Conjecture and Question
 

Hi Everyone:

I have a conjecture that I believe is true and I have reasons for believing that this is true. But I'm hoping that others can explain the reasons a little better than I can, or perhaps explain why my conjecture is wrong. Here it is:

Suppose you are a great tournament player. Perhaps one of the best. You enter a $10,000 buy-in tournament and when the first hand is dealt, since you're such a great player your expectation is $40,000 even though you only have $10,000 in tournament chips. Now a very unusual first hand takes place and you double up. That is you now have $20,000 in tournament chips. My conjecture is that your expectation does not double even though your chip count has. So instead of having an expectation of $80,000 it may only be $78,000, or $75,000, or some other number less than $80,000, but it will definitely be less than $80,000.

So if my conjecture is correct, I would like to hear good reasons why this is the case. If it's not correct, I want to hear those reasons as well.

This should make for a good discussion and thanks in advance for the help.

Best wishes,
Mason

Re: Conjecture and Question
 

This has been discussed in the past, and I for one, believe this is true. I came up with one example

in this thread

to try to prove why doubling your chips = doubling your expectation can not be true, but the discussion died in that time with no clear answer.

Re: Conjecture and Question
 

This just seems obviously true.

1) Every chip you win in a tournament is less valuable than the chip before it.

If your equity is 40k at the start, that does not even come close to implying that your equity at any point is $4 per chip. I think this is obvious, but if needed I will explain this further.

The 40k equity you start with, that includes play for the whole tournament. For example, maybe you make blind level X 60% of the time. Doubling your chip stack will obviously not double your chances of getting that far.

Also, since you are better than the field, it is safe to assume that you will double up early on more than 50% of the time. If this is something that happens more than 50% of the time, and your resulting equity is 80k, that means your starting equity would have to be more than 40k. This logically doesn't follow since the 40k number is a given, therefore, the 80k number would be an overestimate.

I don't know if I explained myself clearly or not, I'll try again later if needed.

Edit: FWIW, even the 75k figure is an overestimate.

Re: Conjecture and Question
 

This is what Atticus was saying, and also what TPFAP says.. the more chips you have the less each are worth.

I think i agree with this, because i think the closer you get to being knocked out, the more important each chip is.

bleh i'm tired, i'll reply to this again in the morning.

Re: Conjecture and Question
 

The reason I always assumed this to be true (not sure if it is correct) is because when you assume you have a 40k equity at the start of the tournament because you have a certain probability to finish in every position (ie an A% chance to finish 1st, B% to finish 2nd.... X% chance to finish in Xth place) doubling up does not double the probability of each occurrence. Although your % chance to finish in 1st place may be 2A% (it could be more or less than double also), other money finishes will not neccessarily double, and may decrease to make up for your increased likelihood of finishing higher. Not sure if this is correct but thats why I there is a diminishing return of chips. However, as the tournament field increases, I think the gain in $EV increases as well.

Re: Conjecture and Question
 

[ QUOTE ]
Now a very unusual first hand takes place and you double up. That is you now have $20,000 in tournament chips. My conjecture is that your expectation does not double even though your chip count has

[/ QUOTE ]

When I double up early (not sure why you say "very unusual"), my expectation actually MORE than doubles.

Just one of the benefits of not being a math guy.

Re: Conjecture and Question
 

Here's a proof:

Given that you have $40k equity from t10k. Let's say there are 10 people in the tourny for simplicity. Clearly, when you have t100k, you can't have $400k equity because the total equity is only $100k.

If you then apply this argument to all the other stacks at the table, you can see that your equity cannot be linearly related to the equity of your chip stack in a skill-less tournament.

Earlier, I proved that your equity also cannot be simply calculated by your stack, the average stack, and your skill level. In case you care: Go Here.

Edit: I think I proved more than you wanted. ICM calculations already dictate that doubling up shouldn't double your EV. This is easy to see if you look at the extreme case of the bubble. Here, doubling up obviously does not double your odds of simply getting through the bubble (that's almost always more than 50% already), so your equity can't possibly double when you double up. In the early stages of a tournament, you're basically at a much less extreme bubble.

