Bankroll Management in Limit Poker and Fractional Brownian Motion

[jason1990]

Consider a typical "winning" small stakes online limit holdem player. Let X(t) denote his bankroll in BBs after t hands. If we assume that X is a Brownian motion with drift, and compute (under that assumption) risk of ruin, confidence intervals for winrate, and many other statistical quantities of interest, then we are led to all of the classical formulas that we know and love. This makes sense, of course. Brownian motion is a process whose increments are stationary, normally distributed, and independent, which are all common assumption made about the results of poker sessions.

Looking at my own graphs, though, I seem to have more swings than I ought to have if that assumption were reasonable. Granted, I am only making an "eyeball" assessment on the number and size of swings. But it seems reasonable to believe that the increments in my bankroll graph are not independent. Things such as my own personal reaction to swings as well as table image can cause correlation. If my opponents use PokerTracker or player notes, then the concept of table image can extend well beyond a single session.

A process with Gaussian increments which are positively correlated, and which become less correlated the farther apart they are, is fractional Brownian with Hurst parameter H>1/2. (See http://en.wikipedia.org/wiki/Fractional_Brownian_motion.)

My questions are these: Suppose I assume X is a fractional Brownian motion with drift. How do the classical formulas change? What is the risk of ruin formula? How about confidence intervals? In particular, if B is fractional Brownian motion, then E|B(t)|^2=t^{2H}. In other words, the standard deviation increases like t^H, which is a faster increase than Brownian motion. This would seem to indicate that confidence intervals are less reliable under this model. Any thoughts?

[pzhon]

This is too interesting not to get a response. I'm not sure that fractional Brownian motion is a better model for your observations, but it should be worth analyzing anyway.

Is there a discrete model for fractional Brownian motion, or a way to obtain one from a standard Brownian motion?

[AaronBrown]

Fractional brownian motion is almost impossible to work with empirically. That is, you can specify a model and solve it, but it's not practical to look at real data and fit it back to a model. Tiny, high frequency variations will dominate your estimation. For one specific example, the periodic blind posting and position changes would defeat your fitting.

A much more robust model is to assume the drift rate follows a random walk. Even this is not easy to fit with any reasonable amount of data, but at least it tends to calibrate to the large swings, which is what you want to explain, rather than microstructure.

[BillC]

As a practical matter, why don't you just increase your variance to account for higher swings? This is the main remedy or easy out to the applying diffusion models.

After all, it is idle speculation that you increments are not independent, right? And there are dependence factors that might negate each other e.g. table image vs. one's emotional response (or maybe these reinforce each other)

Otoh, as pzhon says, the topic is interesting to us math types.

[jason1990]

[ QUOTE ]
Is there a discrete model for fractional Brownian motion, or a way to obtain one from a standard Brownian motion?

[/ QUOTE ]
I don't know of any "canonical" discrete process which approximates FBM. But FBM can be written as

B_H(t) = \int_0^t{K_H(t,s)dB(s)},

where K_H is a certain kernel and B is a standard BM. I guess you could just approximate the BM in the integral by a random walk to get a discrete approximation to FBM.

[jason1990]

[ QUOTE ]
Fractional brownian motion is almost impossible to work with empirically. That is, you can specify a model and solve it, but it's not practical to look at real data and fit it back to a model. Tiny, high frequency variations will dominate your estimation.

[/ QUOTE ]
Why do you think this? People use Brownian motion to model many things and they fit specific data to the parameters of their models. Why would it be more difficult to work with FBM?

In particular, it seems clear that the "smoother" a process is, the easier it is to work with. For FBM, if the Hurst parameter is greater than 1/2, then the process is "smoother" than BM (in the sense of Holder continuity).

[ QUOTE ]
A much more robust model is to assume the drift rate follows a random walk.

[/ QUOTE ]
Could you elaborate on the model you have in mind?

[jason1990]

[ QUOTE ]
As a practical matter, why don't you just increase your variance to account for higher swings?...After all, it is idle speculation that you increments are not independent, right?

[/ QUOTE ]
I don't think I understand what you mean here. With a large enough database, I can estimate my variance to within a very small margin. So it is a given and I cannot change it. If the observed number and size of swings is not consistent with the observed variance, then some basic assumption about the model is wrong. Yes, it is speculation that the flawed assumption would be independence. But at the moment, I cannot see any more likely candidate.