confused?? stack-size and EV

[MicroBob]

this is a cross-post from the MTT forum.
i got zero responses over there and i'm fairly curious about this so instead of bumping it over there i decided to try you guys and see if there someone can help me out.


this may have been covered in TPFAP (which i have only skimmed) and/or i might have missed it in GTAOT or elsehwere on the forums (i don't come by the MTT forum too often) so if this question is a repeat i apologize.


there is an aspect of tourney-theory that i find a bit confusing.

shorter stack = more value per chip..correct??
(larger stack = each chip has less value)

but in other parts of these books...it appears that the chips are given the same value for each stack.


for example - 2 players are remaining....both of equal playing ability.
player A has 2k in chips and player B has 1 k in chips.
we are told that player A has 66.7% chance of winning since he holds two-thirds of the chips.

shouldn't player-B's chances be slightly better than 33.3% because each of his chips has a slightly increased value??

these concepts are discussed regarding making deals, whether or not to get out of the way if there are 3 players left and 2 of them go all-in, and i think a variety of other situations.



am i just inappropiately combining two entirely different concepts??

input appreciated.

[PrayingMantis]

TPFAP really deals with the question you're asking in quite a thorough way. I'll try to make it clear in a few simplified examples:

Suppose 2 players get paid in a tourney, (say: 70/30). If there are 3 players left, 2 players with equally big stacks, and you with only one chip left, your $EV will jump very high when the two other stacks are playing against each other, since if one of them busts, you are winning 30% of prize pool, even if it's your last chip. So basically, you can win a lot even with a very short stack. It's obvious then, why your one chip is worth much much more than any specific one chip in any of the the other players' hands, IF it's not a winner-takes-all.

However, if only one player gets paid (100%), then even if one of the big stacks busts (like in the previous scenario), you gain VERY little, since you have practically 0 probability to win anything at all. That's why winner-takes-all tournaments should be played in a very different way than otherwise. Being a short stack and surviving to get into the money, doesn't help you at all : there is no "in-the-money" - only one player wins.

Now, back to the 3 players left in a (70/30) structure. If one of the 3 busts, you are left to play against the other one. Notice that *each* of you has already won 30% of the prize pool, as you're fighting for 1st place, which means you're fighting now for the remaining 40% of the price pool.
Only one of you will win it, therefore: every HU tourney situation (whether its in the end of a 2600 players tourney, or simply a HU match), can be consireded as a winner-takes-all, in which you win *nothing* unless you have *all* the chips at the game. So, here, chip EV is exactly the same as $EV, for winning the last 40% of the prize pool. In other words, in HU there is no meaning to survival, only to CEV (although there is still the princinple of avoiding certain +CEV situations, if you believe you'll get better ones later on, against this specific HU opponent).

I hope this helps.

[MicroBob]

your explanation of 'survival' with the last chip is helpful (although i do remember that part)....especially considering the pay-out structures.



there were some examples in GTAOT that i remembered having questions about....i'll look them up and get back to you if your explanation doesn't seem to fit the bill for me.


let me ask a ridiculous hypothetical question....
it's the first hand of a 1,000 player MTT.
i win a 10-way all-in and increase my stack from 1k to 10k.
every other table folded around to the BB (so there are some players with 1005, some with 995, most with exactly 1k).
does this mean that i am now technically 10x more likely to win the whole thing as any other player??

i went from holding 0.1% of the chips to 1% of them.

[MicroBob]

also...this would seem to contradict an idea that i had previously held that a small stack actually has a slightly greater chance of making it to the money because of many factors including the ability to catch that winning card on their all-in (especially so in a multi-way pot).

your post seems to indicate that the small-stack is only stronger when you are near the bubble...
but what if you have the small stack at the extreme early stages of the tournament? i believe the theory is that the small stack's chances are not nearly as dire as one might initially think. and i believe this was the point that DS and MM were making with regard to the value of individual chips increasing when you have less of them (and vice-versa).


do we just apply the ideas you described regarding 'surviving until you're in the money' even at the early stages and assume that surviving until you reach the money is the reason the single-chip can have more value.
this is unrealistic if you are nowhere near the money imo and shouldn't really be considered a factor.

thus, i think there are other factors at work here then the ones you described.


i have had several tourneys where i have rallied from an insanely small stack to place well in the money or even take the whole thing.
obviously there was some luck involved in these....and i hate making assumptions using the word 'seems' but i'm going to anyway:
it 'seems' like i have had more of these amazing type comebacks then one would think is realistically possible if my chances were REALLY that slim with such a short-stack.



anyway, thanks for reading (and responding...)
i'm still working on my understanding of strength of stack-sizes and chances of winning, etc etc.

