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.