Exploring how marginal chip value changes with stack size

[bbartlog]

Assuming that ... player r's skills are equal to the skills of the field, or are superior, we would ordinarily expect the following to be true at the start of the tournament: F_r >= Z/n

I don't think this is true (and that has a lot to do with why I don't play tournaments, generally). Because the tournament payout schedule is quite top-heavy, you need to be on a par with the best of the field in order to have positive expectation. Even if 80% of the field is dead money and you're around the 90% mark in overall skill, you might have negative expectation - though this does depend on just how great an advantage the top players have over you.

However, I don't think this actually affects your other observations.

[WRX]

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Assuming that ... player r's skills are equal to the skills of the field, or are superior, we would ordinarily expect the following to be true at the start of the tournament: F_r >= Z/n


I don't think this is true (and that has a lot to do with why I don't play tournaments, generally). Because the tournament payout schedule is quite top-heavy, you need to be on a par with the best of the field in order to have positive expectation. Even if 80% of the field is dead money and you're around the 90% mark in overall skill, you might have negative expectation - though this does depend on just how great an advantage the top players have over you.

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This is a thought-provoking and important observation or theory. Whether or not it holds true would seem to depend a great deal on the makeup of the field in an individual tournament. If true, it has a number of implications. If a large majority of the field--80% or maybe a lot more, in your view--has a negative expectation going into a tournament, it implies that the top players have quite a large positive expectation. In a zero-sum game, a lot of small losers add up to a few big winners.

How one would test this theory is hard to say. If one could get a large enough unbiased sample of players to cooperate in reporting all their tournament results, one could certainly draw some conclusions about the distribution of wins and losses, which should translate into the distribution of starting EV. But it's hard to get poker players to give their real identities, much less be diligent enough to report results consistently. It seems it would take an expert statistician and a big effort to put together a valid study.

Now, if most players in a tournament have a negative starting expectation, it is conceivable that many of them will have increasing marginal chip value across a broad range of stack sizes, for reasons David Sklansky has alluded to from time to time. Basically, they are likely to be camped out on a curve that is far below what a random model like the ICM would imply. Their curve may be "convex" across a broad range, meaning that "coin flips" are positive $EV for them. The underlying reason for this would be that they would be so deficient in the play of hands in most situations that they would not have many +cEV situations available to them, so that their best shot at finishing in the money would be to get into as many neutral or even slightly negative cEV situations as possible.

I do have a problem with the suggestion that this leads to any useful tournament advice for inferior players. The problem is, if you are poor at playing poker hands, and thus poor at creating +cEV situations for yourself, it is quite likely that you are also poor at recognizing "coin flip" situations when they come up. The very lack of experience and judgment that makes it hard for you to create edges also makes it hard for you to neutralize opponents' edges by gravitating toward coin flips. I suppose you could have an expert sit next to you, and advise you about when these opportunities come up--but that kind of expert help would probably be better directed to making you a better all-around poker player.

If it is true that in a given tournament, most players are negative expectation, and a few players are hugely positive expectation, it implies, for those few experts, that the range across which a chip utility effect could give them increasing marginal chip value is quite limited. The reasons I say this are those that I gave at the start of this thread. What it seems like it boils down to is this--if you start with a huge advantage over your opponents in the play of stacks of a variety of sizes, there is less room for increasing that edge by getting into deeper-stacked play.

Any more thoughts?

[bbartlog]

Well, there are a couple of other simplifications in the model that distort the conclusions, but I don't know how to fix them (at least formally). One problem is that the model treats someone's advantage over the field as fixed over time; in reality, the skill level of the players will increase with time as the weaker players tend to be eliminated more. This implies increased chip value early on, since more chips are needed to take full advantage of certain situations that will be less likely to occur later.
As regards inferior players recognizing coin-flip situations, I think the problem for them is more 'meta' so to speak, in that the vast majority of them don't realize they are inferior. Further, most inferior players won't understand the theory outlined above, though from observation it does seem that most people instinctively understand that if they're playing against someone like Daniel Negreanu their best chance is to gambol it up and go all-in even if the odds might not be the best.
Anyway, if we assume that most players overestimate their skill in a systematic way *and* that they understand the 'coin flip aversion' results of the above model on some level, it has further (perverse) implications for optimal play. (You can obtain the same results if you assume simple irrational loss aversion on the part of most players). The conclusion is that if you can threaten opponents with elimination of their entire stack, you can force them to back down from coinflip type situations (even ones where they might have a slight advantage). This means that you need a big stack yourself. If your stack is small compared to the one you are threatening the curve is too flat (not convex enough locally) to encourage folds in marginal situations. This once again suggests increasing marginal chip utility if you're a shorter stack, with decreasing utility if you're already big enough to threaten others (but the stack sizes keep increasing...). It's also consistent with the real-world observations of my brother, who plays professionally.