Generalizing the diffusion model for tournament equity

[eastbay]

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?

eastbay

[durron597]

I spent some time thinking about it, and I realized that there are too many factors that need to be factored in here to accurately model one's performance in a poker tournament. For example, you have to include the blind levels affecting larger swings in one's stack as time goes on, a players relative skill at various levels in a tourney, the skills relative to the other players as time goes on in a tourney. Also don't forget that the amount of chips you have is discrete, not continuous, so applying calculus concepts probably won't apply here.

I'm trying to think about how to model it as a random walk over a large (understatement of the week [img]/images/graemlins/grin.gif[/img]) Markov chain, where states both have information about blind sizes and stack size (and # of players remaining?)

I think a good model for this might be a good topic for a Ph. D. thesis in Computer Science or Math... You don't happen to be a grad student do you?

[SossMan]

[ QUOTE ]
I spent some time thinking about it, and I realized that there are too many factors that need to be factored in here to accurately model one's performance in a poker tournament. For example, you have to include the blind levels affecting larger swings in one's stack as time goes on, a players relative skill at various levels in a tourney, the skills relative to the other players as time goes on in a tourney. Also don't forget that the amount of chips you have is discrete, not continuous, so applying calculus concepts probably won't apply here.

I'm trying to think about how to model it as a random walk over a large (understatement of the week [img]/images/graemlins/grin.gif[/img]) Markov chain, where states both have information about blind sizes and stack size (and # of players remaining?)

I think a good model for this might be a good topic for a Ph. D. thesis in Computer Science or Math... You don't happen to be a grad student do you?

[/ QUOTE ]

english only, please.

[eastbay]

[ QUOTE ]
I spent some time thinking about it, and I realized that there are too many factors that need to be factored in here to accurately model one's performance in a poker tournament. For example, you have to include the blind levels affecting larger swings in one's stack as time goes on, a players relative skill at various levels in a tourney, the skills relative to the other players as time goes on in a tourney. Also don't forget that the amount of chips you have is discrete, not continuous, so applying calculus concepts probably won't apply here.

I'm trying to think about how to model it as a random walk over a large (understatement of the week [img]/images/graemlins/grin.gif[/img]) Markov chain, where states both have information about blind sizes and stack size (and # of players remaining?)

I think a good model for this might be a good topic for a Ph. D. thesis in Computer Science or Math... You don't happen to be a grad student do you?

[/ QUOTE ]

I have for now abandoned a PDE approach and decided to try a more brute force simulation technique for trying to answer the question:

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

GRRR! I just wrote up a long-assed description of a model I wrote today, and then somehow misclicked and lost the whole thing.

Here's it in a nutshell, for straw man purposes:

N players P_n start with S chips
Each player has a "rating" r which is a small real number
2 players are randomly chosen to play a "hand"
The pot size is modeled in some way for each hand.
The winner of the hand is:

u=random[0,1]
if (u < 0.5 +r_1 -r_2) P_1 wins; otherwise P_2 wins;

That's it. Play continues as players bust until somebody wins.

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

If you put a blind progression into the model, you'd obviously want to make pot size a function of blind size.

I calibrated edge in a crude way by setting up 10-player tournaments, giving one player an edge over a uniform field of r=0, and adjusting r to give an ROI of 40% (a skilled player at 1 table SnGs). This is assuming winner-take-all.

From there you can scale up the field size and see how ROI changes. The results are kind of depressing.

Let me post this before I misclick and lose it. Results to come.

eastbay

PS Once upon a time I was a graduate student. Not any more, thank god.

[eastbay]

N=10
r_k = 0
r_0 = .03

winner-take-all, no fee

50,000 samples at each field size (note that may not be enough for the bigger field sizes)

N vs. ROI
---------
2 0.10
4 0.23
8 0.37
10 0.38
16 0.50
32 0.59
64 0.92
128 1.00
256 1.30
512 1.68

I kind of expected a faster progression in ROI vs. field size. If someone wants to give me a reasonable expression for time-of-play vs. field size, I can convert this into $/hr.

Observation:

Replacing a uniform r=0 field with a random distribution of r_k<0 against my r=0 - with the same mean difference - doesn't change the answers much.

Question:

If I put a reasonable payout structure assumption in here, would that change things radically? I think not, but can't say why convincingly.

eastbay

[woodguy]

[ QUOTE ]
If someone wants to give me a reasonable expression for time-of-play vs. field size, I can convert this into $/hr.

[/ QUOTE ]

Would you want to take into account variability of this expression due to the speed of which the blinds increase vs. starting stack size?

regards,
woodguy

[eastbay]

[ QUOTE ]
[ QUOTE ]
If someone wants to give me a reasonable expression for time-of-play vs. field size, I can convert this into $/hr.

[/ QUOTE ]

Would you want to take into account variability of this expression due to the speed of which the blinds increase vs. starting stack size?

regards,
woodguy

[/ QUOTE ]

Yep. Any such expression would be blind structure specific. But this is all a rough approximation anyway.

eastbay

[woodguy]

So what do you want to use?
My guess is no re-buy to keep it easy.
Which site are you modeling?
I think Party usually equals T1000 vs. 20 min? (a guess, please correct if wrong)
Wanna use that?

regards,
woodguy

[durron597]

This is definitely a much better way to go about doing this than both the PDE way and my way. My way is way overly complicated, but is probably the most exact way to do it; I think your way has the beginnings of a good model. Anyway:

You definitely have to factor "r" into the pot size. If Player P has an advantage over player Q, then P will tend to win larger pots from player Q, whereas player P knows when to fold and player Q will only win small pots. But you have to leave in the remote possibility of AA vs. KK situations, etc. etc. where the less competant player can take a big pot off of a better player. Also, blind sizes are SO important when trying to determine what a pot size is likely to be, so I'm going to try to factor that in too.

So let's try something like this.

III)
k = r_winner - r_loser
BB = current blind size
L = large stack
S = small stack
u = (int (random[0,1] * (L/BB))) + 1
m = random[0,1] ^ (abs(k) * 50)
If (k < 0) m = 1 - m
p = int((m - .5) * (u/2))) + u
p = min(p, S) * BB

For those who aren't CS geeks, int = greatest integer function.
I think that this is still a poor approximation (assuming I haven't made any mistakes, I did it mostly in my head), but a bit better than what we have so far. I multiply k by 50, because I want the exponent there to be something along the lines of 2,3, etc. though much bigger than that will skew the pots way too much. Though I suppose you could consider that to be a "great read" on a very predictable player.

A few things this doesn't take into consideration; multiway pots, pots with dead money, that sort of thing. Also this whole model doesn't take into consideration is tightness vs. aggressiveness... the player who only plays AA/KK/QQ/AK ever is a poor player, but will never lose (nor gain) much money, but still loses in the end due to blinds.

Also, eastbay, feel free to PM me your program, I'll take a look at it.

[La Brujita]

[ 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?

[/ QUOTE ]

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]