Re: Conjecture and Question
 

I agree that this is fairly irrefutable, I believe it is an implication of Mason's published argument that due to tournament payout structures the more chips one holds, the less each chip is worth, and vis-versa. If we accept that as true (and I think the majority here do), the conjecture necessarily holds by extension.

Given that a player's expectation is contingent on his chip stack and his relative skill to the field the conjecture must hold for the same reason that in mathematics any factor which weighs on an argument of an equation must weigh on the equation's result.

(In this case, the result is of course the expectation, and the arguments of the equation that derives that result are a player's relative skill and his chips, the latter being conditional on their number but not in a straightforward way which by extension means the result is also not correlated in a linear way with the amount of chips).

Or perhaps I'm missing something. I would like to see someone try and tackle the nay side of this argument.

The Coinflip Game!
 

Let's say instead of poker, you're playing the coinflip game. You have a really good coin that wins 60% of the time. This is where your edge comes from.

Your equity is 40k. 60% of the time, you will win your first flip and double up. 40% of the time you will lose your flip and be out.

Since your initial equity was 40k, your equity after the first flip (with undetermined results) will also be 40k. If you lose the flip, your equity is 0.
.4x+.6y = 40,000
x=0
.6y=40,000
y= 66,667

In this scenario, after your first flip, assuming you win, your equity is $66,667

Although this is not poker, the parallels to the given example should be obvious.

Re: The Coinflip Game!
 

I really like this example. I think that atticus is close to an exponential model that may give correlation to skill/chip amount/ chip average.

Re: The Coinflip Game!
 

I just want to point out that my "proof" doesn't even factor in tournament payout considerations. What I mean is, my proof would be more accurate in winner take all tournaments. Having a flatter payout structure would only decrease the solution (66k) and reinforce Mason's conjecture further.

This means that both payout structure and number of players will affect the rate at which each chip gives you equity.

Re: Conjecture and Question
 

1. Your equity has an upper bound of the 1st place prize. Your expectation cannot increase linearily when there is an upper bound. I am too far out of 9th grade math, but this is called an S-curve? Your equity cannot increase to infinity.

2. First place isn't worth the sum of all the chips: Win you win the 1000 man tourney, you will have 10,000,000 chips. However, 1st place is a mere 25% or $2.5M. Even though chips were issued at a 1chip = $1 scale, they don't pay that way.

3. Part of your equity comes from your seat, part of it comes from your chips. The part that comes from your seat does not change at the same rate as your chip stack. Seat eq is based on #of players left, chip eq is based on %of total chips.

4. Part of your equity comes from the opportunity present at the table. If you have 20K chips, but no one else at your table has more than 10K, 10K of your chips have no opportunity attached to them. The shortest stack at the table has the most opportunity equity.

5. You are expected to quadruple your buy-in. While it is uncommon to start with a double up, it is, to some degree expected. You have a 4X edge because you find ways to double up. Your equity does go up (obviously) but because you're expected to increase your chip stack, it cannot double. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.

Re: Conjecture and Question
 

[ QUOTE ]
5. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.

[/ QUOTE ]

This is something that not many of us have thought of because we like to consider ourselves winning poker players, but I think that this may help find some conclusions. Are you saying that somebody with a $4999 expectation can have an expectation over $10000 after doubling, or somebody with a $5001 expectation still will not have an equity over $10000 after doubling?

Re: Conjecture and Question
 

[ QUOTE ]
[ QUOTE ]
5. If you were expected to half your buy in, doubling your chips would have a greater effect? This doesn't sound right.

[/ QUOTE ]

This is something that not many of us have thought of because we like to consider ourselves winning poker players, but I think that this may help find some conclusions. Are you saying that somebody with a $4999 expectation can have an expectation over $10000 after doubling, or somebody with a $5001 expectation still will not have an equity over $10000 after doubling?

[/ QUOTE ]

The amount of skill you have changes the rate of equity change when you double. See my example if you want to try to apply some math to it. Change the 60% figure to 30% or 80% and see how the outcome changes.

Or, if you're too lazy to do the math, someone with max equity (guarenteed first place) will gain zero from a double up.

Re: Conjecture and Question
 

The really simple answer is that our skill edge doesnt multiply along with our chips.

Basically, we calculate EV by (for an average player)

The sum of:
% of total chips * payout for 1st place
% of total chips * payout for 2nd place
.....