[SumZero]

With the 1000 person tournament situation this is an example of CEV != EV. So you have a less than 1% chance of winning the tournament even though you have 1% of the chips. And those people left with 1000 have a better than 0.1% chance of winning. But since there are 991 people left the "less" and "better" are still pretty small. It is only when you get nearer the bubble that those "less" and "better" tend to become very significant.

Assuming that you are equal in abilities to everyone else in the tournament, and that you play large and small stacks equally well, you should win more money in 10 tournaments in which you are (essentially) tied with 900 odd others at 1000 after one person has 10000 chips like you describe than if you got to play 1 tournaments where you started with 10000 chips but still had 990 other opponents left (although the differences are small, and the varrience is big, so you'd want more like 1,000,000 situations like the first against 100,000 like the second).

It feels like this would be something one could simulate, but I'm having trouble figuring out how to simulate the way a bigger stack could bully a smaller stack so it isn't a straight random walk.

[Cosimo]

[ QUOTE ]
it 'seems' like i have had more of these amazing type comebacks then one would think is realistically possible if my chances were REALLY that slim with such a short-stack.

[/ QUOTE ]

As per TPFAP, the +$EV-per-chip factor of a very small stack is very high.

Another factor is skill. Also, AK is stronger if it can see all five board cards; you won't be forced to fold if you miss the flop. All that matters is preflop hand strength, and it's tough for any hand to be too much of a dog, especially if you get to choose which hand that is (out of, say, the next eight hands).

Your chance of taking first place (assuming all players equally skilled) is proportional to your stack size. Sklansky details this in TPFAP. What changes value is prize-money equity of a chip. The larger your stack, the less each additional chip contributes to your equity in the prize pool (assuming multiple spots pay out). It's not that your chances of winning the whole thing isn't proportional to your stack size, because it is.

The easiest way to envision this is to extrapolate from an extreme case. Ten person SNG, $1000 buy in, top three pay out 50/30/20. At the start, your chips are worth $1 each. You move from 1000 chips up to 5000 chips, yet the other nine players are still in the game. Do you have a 100% lock on 1st place? When it gets down to two-handed, those 5000 chips give you a 50% chance of winning the remaining $2000 that you're fighting for, so now the chips are worth 20c each.

[Cosimo]

[ QUOTE ]
It feels like this would be something one could simulate, but I'm having trouble figuring out how to simulate the way a bigger stack could bully a smaller stack so it isn't a straight random walk.

[/ QUOTE ]

I'd start with the assumption that the bully-factor is proportional to the trap-factor of a small stack slowplaying. Unrealistic, but that's what simulations are about. =)

I'll try this tomorrow. I'm most interested in finding out the equity of different stack sizes on the bubble. Probably the easiest puzzle is equity in a 3-person game.

[PrayingMantis]

[ QUOTE ]
let me ask a ridiculous hypothetical question....
it's the first hand of a 1,000 player MTT.
i win a 10-way all-in and increase my stack from 1k to 10k.
every other table folded around to the BB (so there are some players with 1005, some with 995, most with exactly 1k).
does this mean that i am now technically 10x more likely to win the whole thing as any other player??



[/ QUOTE ]

For winning "the whole thing", the answer is yes (assuming all are equally skilled players, and neglecting some more minor details). However, you should notice this fact:

You have a 0.01 chance to win the WHOLE thing, which means you have 0.01 to win *1st place*. If it was a winner-takes-all, you can say you "have" now 1% of the prize pool (in terms of $EV). BUT, when there are more places that get paid, you now have less than 1% of the prize pool in terms of $EV, since winning 1st will not give you 100% of the prize pool, and by definition you cannot win 1st AND another place (or better - you cannot win ALL the places in-the-money). So in case of a structured pay-out, holding X% of the chips in play, isn't the same as having X/100 probability to win 100% of the prize pool, but less. Usually - a lot less. As a matter of fact, you can NEVER have $EV which is more than what 1st place is getting paid. In a structure of (25/15/10/5/3.....), you simply can't win more than 25% of the prize pool, even if in some point you hold 99% of chips, and all others hold 1%. Notice, however, that you are almost guarenteed a 1st place win.

That's why deal-making calculations for several players in a final table can get so complicated, and to get a meaningful answer you should use X or Y model (there are a few), or use simulations, which usually don't take in differences in skill.

