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Advanced Poker Mathematics: Win Rates & Variance

11,401 Views on 3/11/15

More advanced poker mathematics delves into win rates, variances, and focuses on more specific math situations.

The win rate we achieve in poker not only defines our profitability, but also forms the basis for many decisions we make off the table. The win rate can inform our choice of variant, the stakes we should play for, the number of tables we play, and how we manage our bankroll.

There seems to be some debate in parts of the poker community about just how to attach significance to this number, so here we will consider a simple statistical model that address the issues involved.

We will start with an example: Player A determines that he is winning at a rate of 5 bb/100 over a 5000 hand sample. What significance can he attach to this win rate? In particular, can he state that he is a winning player at his stakes, and express a degree of certainty?
Standard statistical techniques can address this question. We are attempting to show that Player A's win rate is not random, that is, the win rate is produced by his skill advantage, not his card advantage. To do this, we can compare Player A to a Break Even Player.
We can visualize a Break Even Player as someone who is playing against clones of himself – thus he is unable to establish any skill advantage against the table. All of the observed variation in the Break Even Player's result will be based on luck. We can thus treat the Break Even Player as if he was flipping a coin; he will have an equal probability of winning or losing any flip. While we expect his win rate to be zero, we also expect some variance.

Our goal is to see if Player A can separate himself, by comparison, from this variance. We should take note of what this statistical model does not tell us. We make no claim as to Player A's 'true' win rate. It may be 1 or 10 bb/100.  We only seek to establish that, based on the observed win rate that he is winner at all. We also can't claim to know how difficult it might be to become a break even player (in fact it is probably quite difficult). And we should always keep in mind that our results are going to be expressed probabilistically as in a likelihood estimate.

Before we can proceed, we must remember that Player A has expressed his result in terms of actual money won. The Break Even Player's deviation from zero is given in terms of number of hands won. To equate these two results we need to know the average amount wagered by the Break Even Player on any given hand. We will refer to this number as the volatility.

The volatility in poker simply tells us the amount, in bb/hand, which a player has at actual risk in any given hand. This is sometimes referred to as the average turnover. Clearly there will be some variation in this number. We can imagine that the value will depend on stack size, how loose or tight the various players might be, as well as many other factors. Nevertheless, it is possible to determine this number experimentally. We can examine a sufficiently large sample of games and arrive at an estimate of this value.

When this is done the value is usually found to be between 2 and 3 bb/hand, centering at 2.5 bb/hand. The observed volatility usually converges to its nominal value quite rapidly; after only a few hundred hands. If we were so inclined, we could treat the volatility itself as a random variable and subject it to an analysis of variance. However, for our purposes, we will accept the value of the average amount wagered on any hand as 3 bb/hand. This way we will err on the side of the maximum volatility that we might reasonably expect.

With this number in hand, we can calculate the expected variation and compare it with Player A's advertized results:

w*n = v * z * .5 * √n

Where w = win rate in bb/100, n = number of hands, v = volatility in bb/hand, and z = standard score.

On the right hand side .5*√n is the standard deviation of the break even player (which is binomially distributed – neglecting a correction for continuity). The Z score can be used to find the cumulative probability of exceeding Player A's profit.
After substituting our numbers in the above equation we see that Player A is in fact at the 2.4 sigma level of confidence, below the 1% criterion for demonstrable significance.

A Note on Alternate Methods

The methodology presented here is intended as a starting point for discussion on the significance of win rates. There is certainly room for debate, particularly concerning the determination of volatility and its use in analysis. If the reader chooses to hunt down other informal published material on the web, he will find in any number of blogs, forum posts, website articles, and other treatments that seriously contradict the conclusions presented here.

Insofar as the author understands these treatments, here is my critique of them.

The typical method proceeds as follows: a sample of hands is subdivided into subgroups of one hundred hands. From each sub grouping a win rate is extracted. The standard deviation of the win rates so collected is determined. This standard deviation (typically between 80 and 120) is used as a basis for simulation. These simulations, rather predictably, suggest that extremely large sample sizes are needed to establish confidence in any given win rate.

This method is seriously flawed. A win rate is an attribute of the individual who possesses it.  The win rate may itself vary for the individual, and it may legitimately be used in a population statistical study, but it is only a summative description of a sample - it is not itself an observable of that sample.

To illustrate this, consider a one hundred hand subdivision that shows a 40 bb/100 win rate. The next one hundred hand sampling shows a -30 bb/100 win rate. There is obviously significant variance, but by the simple expedient of combining the two subgroups the win rate will become (40+ -30) /200 = 5 bb/100.
We can reverse the process too by considering win rates at the lowest 'grain' possible.  At the level of a single individual hand, we might see a player win 3 bb. This would translate into a win rate of 300 bb/100. Riches indeed beyond the dreams of avarice.

All such win rates, so calculated, are a product of a strictly arbitrary subdivision of the sample space.
There is no reason to believe that a one hundred hand unit constitutes any natural length scale. It is true that poker players will generally describe their win rates in terms of bb/100, but this is strictly a matter of convenience and convention.

I believe the source of the confusion may be that the major vendors of poker database software include this number in their stats section as "std deviation in bb per hundred". Perhaps the availability of this number instigates its use. If so, it would be best if it were eliminated in the next release.

If you enjoyed reading this article, take a look at David Nachowitz' coaching video - Am I A Winning Poker Player? - And let us know your thoughts in the comments below.

Or take a look at some other poker mathematics related content!