= Our Expected Value of playing the tournament


So, something else has to be at play here for a better player.

They calculate their EV by:

% of total chips + SKILL ADJUSTMENT * 1st place payout
% of total chips + SKILL ADJUSTMENT * 2st place payout
..........

So for Mason's example, the player has a skill advantage of
X. Which is just how much more on average they expect to finish in certain spots.

So, when we double our chips, we double the portion of the above calculation that is from % of total chips. If we do not also double the "SKILL ADVANTAGE" Factor, the EV does not double along with it.

Intuitively, this makes sense. The further we progress through a tournament, the more our % of total chips effects the outcome. If we did in fact try to double or "skill adjustment" along with total chips, we would quickly reach an unreasonable EV.

So, it comes down to the decay of the skill adjustment factor. It is mathematically impossible for it to double, but if it were to remain constant, It quicky becomes unimportant in decesion making.

Actually, that doesnt feel very different from how many pros play. The edge they require before risking all of their chips getting smaller and smaller as the field condenses, converging to anything greater than 0, at a point.


That being said, I find it hard to believe that many people have a 4x edge in a large MTT, and 99% of the players on this board are going to use this as an excuse to play weak tight

Re: Conjecture and Question
 

Okay, I'll try the other side since nobody else has, and because it's what I believe. First let me state my conjecture: In the situation Mason described, your expectation probably does not double if your chip count does. However there are times when it can double and more. Therefore, Mason's conjecture that it definitely does not double is incorrect.

Basically, I think that the SKILL ADJUSTMENT factor in A_PLUS's equation can change value when your chips stack does. I don't think anyone will argue that having a bigger stack allows you to make more +EV plays, especially on the bubble.

If the game is such that the gain you get from having a big stack at bubble time (or at other times) is sufficiently large, your expectation can double (or even triple, theoretically) when you double up. This gain may be different depending on tourney structure, or the tendencies of your opponents.


As an extreme example, let's say we're 25 handed, with 20 spots paying. Blinds 100/200.

You have 6000 chips and everyone else's stack is between 3500 and 4500.

Everyone is trying to make the money desperately. So desperately, in fact, that if the pot is raised in front of them by someone who covers them, they will fold any hand but AA. If they cover the raiser, they will play a much more 'standard' game.

Of course this is a huge exaggeration of most real players' tendancies, but think of the massive advantage you gain in having 6000 chips compared to having 3000. This hopefully shows that a much smaller but very real advantage exists when you have a big stack in more realistic scenarios. This advantage may sometimes be enough to counter the effects of the standard equity assumptions most of us have.

Edit: 2 more points.

Your edge doesn't stop when the bubble bursts either, as people are attempting to climb the prize ladder and will still fear you.

If what I'm saying is true, the implications would be strange and counter intuative. In a situation where doubling up would more than double your equity, it would be +EV for two players next to each other to decide to go all in blind the first time it was folded around to them in the blinds. This makes me doubt my conclusion a little bit, but I'd like to hear what others have to say.

Re: Conjecture and Question
 

***Feel free to disregard this post. I know it is rambling, but I believe I have a better chance of learning something here for having engaged the problem.***

Bravo, Shane, and for what it's worth (not much I'm sure)I agree, but in this hypothetical I am... "a great tournament player. Perhaps one of the best."

With this as the given my instincts tell me doubling on the 1st hand certainly couldn't have more than doubled my expectation and it is probably a bit less than double due to my believing that each chip I win is worth less than the one before.

A guesstimate would put my new expectation between 72500 and 77500, but that is just a feeling which I'm sure doesn't help with Mason's question all that much. Really interested in seeing some more of these replies.

-sloth

Re: Conjecture and Question
 

I agree with this. Your expectation less than doubles.

However, in most situations where you have an average to twice average stack from the point where half of the field has been eliminated onward, doubling up results in much more than a doubling of your expected payout, regardless of your skill level.

Re: Conjecture and Question
 

[ QUOTE ]
So if my conjecture is correct, I would like to hear good reasons why this is the case. If it's not correct, I want to hear those reasons as well.

This should make for a good discussion and thanks in advance for the help.