[PrayingMantis]

[ QUOTE ]
your post seems to indicate that the small-stack is only stronger when you are near the bubble...
but what if you have the small stack at the extreme early stages of the tournament? i believe the theory is that the small stack's chances are not nearly as dire as one might initially think. and i believe this was the point that DS and MM were making with regard to the value of individual chips increasing when you have less of them (and vice-versa).


[/ QUOTE ]

I think you are talking about 2 different concepts here. The first we already discussed - as a short stack, especially near or in the money, your probability to win any share of the prize pool does not correlate to your actual stack size, since you can win some even if you're merely surviving, in certain circumstances.

The fact that a short stack can have a better shot at winning showdowns, because he can't be pushed-off a pot, and might actually pay less (than bigger stacks) to see the whole board if he's all-in - well, this fact has some important implications on any kind of NL play (ring or tourney), but being a short-stack, per-se, especially in the early stages of a MTT, is definitely a disadvantage. The money is still far enough so that suvivial and bubble considerations are fairly unimportant, and as a short stack you simply have LESS chips, and therefore, smaller $EV. Remember you have to double up a few times, and win a meaningful amount of showdowns, in order to win a significant prize in a multi. Being a short stack in early stages can reduce your chances in a big way, but of course, as long as you're there (a chip and a chair), you're there.

[ QUOTE ]
do we just apply the ideas you described regarding 'surviving until you're in the money' even at the early stages and assume that surviving until you reach the money is the reason the single-chip can have more value

[/ QUOTE ]

Not exactly. The "single chip" idea is not merely in regard to surviving, but also to the structured pay-out (see my other post here). These two reasons for the "single chip relative value" are mingled together, and are dependened on each other.

[ QUOTE ]
i have had several tourneys where i have rallied from an insanely small stack to place well in the money or even take the whole thing.


[/ QUOTE ]

That's not very rare, if you're talking about SNGs. In a way, SNG is a bubble situation right from the start, because you are very close to the money, even if you're 10th. MTT are VERY different in this regard. Notice that you can win some money in an SNG simpy by surviving, and not doing ANYTHING (this is rare, but could happen in lower buy-ins). This will not happen in any normal MTT, unless in some very very bizzare circumstances.

[ QUOTE ]
it 'seems' like i have had more of these amazing type comebacks then one would think is realistically possible if my chances were REALLY that slim with such a short-stack.


[/ QUOTE ]

Again, SNGs and MTTs are very different in this aspect. But of course, if you're lucky enough AND know how to use a short stack, you alway have a chance, especially against weak opposition. For instance, in the last stages of an SNG, winning two showdowns can move you from 4th to 1st. This is not rare.

[MicroBob]

[ QUOTE ]
I think you are talking about 2 different concepts here.

[/ QUOTE ]


i definitely was...thanks.


i have indeed had a couple of these types of comebacks in larger multi's....but i appreciate your point that SNG's practically start in a 'near the bubble' situation.

one of the multi's that i am thinking of wasn't THAT much a multi at all.
it was a 60 player (approx) KOTZ tourney. early on i was the short-stack with around 50 players left.
i was still the short-stack at the 1st break with 45 players left or so.
at the 2nd break i was the chip-leader with 15-20 left i think. i finished 2nd.
obviously there was quite a bit more luck involved than skill.

i've had a handful of MTT's where i was around 95th place with 100 remaining, won my all-in on the blinds...and somehow battled my way to finish in the top 10 or so.

again, i think these are more attributable to luck of course.
but i also think it points to the potential one has as a short-stack which is related to that whole bit about usually not being THAT much of an underdog on your all-in hand....
as well as the general idea that it is better to hang on for as long as possible and go out with a whimper (as opposed to a bang)...i believe both MM and DS mention this idea in GTAOT and TPFAP if i recall correctly.


anyway, those are just some general observations that are only peripherally related.

you guys have helped me straighten out the different ideas that i had previously been incorrectly lumping together.
thanks.

[Cosimo]

No idea if this is useful here, but I ran a simple sim with three people, stack sizes 2x, x, x. The 2x stack indeed won 50% of the time, but also came in second a bit more than the other two (of course). Proportions of 1st, 2nd, and 3rd were:

* 2x stack: 50% 36% 14%
* 1x stack: 25% 32% 43%
* 1x stack: 25% 32% 43%

Maybe latah I'll try more players, but I gotta go kill some terrorists.

[PrayingMantis]

Take a look at this thread:

The old coin flip debate (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=819750&page=6&view=co llapsed&sb=5&o=&fpart=1#819750)

Some of it deals with very similar problems (and don't miss Bozeman's post, at the end of it. Extremely helpful.)