Generalizing the diffusion model for tournament equity

[pzhon]

I think your basic model is ok, but I think you need to include opponents of different skill levels. Otherwise there is really only one opponent with the rest of the chips, and all of the other stacks are only there to determine the size of the pot.

I prefer pot-size model II (based on the average stack) to model I (all-ins are very likely at the start and end of the tournament). Did you get similar results from the two?

The calibration you used to a 40% ROI for 10 players is suspicious. If SNGs were winner-take-all (and had no rake) I think the ROI for a good player would be much higher.

[ QUOTE ]
How does ROI scale with field size given some fixed "edge" of a player relative to his average opponent?

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Another question might be addressed easily by this method: How does changing the initial stack size for one skilled player affect the player's average win?

[schwza]

that's interesting... i was thinking about a related problem the other day. i think the most fruitful way of approaching this problem would be to focus more on collecting lots of data from real online tournaments. then you could approximate a probility distribution of change in chips given the current state of the tournament (blinds, button position, chip stacks, payouts). then, once you had the probability distribution you could run monte carlo simulations to find expected value of being in a particular state of the tourney.

i would start with assuming that players are homogenous and equivalent to the average player in the tournament data set. (i would also start by looking at sit n gos for simplicity).

the main purpose i wanted to do this for was to determine the extent to which you should value survival over value. for example, suppose there are 4 players left, each with 1000 chips and a 50-30-20 payout structure. folded to the SB who pushes. you're sure that you have a 55% chance to win. if there was a decent model that could tell the EV of having 2000 chips against 2 opponents with 1000 each, then it'd be an easy decision.

i don't really know if it's possible to get your hands on a lot of tournament data, and i also don't know how to magically transform a whole bunch of data into a state-dependent probability distribution of changes to your stack (but i think it'd be pretty easy to go from there to finding EV of a given state). anyway, my 2 cents.

[Ian]

Your original approach was creative, but there are much more direct means to model tournament EV IMO. As for the simulation, I don't understand your model (or the results for that matter -- how can n10 = 0.38?). But the relationship between n and EV seems about right and consistent with the financial models I have run. Which also makes me believe your simulation is off a bit because my model accounts not just for the compounding effect of larger fields, but the advantage of using optimal (i.e. higher variance relative to time) strategies as fields grow. If you really want to see some funky stuff, check out the volatility measures as n grows, then back-fill variance to calculate NPV using a reasonable T for tournnaments played -- e.g. try T=20 for example if you are talking WSOP championships. Scary stuff! Makes it harder for me to justify using my 401k for entry fees -- lol.

[tripdad]

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english only, please.

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here, here! Hold'em was invented by a good ole boy like me from the great state o' Texas, not some pure bred Harvard MBA from massychassits! let's keep it real in these here forumns, ya hear?!

cheers!

[Whitey]

Do these guys realise poker is about people not x=y/z to the power of 10 squared: [img]/images/graemlins/tongue.gif[/img] [img]/images/graemlins/tongue.gif[/img] (joking [img]/images/graemlins/grin.gif[/img])

[poboys]

[ QUOTE ]

How does ROI scale with field size given some fixed "edge" of a player relative to his average opponent?


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What a fantastic question! Interresting concept to model.

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The modeling of the pot size is the key component of the model. I have tried two simple models:

I)
u=random[0,1]
p=u*big_stack
p=min(p,small_stack)

II)
p=C*avg_stack
p=min(small_stack,big_stack,p)
C is a free parameter in this model



[/ QUOTE ]

I snipped a bunch of the strawman description for space reasons. It seems to me that you'd have to scope the problem in order to more accurately model a MTT. It seems like each 'part' of the tourny would require a different model (or at least entirely different parameters to the model).

- Early rounds, where each player (or almost all players) have around 30+BB, and generally there are more than one person involved in a pot.

- Bubble-time, where you generally see heads up, pre-flop all in plan

- In the money, where you generally see wildly aggressive play, as people try to build a large stack and shoot to make the final table.

- Final table.

