Hm...

So if one could simulate tournaments of finite length with equal skilled players (ignoring position) whose stacks would be on average the same after each hand, one might get quite a decent approximation of real TEQ?!

Did I get you right?

EDIT: If I'm totally misunderstanding, ignore the following.

So I’m gonna present the model I came up with about a year ago since I think it makes sense in conjunction with pzhon’s post about martingales.

Transition model

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Let’s assume N equal skilled players with stacks S1, S2, …, SN are playing a poker tournament with randomized positions and a payout structure (f1, f2, …, fk) [k < N; f1 >= f2 >= … >= fk]. Let’s assume further these players do know their exact real TEQ and chose their shoving and calling ranges always exactly in the real TEQ nash equilibrium.

Real TEQ has to be a martingale. So in their game with randomized positions, on average the expective value of their stacks after playing a hand has to be equal to their stacks before playing the hand (This is true since the stacks are the only difference between these players so the TEQ only depends on the stack distribution in a fixed payout structure).

What happens in the game of these N perfect players when they play a hand is that some players win some chips and some other stacks lose chips (of course the total number of chips is conserved). But on average neither one will have a change in his stack since TEQ is a martingale.

This can be used to design a simple model. For the sake of simplicity, I assume hands to play out the simplest way possible and that is to just assume a “hand” as transition of states where the stack Si gains C chips and one other stack Sj loses C chips.

(…, Si, …, Sj, …) --> (…, Si+C, …, Sj-C, …) (*)

If one chooses the transition-probabilities equal for each i and j (this means the probability of gaining/losing chips in a hand is 1/N for each player), it is guaranteed to not change the stack distribution on average! So the change of state (averaging over all transition cases) does not change the TEQ of any stack. So this simple model kind of creates the situation of “perfect players” playing hands against each other with the restriction of simplifying to only allow transitions like (*). Of course if one stack busts, he can’t gain chips anymore and N is reduced. If one lets them “play” a lot of tournaments (I usually take 100,000) and watches the outcome, one should get an idea of how much equity each stack has. Note that we don’t have any knowledge about their ranges (if we had, we wouldn’t discuss this ^^)! The only thing we use is knowing TEQ is a martingale and that playing a hand in poker means that chips usually change their owners.

So after reading pzhon’s post on martingales, I think this model may yield a decent approximation of the real TEQ in the case of randomized positions and equal skilled players.

If you think that it might be worth a try I could run the model to calculate its TEQ predictions for lots of different stack distributions so that we could try to parameterize the results and then have the outcoming nash ranges calculated by jbpatzer’s program and see if they stand a chance against ICM.

Sry in advance if this turns out to be complete bullshit