Four Way Tournament finish probability

[Bozeman]

First, if this subject is at all interesting to you, you should look at tourney finish probability if you haven't seen it.

I have made another (small, slow) step in my quest to understand how tournament finish place correlates with chips.

I have been using the diffusion model, which I think gives proper probabilities of each finish for a poker tournament between equal opponents. The solution method I have found most useful is successive finite element approximations of the result over a triangular grid in the (n-1) dimensional triangle with appropriate boundary conditions. For the four player model, this is a gridded tetrahedron. For 4th place the boundary conditions are 1 on one face and 0 on the others, while for other places the faces are the results for the 3 player problem (solved previously by myself with this technique, and done exactly by Tom Ferguson ("God?")). For the four player problem, 4 hours of computation yielded results valid to 5 significant figures over a grid with 100 points in each dimension. Results between these can be linearly interpolated to ~4 sig. fig.s. Some small difference from the continuum result occurs within ~1-2 points of the faces. For our purposes it is quite tolerable, since finite (and usually large) stakes are used for poker tournaments. In addition, position considerations (ignored here) will swamp these errors.

I don't think I'll solve the 5 player problem, the numerics get long, and I had enough envisioning the 3 dimensional interpolation (tertrahedra do not cover three-space, you need octahedra as well).

However, at this point I have enough information for a couple of projects: $EV on the bubble in a SnG, approximate errors of approximate models.

Estimating the errors of other models is quite useful because the other models are not very computationally difficult, even for much larger numbers of players. If someone could come up with an extension of the correct results (or even a better approximate model) for ~10 players, this would be very useful.

First, note that all models give or use the basic result that probability of 1st is proportional to number of chips. So errors are only present in finish places below 1st.

The best of the approximate models is what I have called the Malmuth model (apparently presented in GTAOT) in the past, but because this model predates Mason, I will call it the independent chip (IC) model. It gives the results you would predict if a stack (n chips) operated like independent one chip stacks (n of them), with the player's result being that of his best finishing chip.

The IC model favors small (less than average) stacks. How big are these errors, and when are they most important?

Here I show the results for the diffusion and IC models for three equal stacks and one other stack as a function of the size of the unique stack.

Here I plotted the difference between the exact result and the IC approximation:

Here are the results for the situation where there are two stack sizes, with 2 players having each.

Finally, I looked at the results for stacks of .3T,.2T,xT,(.5-x)T, for the 3T and xT stacks.

So, the biggest differences occur for small stacks' (5-10% of the chips) probability of taking last, and are no bigger than 6%. However, real money will be linear in these probabilities and the pre-factor (amount of money for that place) will be largest for 2nd place. Thus, these errors are not very large, and will usually decrease with number of players. The following graph shows that for standard SnG payouts (50%,30%,20%), errors are less than 1.5%, or 1/20 of second place money.


On another note, would anyone be interested in my program to calculate these 4 way probabilities? Would someone be interested in converting it (or helping me convert it) to a graphically interfaced executable?

Thanks,
Craig

[karlson]

Craig -

Looks like really fun stuff. I'm curious, do you have the curves for the IC model for more than 4 players? It seems that that model should not be particularly computationally intensive.

I'd play around with your program if you share it, but I'm probably not a good person to ask to make a GUI.

[Bozeman]

Yeah, I've done it up to 20.

Craig

[eastbay]

This looks intriguing, but I'm a little slow picking up on this. Can you write out the 4-player diffusion equation you're solving?

(If you're looking for performance improvement on a diffusion problem, multigrid is the way to go. You can get orders of magnitude in work reduction using multigrid.)

eastbay

[Stagemusic]

OK, I will now quit playing poker. My mind went numb while reading this and I no longer have any desire to compete, knowing that because I can't figure out A. What you did or B. Why this is important to drawing to a third J on the turn, it must be the cause as to why I do not win every tournament I play in. [img]/images/graemlins/tongue.gif[/img] I will just have to content myself to being able to figure out pot odds and such on my trusty abacus.

[eastbay]

[ QUOTE ]
First, if this subject is at all interesting to you, you should look at tourney finish probability if you haven't seen it.

I have made another (small, slow) step in my quest to understand how tournament finish place correlates with chips.

I have been using the diffusion model, which I think gives proper probabilities of each finish for a poker tournament between equal opponents. The solution method I have found most useful is successive finite element approximations of the result over a triangular grid in the (n-1) dimensional triangle with appropriate boundary conditions. For the four player model, this is a gridded tetrahedron. For 4th place the boundary conditions are 1 on one face and 0 on the others, while for other places the faces are the results for the 3 player problem (solved previously by myself with this technique, and done exactly by Tom Ferguson ("God?")). For the four player problem, 4 hours of computation yielded results valid to 5 significant figures over a grid with 100 points in each dimension. Results between these can be linearly interpolated to ~4 sig. fig.s. Some small difference from the continuum result occurs within ~1-2 points of the faces. For our purposes it is quite tolerable, since finite (and usually large) stakes are used for poker tournaments. In addition, position considerations (ignored here) will swamp these errors.

I don't think I'll solve the 5 player problem, the numerics get long, and I had enough envisioning the 3 dimensional interpolation (tertrahedra do not cover three-space, you need octahedra as well).

However, at this point I have enough information for a couple of projects: $EV on the bubble in a SnG, approximate errors of approximate models.

Estimating the errors of other models is quite useful because the other models are not very computationally difficult, even for much larger numbers of players. If someone could come up with an extension of the correct results (or even a better approximate model) for ~10 players, this would be very useful.

First, note that all models give or use the basic result that probability of 1st is proportional to number of chips. So errors are only present in finish places below 1st.

The best of the approximate models is what I have called the Malmuth model (apparently presented in GTAOT) in the past, but because this model predates Mason, I will call it the independent chip (IC) model. It gives the results you would predict if a stack (n chips) operated like independent one chip stacks (n of them), with the player's result being that of his best finishing chip.

The IC model favors small (less than average) stacks. How big are these errors, and when are they most important?

Here I show the results for the diffusion and IC models for three equal stacks and one other stack as a function of the size of the unique stack.



[/ QUOTE ]

What is the y-axis on this plot (probability of win?)? Why are there three curves (and not four, one for each stack)? Shouldn't the results be the same for each of the three equal stacks? It seems like the problem would have to be symmetric with respect to the three equal stacks.

eastbay

[ravenight]

The y-axis is probability of achieving that place, the x-axis is the size of the unique stack in terms of total chips, and the 3 different curves represent the probabilities of 2nd, 3rd and 4th place, respectively. 1st place is not plotted, since it is assumed to be a straight line from 0,0 to 1,1

[Bozeman]

I have put the source code here:
4Source

lit4.c: looksup up the answer in the lookup tables and interpolates for user specified data (input separated by returns)

it43.txt and it44.txt: lookup tables required by lit4.c (these are quite big, especially since I used text)

it44.c: code for generating it44.txt

ICmethod.c: 16 way (not very fast) implementation of independent chip method

WARNING: NO COMMENTS ARE INCLUDED (I am a terrible programmer)

lit4 ought to compile and work as is, if you have it43.txt and it44.txt in the right directory