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Thanks for the remarks, CityFan and everyone else.

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I don't think my function F has any mystical significance, except that I wrote it in response to the challenge that "chip value is not a meaningful concept".

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Nothing mystical about it, but useful. The function assumes one algorithm for adding chips to x_r, which is removal in equal parts from all other stacks. It strikes me that certain results could probably be extended to an even more general function based on an arbitrary algorithm allowing for removal from other stacks in any other manner. However, I doubt it's worth a big effort to try to do so at this juncture.

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Arguing about the properties of F is precisely what Sklansky, Snyder and people on here have been doing to death for the past x weeks.

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Which, admittedly, is what got me started. It's an effort to settle or at least narrow some of these debates, and if it luckily produces other results, too, great.

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I think though, unless we're going to do some serious maths, bringing all kinds of multivariate functions into play is probably not going to advance any debate. We're merely obscuring the wood by identifying a few trees.

So, sorry, I'm not going to be drawn into any great discussion about this F, unless there are questions that really need the maths.

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Well, analyzing a bunch of multivariate functions would be way, way beyond my capabilities. I wouldn't ask you get into that, and I'm dubious that it would be practical. Possibly one could draw out some general results as to the effect of changes in opponents' stack sizes while your own held steady, but I don't know.

My goals in looking at this problem have been quite modest, but I consider the following results significant, if they stand up to scrutiny:

o First, I wanted a rigorous proof that for an average or superior player, marginal chip value must begin to decline at some point, as the player's stack increases and other tournament conditions hold steady. A lot of people consider this intuitively obvious. However, some have disagreed. (And what is intuitively obvious is sometimes wrong!) For that reason, I think that a proof is worthwhile. Anyway, it's always best to have one's theories built on a rock-solid foundation. The demonstration of this point probably could have been put into fewer words, but stating the problem as one of solving your function lays the groundwork for other observations. If anyone here can do a better job of a formal proof, that would be great.

o For an average or superior player, marginal chip value must begin to decline before the point at which the player has built a stack that bears the same ratio to all chips in play as first prize bears to the total prize pool.

o The greater the player's initial $EV, which is to say the greater his initial advantage over the field, the sooner his marginal chip value must begin to decline.

o While these considerations apparently limit the range across which a "chip utility" effect could result in increasing net marginal chip value, they leave open the possibility that this could hold true for a substantial increase above the initial stack in a tournament with a sizeable field.

o The spreadsheet solving the specialized, flat-opposing-stack case of the ICM may be useful for studying other problems. With the program I'm using, it's capable of modeling tournaments with up to 13 payout places, and an essentially unlimited number of active players. It could probably be rewritten to handle more prizes, but it would be a bear. I haven't previously seen ICM calculators that will handle anything but final table problems. It's hard to come up with anything truly original--probably someone, somewhere has done this before--but I can't find it.

Some of these observations depend on assuming that once the curve goes convex, it stays convex. Of course it's conceivable that the curve could follow a wavy form, going convex, then concave, then convex again, etc. However, it is very hard to imagine that such an odd pattern could result from a chip utility effect. That is, if having an increasingly large stack gave one more profitable opportunities to use one's chips, across some range of stack sizes, resulting in increasing marginal chip value, one would expect that effect to continue and maybe peter out at some stack size, but not to stop, and then restart later with increased vigor.

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I like the observation that for an expert player F should always be ABOVE the curve you've plotted which assumes equal skill - though that might not be true if that player simply couldn't be bothered grinding with a short stack and therefore played poorly.

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In that particular game, the so-called expert wouldn't be a superior player at all. Maybe Chip Reese, Doyle Brunson, and Daniel Negreanu have days they can't be bothered to play good poker--if so, bring'em on!