[/ QUOTE ]

Hello Mason,

With respect to the start of the tournament, IMHO, it is intuitively obvious that your conjecture is correct. Typical tournament payouts are more or less linear whereas doubling up is exponential. At some point, the maximum expectation will be reached regardless of the amount of doubling. Stack size will increase at a faster rate than expectation.

With respect to latter stages of the tournament, I don't think this is so much the case. For example, does doubling up further increase expectation once into the money? I would think that it most definitely would (or is that stating the obvious??).

When I was thinking about your conjecture, the immediate analogy that popped into mind was Tiger Woods in golf. After all, he is the greatest tournament player of the current generation, if not all time.

For 2005, he played in 21 PGA events and made 13 top 10s and 6 wins. As an aside note, he also dogged it and missed the cut in 2 events.

So let's say Tiger is roughly 2:1 or 2.5:1 to win the event (as he usually is rated by the bookies here in the UK). He then promptly goes out and eagles the first hole to take a two shot lead on the field. Does that increase his expectation? Maybe a little but not too much. He is after all quite a bit better than even money to finish in the top 10. But it is also very early in the tournament.

Now lets say it is Saturday afternoon, and he has already made the cut (i.e., in the money) and is now tied with 9 other players for the lead. These 10 players together are 4 shots clear of the rest of the field. Now Tiger goes out and eagles the first hole to take a two shot lead over his 9 closest competitors and 6 on the rest of the field. Does that increase his expectation? I would say yes, proably by a lot.

So now back to poker, the greatest tournament player is now already somewhere in the money. Does a double increase his expectation more so than at the beginning of the tournament? Yes, I think intuitively that it would have to. Or maybe this is restating the obvious as well??

Just my own wrong opinion FWIW ...

Cheers,
Spee

Re: Conjecture and Question
 

lots of golf analogies on this forum lately.

Re: Conjecture and Question
 

Consider this extreme situation. Let's say that you miraculously acquire someone's entire starting stack on every hand you play and no one else gains or loses chips (ignore blind and ante money for simplicity). You must knock out X% of people, minus 1 for yourself, to have equal equity as the X% payout for first place if you assume that your equity doubles with your chip stack. This essentially equates to 1st place payout equity if you start with X% of chips in play. Even in such an advantageous position you surely cannot be guaranteed to win the tournament 100% of the time.

This only assumes that you have $10,000 equity in a $10,000 tourney. Suppose you have Y x buyin equity (in your example Y=4), then if the conjecture is true you only have to knock out X/Y - 1 people before you are "guaranteed" to win.

Re: Conjecture and Question
 

One thing I hear a lot here is the whole "decreasing marginal value" of extra chips. But I think there's something that is being missed here:

If you are an excellent player, one of the best, then your ability to take advantage of a very large chiplead should be in your skillset. Players will know that you are less afraid of them than other players would be because if you lose a big pot to them you still have a nice stack.

Thus, I think for the very best players, having a relatively large stack for your table actually INCREASES your EV for the tournament because of the difficulty for you to go broke - example. Say in the very next hand after doubling up, you get allin with AA vs. KK. If you had not doubled up the previous hand, 20% of the time you will be out of the tournament. However, since you've doubled up, 20% of the time you will have an average stack for the tournament and the other 80% you will have an even bigger chip lead.

Thus by having a stack that's much larger than those around you greatly increases your EV because of the greatly decreased risk of gambler's ruin. Suddenly you can push small edges to the fullest because you know that even if you lose your small edge, you are not out.

Does this make sense? Here's a very contrived but alternate scenario. Say in the first few hands of the tournament, you and 8 other people double up. Then by some bizarre coincidence, within the next few hands all of you are moved to the same table. So now you have a table full of 20,000 chip stacks very early in a 10,000 starting stack tournament. In *this* situation I feel that your EV drops as per Mason's conjecture... now you no longer have an advantage over the rest of your table, you have to worry about any one of the stacks busting you out of the tournament in any one hand.

Note also that the initial premise specifies that Hero is one of the best players. I want to make clear that this is key for giving Hero the increased EV - he must be able to take full advantage of his big stack and not squander the extra chips making loose, speculative plays.

