MTT -- math point of view

Question

Preface

First of all I want to say, that I am not familiar with poker games other than No Limit Texas Hold'em. However, my theory should be extended to a wide class of MTT games. The information provided in my scheme is quite doubtful, i.e. it does not provide any strategies like "how to win in the MTT" or "how to beat some hand", although it could be helpful to make better decisions in marginal situations. Furthermore, I do not even know if someone has already tried something similar, because my experience in poker is poor. If anything I have written is already well known, please provide me a link. I do not have a certain question I just post my ideas (which I am going to update over time) and hope to get a feedback.

P.S. I will deal with some basic math, but I did not find an integrated LaTeX editor here. That is I will try to write as clear as possible.

Main part

Suppose we have a MTT with P0 entries and S0 starting chips. Define two functions: A(t), B(t) -- values of ante and big blind at the time t. These functions are known and can be viewed at the tournament's structure page. Suppose, P(t, r.p) = P0 * exp(-r.p * t) is a number of active players in the tournament at the moment t with an outting rate r.p>0, which is specific for a certain tournament and could be estimated beforehand. Let H(t, r.h) = r.h * t be a number of hands we have seen up to t with a handling rate r.h>0, which is specific for a certain table and could be estimated online. Now we are interested in a function S(t) -- a number of chips in our stack. In order to obtain this function in a closed form, one have to solve a differential equation:

dS = dX - [A + 1.5 * B / p.t] * r.h * dt,

where p.t is an expected number of players at your table (which depends on P) and X is a pure jump stochastic process depending on both t and S, which describes your active moves in game. I am thinking about dX part for now, however throwing it out still may give us useful information. In particular, one can estimate his place in the tournament if he starts folding each hand from some moment (using P and solving differential equation with dX=0 and suitable boundary conditions).

I hope someone find this thoughts helpful. Feel free to ask any questions, any feedback will help me. Thanks in advance!

Answer 1

You May Find These below concepts useful:

What you should do with your stack with your stack according to blinds + antes: "M Theory" https://en.wikipedia.org/wiki/M-ratio

The Strength of your stack in relation to the average stack - https://en.wikipedia.org/wiki/Q-ratio

Answer 2

This is mere mental masturbation my friend. I'm down voting this non-sense. Smoke less pot.

Yes, this has already been done: See Dan Harrington's concepts of Q and R. Which are MUCH better predictors than what you are attempting to do. You ignoring specific strategies, which is fine because you say you're doing that. Harrington considers the elimination rate AND gives strategy.

You're calculations vis-a-vi the blind structure and antes are irrelevant, since you also have a "outing rate". The outing rate can only be shown by having a given data point. In other words, to know the outing rate [because you don't know their strategy], you simply have to watch a lot of tournaments and physically COUNT the rate. If everyone has an aggressive strategy the outrate is high, if everyone folds every hand, the outing rate is low.

OK, so assume you do that; count the outrate, then your formulation is redundant. In other words, you seem to say that A(t), B(t) -- is somehow important but we must also be GIVEN the "outing rate", making your formulation useless. Although the concept of an "outing rate" might be useful in a general solution.

Harrington not only considers the outing rate, but gives solid strategic advice as to how to play based on where you are in the funnel. i.e. with a Q < 5 you should open shove any ace - Harrington.

This is a svengali.

You seam to be trying to find a general solution based on the elimination rate. The rate is irrelevant. Mason Malmuth has already shown that if you eliminate strategic concepts, by definition you're equity is merely the chip count. If you're first in chips, you're winning. 2nd in chips, you're 2nd place. If the tourney take ten minutes or ten years, if you don't have any skill deferential [because you're not considering strategy], you have nothing else to base anything on.

Don't feel bad though, I know lots of guys in Vegas with a "system". :)

Answer 3

As I read the question, you are trying to find the expected number of chips in a specific stack? This is not answerable, we may however estimate a distribution for the number of chips over all stacks. We know we begin with P0*S0 chips, and that at any given t, there are this number of chips still in play. So you are asking for the distribution of these chips.

Which I don't know, but as P is exponential, I would guess we are looking at an exponential distribution.