ask you to never stop "thinking aloud" on these forums. /images/graemlins/smile.gif

By grouping terms in just the right way, I was able to extend my proof to cover tournaments that pay 3 places. Under the independent chip model, when at least 3 remain in the tournament, it is ALWAYS negative EV in terms of tournament winnings to take part in a bet with one other player for which your chip EV is 0 (or less). Without mathematical typesetting available, I don't think I can convey any details in this forum. (The algebra is sufficiently involved that I used a symbolic manipulation package to double-check my work.)

Hopefully, more to come. If successful, I'll probably eventually write it all up in a format I could send out at some point, although 1st semester calculus and the idea of partial fractions might be a prerequisite for reading it.


[ QUOTE ]
Here's what I know so far.

1. Assume you have the option to call (but not raise) the final bet by the
only other player still in the pot. Imagine a bet putting one of you
all-in or a bet on the river.
2. There may be dead money in the pot from blinds or earlier bets.
However, assume the bet is "fair," i.e. the pot odds equal the odds against
your winning the bet.
3. And so far, unfortunately, assume the tournament pays only two places with at least three still in it.

The ICM then implies that calling the bet has negative EV.

Sketch of the proof. Your total EV is comprised of several terms. It is
fairly easy to check that your expected value for finishing first or for
finishing second when anyone except the bettor finishes first is the same
whether or not you call the bet. The only remaining term is when the
bettor finishes first and you finish second. Here your EV decreases if you
call the bet. In a nutshell, this is because f(z)=(y+z)(x-z)/(1-y-z) is
concave down (x represents your percentage of the total chips, y the
bettor's, and z your loss, which could be negative).

This same proof shows that raising on the river if you are certain your
opponent will call is worse than calling.

[/ QUOTE ]

No because of the declining value of chips in a tournament.

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No because of the declining value of chips in a tournament.

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This is a pretty vague statement. For instance, it's easy to come up with an example where, in a single hand, your chips have increased AND your tournament $ EV per chip has increased, i.e. the average value of your chips has gone up as your stack has increased. (My example is an all-in where you are short-stacked, win the main pot while the big stack busts several others out.) Can you define "value" and "declining value" precisely? (Even better would be a reference that has a proof.)

In the context of the OP, I think I can now show that $EV can only go down in tournaments that pay up to 4 palces. The jump from 3 to 4 wasn't as easy as I had hoped at the time of my earlier posts.