It also seems that your edge may be a decaying function. As fewer and fewer players are left, your edge would decrease. As you mention, as the blinds-to-stack size changes you'd want to change your requirements to go all in.

Obviously this is a complicated problem (understatement, huh)? Wonder if anyone else has tried developing a similar model...? This is a great idea, and I'll definately be keeping my eye on this thread for future updates!

[eastbay]

[ QUOTE ]
[ QUOTE ]
If Laplace's equation gives the probability density associated with a random walk, what equation gives the probability density associated with a semi-random walk that has a bias in one direction? Is it a Poisson equation? How could you formulate a bias in the walk to model a player's edge, and what does the resulting RHS in the Poisson equation look like?

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Eastbay is this your attempt to move past Bozeman as the smartest poster on the forum?

I bet M.B.E is sitting somewhere chomping at the bit [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

I know you're kidding, but...

No. If I thought I was the smartest here wrt this stuff, I would just work in secret and use the results against everybody (no offense, but I do play to win.). My posting here acknowledges that I think there are others that are smarter than me who can improve my results.

eastbay

[eastbay]

[ QUOTE ]
If Laplace's equation gives the probability density associated with a random walk, what equation gives the probability density associated with a semi-random walk that has a bias in one direction? Is it a Poisson equation?

[/ QUOTE ]

FWIW, I am convinced after doing some experiments that the answer to this question is certainly "no," at least for any form of bias that I worked with.

eastbay

[eastbay]

[ QUOTE ]
Your original approach was creative, but there are much more direct means to model tournament EV IMO. As for the simulation, I don't understand your model (or the results for that matter -- how can n10 = 0.38?).


[/ QUOTE ]

Why not?

And more to the point, what are your direct methods?

[ QUOTE ]

But the relationship between n and EV seems about right and consistent with the financial models I have run. Which also makes me believe your simulation is off a bit because my model accounts not just for the compounding effect of larger fields, but the advantage of using optimal (i.e. higher variance relative to time) strategies as fields grow.


[/ QUOTE ]

So you think my model is "off" because the results are consistent with more complicated models? I would wonder if it is not more likely that such details "wash out" in the results and end up not being as important as you might think.

[ QUOTE ]

If you really want to see some funky stuff, check out the volatility measures as n grows, then back-fill variance to calculate NPV using a reasonable T for tournnaments played -- e.g. try T=20 for example if you are talking WSOP championships. Scary stuff!


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I have no significant training in statistics, so I don't follow your lingo here.

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Makes it harder for me to justify using my 401k for entry fees -- lol.

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Yeah, I mean hell, you look at the sample sizes necessary to converge my simulation for a 1000 player field, and you realize that these things are gambles in a way that probably few of us would like to admit.

eastbay

[eastbay]

[ QUOTE ]
I think your basic model is ok, but I think you need to include opponents of different skill levels.


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I did play with that a little bit for the 10-player case. As I noted, I found that putting in a distribution of r_k such that <r_0-r_k> is constant didn't seem to have a very big effect.

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Otherwise there is really only one opponent with the rest of the chips, and all of the other stacks are only there to determine the size of the pot.


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In a sense, yeah, but I'm not sure why you consider that a problem.

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I prefer pot-size model II (based on the average stack) to model I (all-ins are very likely at the start and end of the tournament). Did you get similar results from the two?


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Qualitatively similar, yes.

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The calibration you used to a 40% ROI for 10 players is suspicious. If SNGs were winner-take-all (and had no rake) I think the ROI for a good player would be much higher.


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Well 40% is real world numbers for PP structure of 50/30/20. I wasn't sure how to adjust it for winner-take-all. If you think it should be higher - why?

[ QUOTE ]

[ QUOTE ]
How does ROI scale with field size given some fixed "edge" of a player relative to his average opponent?

[/ QUOTE ]

Another question might be addressed easily by this method: How does changing the initial stack size for one skilled player affect the player's average win?

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Yeah, this is true. That's another dimension to be explored here.

eastbay