Re: Conjecture and Question
 

[ QUOTE ]
Here's a very contrived but alternate scenario. Say in the first few hands of the tournament, you and 8 other people double up. Then by some bizarre coincidence, within the next few hands all of you are moved to the same table. So now you have a table full of 20,000 chip stacks very early in a 10,000 starting stack tournament. In *this* situation I feel that your EV drops as per Mason's conjecture... now you no longer have an advantage over the rest of your table, you have to worry about any one of the stacks busting you out of the tournament in any one hand.

[/ QUOTE ]

IMHO, this is a great point. As others have hinted at in previous posts, and I'm inclined to agree, the most hay is to be made from the medium stacks rather than the short stacks or big stacks.

If you get stuck at a table where you are 2x the rest of the field, but everyone at the table is at least 1.1x you, then this is a tough row to hoe for a couple of reasons. 1) The chances of this table breaking up or people getting pulled from it are unlikely for at least a little while. 2) Players at the other tables will find much easier going as they catch and pass players at your table.

Even if you are unquestionably the best player at the table, your expectation may be reduced to almost nothing. It is not unlike a one-table tournament within a bigger tournament, because the whole group is likely to be saddled with each other for some time, and possibly only one or two players will move forward from there in a healthy enough state to make the money.

Re: Conjecture and Question
 

[ QUOTE ]
lots of golf analogies on this forum lately.

[/ QUOTE ]

Yeah I saw Jason's earlier post, too (liked it). Couldn't help it here. It just seemed like a really strong analogy with respect to the OP.

Re: Conjecture and Question
 

[ QUOTE ]
One thing I hear a lot here is the whole "decreasing marginal value" of extra chips. But I think there's something that is being missed here:

If you are an excellent player, one of the best, then your ability to take advantage of a very large chiplead should be in your skillset. Players will know that you are less afraid of them than other players would be because if you lose a big pot to them you still have a nice stack.

Thus, I think for the very best players, having a relatively large stack for your table actually INCREASES your EV for the tournament because of the difficulty for you to go broke - example. Say in the very next hand after doubling up, you get allin with AA vs. KK. If you had not doubled up the previous hand, 20% of the time you will be out of the tournament. However, since you've doubled up, 20% of the time you will have an average stack for the tournament and the other 80% you will have an even bigger chip lead.

Thus by having a stack that's much larger than those around you greatly increases your EV because of the greatly decreased risk of gambler's ruin. Suddenly you can push small edges to the fullest because you know that even if you lose your small edge, you are not out.

Does this make sense? Here's a very contrived but alternate scenario. Say in the first few hands of the tournament, you and 8 other people double up. Then by some bizarre coincidence, within the next few hands all of you are moved to the same table. So now you have a table full of 20,000 chip stacks very early in a 10,000 starting stack tournament. In *this* situation I feel that your EV drops as per Mason's conjecture... now you no longer have an advantage over the rest of your table, you have to worry about any one of the stacks busting you out of the tournament in any one hand.

Note also that the initial premise specifies that Hero is one of the best players. I want to make clear that this is key for giving Hero the increased EV - he must be able to take full advantage of his big stack and not squander the extra chips making loose, speculative plays.

[/ QUOTE ]

But if you still have a huge skill advantage then wouldn't the deeper stacks favor you? Since you can therefore make all your chips work for you, and not just half your stack?

Re: Conjecture and Question - Invoking the Gigabet Discussions
 

I agree with Durron, in that having a large stack has its advantages (blind stealing, blind protecting, being a bubble bully, continuation betting, 'putting a man to a decision for all his chips', etc.) - so while your % of total chips increases, the "effectiveness of your stack" (rather than your skill level) also increases. This goes back to the ("Gigabet") discussions that sometimes it maybe be +$EV to take a slightly -cEV play because of the relative worth of various stack sizes.

I do think that someday some 2+2 calculus wiz will figure out the cEV/$EV relationship across the multi-dimensional curve that includes (as a minimum) the variables: #players, stack sizes, and payout structure.

Probably the biggest difficulty in solving this problem (aside from different players having different skill levels) is trying to quantify the advantages of a big stack (which will vary from player to player).


To Mason's conjecture-

In the MTTs where I never double-up my $EV is extremely low - quite possibly 0.

It seems that the earlier a player doubles-up the better his chances are of making the final table, and the longer it takes to double-up the more his chances of making the final table diminish as he is fighting the battle of having a less than average stack (and not having the afore-mentioned big stack benefits) and the vulnerability of being taken out by a larger stack for most of the tournament.

