My present goal is not to convince anyone that I am correct (yet), so this thread has gone off in a direction other than the one I was hoping for.
This is just a first cut at a theory. I'm at the white
board stage, not the white
paper stage. So I haven't been as thorough in my proof and explanation as I would for a formula in which I had some degree of confidence, and on which I had done some testing. Comments about my lack of rigor are not very helpful to me right now. They're sort of like criticising a diagram of a tank because the piece of paper can't fire ordnance. [img]/images/graemlins/wink.gif[/img]
If you simply fundamentally disagree with my non-linearity premise, that's cool. Come back at a later date when I've made some refinements and tested my model on some data, and we can talk then.
At the present, I have more need for people who understand and agree with my intuitive notion, and who can give me ideas about how to refine my model. THEN I can come up with more rigorious arguments in favor of it.
In the meantime, I'll give you a hint at the direction I'm heading in. I found a curve used in biostatistics which is shaped much like my initial idea. It is used to measure rates of population growth. Its premises are as follows:
* the rate of reproduction is proportional to the existing population, all else being equal
* the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the population growth.
If you think about it, these parameters are analogous to your stack in a tourney. Call your stack your "population." The amount of money you can add to your stack at any one time (i.e. population growth) is proportional to its current size, but limited by the total number of chips in play, and the sizes of your opponents' stacks.
Here's a page discussing the logistic curve, with its origins and applications:
http://en.wikipedia.org/wiki/Logistic_curve
Its shape is very similar to that of my original model, but allows for easier parameterization and normalization. Furthermore, and most intertestingly, it allows for easy addition of a parameter for your relative level of skill. Simply adjust your expected growth rate based on how good you think you are.
I'm still working on how to apply it properly, but I think it has significant potential.