Theory again: Let's take a couple of steps back

[AtticusFinch]

Ok, I think I tried to do too much at once last time, so I'm going to take a couple of steps back and look at one small piece of the problem.

I'd like to try to model the rate at which a person of a given skill can be expected to grow his stack over time. I'll make a few assumptions. I you disagree with any of these, I'd love to hear your thoughts.

My assumptions are:

1) The maximum amount of chips you can make at any time is proportional to your present stack size.
2) Your maximum cEV per hand is a function of your relative skill level
3) Your maximum cEV is limited by the sizes of your opponents' stacks
4) (Closely related to 3) Your maximum cEV is limited by the total number of chips in play

I don't claim these are the only factors, but I think they are significant ones. (Blind size is the notable missing one. I'm still trying to figure out how to factor that in as a parameter, so stay tuned.)

These parameters jibe very well with the parameters used in the Verhulst equation, which I mentioned in the prior thread: http://en.wikipedia.org/wiki/Logistic_curve

The Verhulst equation is used in biostatistics to model, for example, reproduction rates of bacteria in envrionments with limited resources. Its assumptions are very similar to those above:

[ QUOTE ]

* 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.


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Call your stack a population, reproduction rate your cEV per hand, and all available resources the total number of chips in play.

Since this model requires relating your stack's (or population's) growth rate to its present size, a differential equation is used:

dS/dh = rS(1 - S/T)

Where S is your stack size, r is your average cEV (measured as a percentage of your stack, not in chips) per hand based on your skill level when you hold an average stack size, and T is the total number of chips in play. (Note that this formula is a lot more elegant than my last one [img]/images/graemlins/wink.gif[/img])

The result is an adjusted estimated cEV that factors in your stack size relative to the field.

If you look at the fuction's behavior, it models things that many of us know intuitively:

1) Your skill level makes less of a difference when your stack is so small that you have to push/fold, as you only get to make one decision per hand.

2) When your stack is deep, you can make maximum use of your skills, as you'll be able to play "real" poker on all streets, BUT

3) #2 only applies if your opponent's stack is also deep enough. If your only opponent is in push/fold mode, then so are you (essentially), no matter how deep your stack is.

The good news about this one is it takes a lot less data to test, as I'm only trying to measure cEV per hand, not win rates per tournament.

To that end, if anyone has a bunch of tourney history I can use to run some tests, I'd be grateful. I haven't played nearly enough MTTs myself to be able to make any reasonable conclusions.