% Chance to Win a Tournament

I'm being slightly rigorous about this, so it's probably more wordy than it needs to be.

Suppose you are in a tournament with a n-1 players all of equal skill level and equal stakcs. You are not necessarily of equal skill, and your stack isn't necessarily the same as everyone elses. Your chance to win is a%, and everyone elses is b%. (Note that I've imposed no relationship between stack size, skill and chance to win.)

Suppose that you are on the BB and the woman to your left is on the SB this hand and it folds to her. You take an amount of chips off of her this hand such that you now have a chance to win of a'%, all the other equal stacks have a chance of b'%, and the woman has a chance of b'% - x%. Of course, the sum of all the percentages must still be 100%, so a'% + (n-1)b'% - x% = a% + (n-1)b%.

Now, if you make the assumption that your % chance to win is dependent only on your stack, the average stack, and your skill level, then b'% = b%, as neither the stacks or skill levels of all the other stacks at the tournament have changed and the average stack has also remained constant. But, if b'% = b%, then a'% = a% + x%. In other words, if these three factors are the only ones affecting your chance of winning, then your % chance to win increases linearly with stack size. This must be true because you gained the amount that the woman lost, even though your stack could've been much smaller or much larger than hers. Much more significantly, your chance of winning cannot be dependent on skill level under these assumptions. This must be true because you simply gained the amount that the woman lost, even though you could be much more or less skillful than her. To see this another way, imagine repeating this scenario multiple times, taking the limit as your starting stack approaches 0 and the total amount gained after all trials gives you a b% chance to win. Clearly, then, you will trade stacks with the woman and also trade exactly your % chances of winning, without regard to skill level.

In other words, this model must be wrong. We have a lot of empirical data, and basic intuition, to suggest strongly that a very skilled player has a better chance to win a tournament than a horrible player.

Therefore, a player's chance to win a tournament must depend on something more than just her stack, the average stack, and her relative skill level. I think that, once skill is brought into the picture, a player's chance to win must depend also on the individual stacks and skill levels of every player in the tournament.

Discuss.

P.S. This is very far from a discussion on the blinds affecting your chance to win, and I think it would be wise to keep it that way for now. Thus, assume insignificantly sized blinds.

[PrayingMantis]

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if you make the assumption that your % chance to win is dependent only on your stack, the average stack, and your skill level, then b'% = b%, as neither the stacks or skill levels of all the other stacks at the tournament have changed and the average stack has also remained constant.

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I don't see why this is necessarily true, especially if you define "skill" in a more complex sense than say, some constant given that is never changed, but rather a complex relationship between players' abilities at different situations or scenarios.

For instance, think about a simplified situation where you have only 3 players, in which you again won some chips from one of them. Now if you are "better handling a big stack", that actually means that the player who wasn't involved in the hand, as the player who has lost chips to you, won't be able to deal with you as in the situation where you had fewer chips (assuming there are differencies in abilities, otherwise we are back to the equally-skilled models, where p(winning)=(share of chips in play)).

That might mean that the increase in your stack will affect BOTH other players' chances to win, although no apparent factor with regard to the 3rd player was changed (i.e, he has same stack, same "skill", and the average stack is the same). Therefore when chips move from the lady to you, the third player also loses EV (and specifically his chances of winning might decrease).

This becomes very clear when you consider ICM, for instance, even without need to "explain" anything, because you can find spots where your EV (although not your chances of winning, for obvious reasons) is changed because of chips moving from one player to another without you being involved (even without anyone busting, the average stack changes, or any change in "skill", which is the same for everyone anyway, as for this model).

[DonT77]

Exactly. Although you only took chips from the woman, your gain in winning expectancy is not just taken from her, but from her and to a lesser extent all of the other players (assuming you play a bigger stack well and your opponents become more disadvantaged playing against you).

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Exactly. Although you only took chips from the woman, your gain in winning expectancy is not just taken from her, but from her and to a lesser extent all of the other players (assuming you play a bigger stack well and your opponents become more disadvantaged playing against you).

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That's the entire point. I.e. your chance of winning is a function of all other stacks in the tournament.

[DonT77]

If your point is...

"I think that, once skill is brought into the picture, a player's chance to win must depend also on the individual stacks and skill levels of every player in the tournament."

...then I agree, and I'm not sure this is much of a revelation unless I'm missing something.
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If your point is...

"I think that, once skill is brought into the picture, a player's chance to win must depend also on the individual stacks and skill levels of every player in the tournament."

The fact that it is a function of the individual stacks not at your table, and not the average of those stack, I think, is significant. It essentially means that you can't come up with a simple formula to describe your % chance to win.

...then I agree, and I'm not sure this is much of a revelation unless I'm missing something.
/images/graemlins/confused.gif

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