So empirically, I think ZJ's coin-flip example has some merit and that a person's $EV after doubling up early may actually be greater than 2x his starting $EV - although I don't have the mathematical wherewithal to prove it.

The Wotmog theory
 

Mason, this is proven by the Wotmog theory which states:

<font color="red"> If your original stack is worth more than fair chip value (because it is in the hands of YOU), then the average value of the other starting stacks must be worth less than fair chip value because they are in the hands of the NON-YOU player. The value of any further accumulated stacks by YOU is a formula involving the NON-YOU stack value, your skill advantage, # of players remaining, and prize pool distribution. </font>

Let’s think of a $10 buy-in 10-player winner-take-all tournament for $100. You are an exceptional player and your expectation is a return of $20.

Everyone has a stack of chips. If you double through to 2 stacks, you might think your expectation becomes $40. Another double through to 4 stacks and it becomes $80. A third double through to 8 stacks and it becomes $160, and you still haven’t won the tournament. But that cannot be correct, since the total prize pool is only $100 (add to that the fact that for a MTT, the maximum prize is significantly less than the total prize pool)

If your original starting stack was worth $20, then the other 9 stacks on the table can be worth only $80, or $8.89 each stack. Wotmog states that when you win the tournament by accumulating all of the chips, you now own all stacks and their original values [(1x$20) + (9x$8.89)=$100] When you double through, you accumulate one of the $8.89 stacks. The value of that added stack to your original stack would be a complex formula involving the components outlined in Wotmog, but cannot be as high as $20 since continuing to add $20 together you will reach $200 at the end, which is twice the prize pool.

The value would be closer to $8.89 than to $20, which sounds like alot less than $20, but even if it were exactly $8.89, that nearly doubles your profit margin from $10 profit to $18.89 profit.

Re: Conjecture and Question
 

this is not provable without data. one side is the TPFAP/ICM chips decrease in value. the other is most associated with gigabet - if i have 2x chips, my chip-generating potential goes way up.

it will vary by person and situation and you can't make a blanket statement one way or the other.

Re: The Wotmog theory
 

Wow there have been a lot of good posts in this thread. I think Eric's example really nails it for me though - nice post.

Re: The Wotmog theory
 

Excellent post.

Re: The Wotmog theory
 

Well, I think you make a good case that doubling-up early in a 10-player, winner-take-all tournament won't double your $EV; but I don't think this line of thinking carries forward to say a 1000-player tournament with escalating, non-linear payouts.

In a 1000-player tournament, if a player repeatedly doubles-up until he has let's say half of the chips in play, then obviously his $EV won't double-up each time - but we (Mason's conjecture) are only concerned about what happens when the player doubles-up the first time.

Let's say there is an $EV multiplier associated with each double-up. Now, according to this:

"Everyone has a stack of chips. If you double through to 2 stacks, you might think your expectation becomes $40. Another double through to 4 stacks and it becomes $80. A third double through to 8 stacks and it becomes $160, and you still haven’t won the tournament. But that cannot be correct, since the total prize pool is only $100 (add to that the fact that for a MTT, the maximum prize is significantly less than the total prize pool)"

each time you double your chips the $EV multiplier should be less each time, However in larger MTTs sometimes (like once in the money where the money is flat but then escalates rapidly) doubling-up more than triples your $EV.

Clearly there is a complex, non-linear relationship between cEV and $EV throughout the course of a large MTT that can not be explained by simple examples given the complexity of the variables involved (including payout structure and the accumulating benefits of a big stack).

Re: The Wotmog theory
 

[ QUOTE ]
Well, I think you make a good case that doubling-up early in a 10-player, winner-take-all tournament won't double your $EV; but I don't think this line of thinking carries forward to say a 1000-player tournament with escalating, non-linear payouts.

[/ QUOTE ]

I disagree - by the same logic, since the $EV of your starting stack is greater, then everyone else's must be worth (slightly) less on average. It works the same way - what was your argument against it? It was unclear to me what you were trying to say.

I don't think anyone is disagreeing that it is a completely different situation (far greater increase in $EV) when you double-up in the money - but we are talking about doubling up on the 1st hand.

