Testing ICM -- some questions for discussion

[the shadow]

I've been rereading some threads on testing ICM, including:

eastbay's empirical equity study,

eastbay's post in ICM: Does it match the results , and

schwza's computer programmers (x-post) .

For those who would like some background on the subject, check out Section 4.1 of my favorite threads list.

It seems to me that the simplest empirical test of a chip modeling hypothesis is whether the equity function -- the relationship between chip value (CEV) and equity ($EV) -- is linear in a heads-up freezeout. Sklansky seems to assert that the relationship is linear (TPFAP p. 151) ("[E]qual players in a symetrical situation must win exactly in proportion to the size of their stacks."). If true, the probability of hero winning equals the hero's chips as a percentage of total chips. If the hero holds 75% of the total chips, the hero's probability of winning is 75%, etc., etc. As I understand, ICM similarly assumes linearity, at least in freezeout format.

eastbay, zephyr, and others have questioned that hypothesis. JNash has proposed a S-Curve Hypothesis . He conjectures that the equity function is convex for small stacks and concave for big stacks, with the inflection point equaling average stack size. JNash speculates that the S-curve becomes more pronounced as blinds increase relative to the average stack. If true, the probability of hero winning would not necessarily equal the hero's chips as a percentage of the chips in play.

As far as I know, no one has tested whether the equity function is linear in a HU freezeout. While many other aspects of chip modeling are both interesting and await empirical analysis, such as incorporating a skill factor, this aspect strikes me as more tractable and interesting. Focusing on 2-person play would avoid many of the computational and data collection problems involved in 3- or more person play.

Even so, the result would still be interesting. As JNash suggests, under the S-curve hypothesis, it would be correct (+$EV) for the small stack to take a perfect coin flip (50/50 CEV), but incorrect (-$EV) for the big stack to accept the same coin flip. To put it differently, it could also be correct (+$EV) for the small stack to take some -CEV gambles.

So, this reasoning leads me to some questions for discussion. First, can anyone else suggest a better (i.e, simpler and still interesting) aspect of chip modeling to test?

Second, if we were to test this aspect empirically, what data should be collected? It seems to me that it would include the following fields:
Data source
Site (Party, Stars, etc.)
Buy-in ($5, $10, $20, $50, $100, etc.)
Tournament no.
Chips for player A
Chips for player B
Winner (1 for A, 0 for B, or some similar convention)
Are there any other fields that should be collected?

Third, what should be the size of the data set? Should n=500, 500,000, 5 million, etc.?

Fourth, related to the second question, what tests should be run to test the data for linearity or non-linearity? I recall several such tests from my econ days, but recall that they dealt with time series. I am not sure of their application to this problem.

Finally, aside from collecting data, are there other ways of testing the hypothesis of a linear equity function in a HU freezeout?

I have some thoughts of my own, but thought it might be best to pose some questions for discussion. I welcome any suggestions and comments.

The Shadow

P.S. If you find it hard to visualize JNash's S-curve hypothesis, take a look at M.C. Escher's print Concave and Convex. A simple S-curve suddenly seem a lot easier. [img]/images/graemlins/smile.gif[/img]

EDIT: Links edited out to fix legibility problem.