14 January 2010

In the last post, Poker Variance, Winrates and Standard Deviations, I discussed the terms we need to understand how our poker variance simulator works.

I'm just going to jump in and start with some examples. Here is a simulation that I ran for a single player (set Num. of trials to run = 1) with a winrate of 10bb/100 and a standard deviation of 100 over a 10 000 hand sample.

Running another simulation with the same input parameters will always produce a different graph:

How these graphs are produced

First the number of hands input is split up into 100 hand blocks, so if we are simulating 5 000 total hands we will be looking at 50 different 100 hand blocks. We then start by choosing a normally distributed random number which is centered about the input winrate and with the given standard deviation. This random number is then considered to be the winnings for that 100 hand block. This process is repeated for each 100 hand block and then added to the results of the previous runs. For example, a simulation for 500 hands might look like this:

The Cumulative value (y-axis) is then plotted vs the trial number (x-axis) to give the graphs output above. Increasing the Num. of trials to run simply increases the number of times the simulation is run for a given input parameter set. The expected line in the plot is just what the graph would look like for a standard deviation of 0 (you can confirm this by setting the standard deviation to 0 and running a simulation).

Limitations

There are a couple of important assumptions made in these simulations including:

a) Game conditions are constant :: the players (and the competitions) mood and skill level never change and hence their expected winrate and standard deviation never change.

b) Your results in a 100 hand block are distributed normally around your expected winrate.

With these limitations in mind we can use the graphs to help us gain some perspective on some questions.

Q :: How long is the long run?

Short answer: Longer than most of us would like to admit.

It is very common new poker players to ask what sort of sample size you need to determine your winrate accurately. Some people will say 10 000 hands is enough, others will say 100 000 hands still others will claim you need a 1 000 000 hand sample to be confident in your winrate. The author of this article on poker variance claims that "100,000 Poker Hands – The effect of chance now all but gone.". I disagree.

The above graph is a simulation of 100 players playing in identical games with identical winrates (6bb/100) and standard deviations (100bb/100). The difference between the luckiest and unluckiest players in this graph is staggering. The luckiest player is running way above expectation and making an impressive 17 bb/100 while a number of players are breaking even or losing. Remember this is purely down to chance. The actual distribution of winrates for this 100 player sample looks like this:

Even after a million hands the difference in results for the players who run good or bad are huge.

So again we have to take these simulations with a grain of salt because of the assumptions, but they definitely help to get an idea of just how crazy variance in poker can be.

Q :: Is my X Buy In downswing normal?

Short answer: If X< 10 absolutely. If X < 30 probably. if X>60 possible but unlikely!

Players often question their play when experiencing a downswing and wonder just how big a downswing they can expect. The variance simulator also analyzes the trials to find what the largest downswing each player went on is and produces the results in a chart that looks like this:

To generate that graph I ran a 100 000 hand sample for 1000 different players (ntrials=1000) who all win at 6bb/100 hands (sd = 100). This is a cumulative histogram of the distribution of downswings from all the runs. The far left (probability of 1) represents the smallest downswing from all the runs (ie. everyone will experience a downswing at least this big). The far right (probability close to zero) represents the largest downswing from all runs.

Out of those particular trials every single one of the players experienced a downswing of ~10bi, 20% of the players underwent a downswing of ~28 bi's and roughly 1-2% of players underwent a downswing of 50 b.i.'s. The worst downswing experienced by a single player is in the neighbourhood of 63 bis. Scary stuff.

Q :: So what does this all mean?

Short answer: that is really up to you to decide.

It is important to remember that these simulations do not represent reality 100% accurately. They are merely one tool to help you come to grips with the fact that variance happens and you should be prepared for it. It is important to note that the truly frightening runs highlighted above are very rare and the majority of us would never experience a 60 buy in downswing under the assumptions outlined above.

Don't forget that the easiest way to reduce your variance is to sign up for a Rakeback Deal at an online cardroom and add some guaranteed monthly income to your winrate!