Re: The Wotmog theory
 

Couldn't it be argued that when you double up on the first hand you have picked up the chip equity from the eliminated player but his seat equity is split between the remaining players? Therefore your equity cannot have doubled.

Re: The Wotmog theory
 

Who's to say that when you double your stack to 20K on the first hand your $EV isn't 80K+? Maybe you've reduced the value of your opponent's chips more than 40K as a whole. Until somebody shows me the exact cEV/$EV curve over an large MTT with varying chips stacks, varying payout structures, etc. I'm going to go with with Shane (Shaniac) on this one based on empirical results that when I double up early I tend to finish ITM and finish deep at least 2x more often than if I don't double-up early.

Re: The Wotmog theory
 

Doubling up early could be a psychological thing for you too, but we are not addressing that - just the stats.

Haha - I certainly won't be the one generating that data for you, but anyway Eric's example makes it intuitively clear to me...I just wanted to know your logic behind that statement. And now I know.

Re: The Wotmog theory
 

I think there are two competing factors here. First is the fact that chip values diminish as your stack increases. Everyone seems to understand this theory, so no more needs to be said. However, the second is that a great tournament player can play a 'big stack' game now, and that will increase his/her equity. As the great player's stack increases, not only does his equity increase but also his ability to either splash around in more pots and outplay his opponents or push people around and pick up even more chips.

All this being said, a double up on the first hand doesn't really make the player's stack SO big that they can go hog wild with big stack play, so in the example given I'd have to say less than double the starting equity. However, let's say the next hand the player takes someone else's stack. Now he has 30k to the average 10k... Not only has his equity increased, but so to has his leverage. So, another thing to consider besides the player's stack and the average stack is the player's stack compared to the total chips in play. As that ratio increases, the player's leverage within the tournament increases, and thus adds to his equity.

So, in the example given... at a 3 table tournament, his relative equity after doubling up on the first hand is obviously higher than in a 500 person tournament.

Re: The Wotmog theory
 

What if you are Sammy Farha, and you double thru Danny Negreanu, knocking out someone else who is a 2x par competitor and leaving just the fish left at your table?

then clearly its possible to more than double, it just wouldn't be the typical result.

very nice post, wotmog.

-g

Re: Conjecture and Question
 

I've posted a couple of threads recently trying to come up with a model to illustrate this.

The most recent is this: http://forumserver.twoplustwo.com/sh...3452&amp;page=

In a nutshell, my theory is that your expectation is based not just on your stack size, but your ability to grow it over time, which has some limiting factors constraining it.

For example, you can never win more in any one hand than the lower of your stack and your opponent's stack. And you can never win more than all the chips in the tourney.

More concretely, if you have more chips than anyone else at your table, you can't "invest" all of your chips in a good hand, so not all of them are "working." Thus your potential growth rate decreases as you get farther and farther ahead of the field, no matter how good you are.

Thus I think the key to estimating your expectation is finding a weighted expected growth rate based on your relative skill, and the constraints of the tourney format.

To estimate your growth rate, I took a formula from biostatistics that's used to measure growth rates of species in environments with limited resources. The parameters are quite analogous to a poker tourney.

I need data to test it, but I have high hopes for it.

I have a lot more to say about this, but I'm at work, so I'll have to add more later.

Re: Conjecture and Question
 

[ QUOTE ]
this is not provable without data.


[/ QUOTE ]

Agreed.

[ QUOTE ]

one side is the TPFAP/ICM chips decrease in value. the other is most associated with gigabet - if i have 2x chips, my chip-generating potential goes way up.


[/ QUOTE ]

This misstates Gigabet's position. He believes the value of your chips accelerates for a while after you pass the middle of the field, but once you are way past the field, it slows down significantly. My recent threads on this topic were in part an attempt to quantify Gigabet's theory.

[ QUOTE ]

it will vary by person and situation and you can't make a blanket statement one way or the other.

[/ QUOTE ]

Certainly, but you can estimate it based on relative skill level. I think that's what we're attempting here.

Re: The Coinflip Game!
 

Okay, so what if by winning the first coin-flip your advantage becomes .61 instead of .60 and after the second coin flip it becomes .62 and so on. Can we then say how winning the first coin-flip affected your expected outcome after n coin-flips (which will be a factor of the size of the field)?