Poker Strategy With Ed Miller: Estimating Your Winrate

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Serious poker players are always trying to estimate their winrates. These estimates help inform all sorts of decisions—bankroll decisions, decisions about taking shots at bigger games, decisions about whether to play full-time. It’s important to be able to come up with a reasonable estimate for your winrate.

The problem is that the way most people try to estimate their winrates is fundamentally wrong.

How would you go about it? Let’s say that you’ve played for 200 hours in a $2-$5 cash game and you won $2,600 in that time. What’s a good guess for your winrate?

Before we answer the question, I want to make it clear that when I say “winrate” I’m talking about the forward-looking rate. “How much can you expect to win per hour in the future?” I’m not talking about winrate in the past—calculating that is obviously trivial.

Ok, back to the question. You’ve played 200 hours and won $2,600 in that time. What is your estimate for your forward-looking winrate?

Chances are you’d start by taking $2,600 and dividing by 200 to get $13. Obviously there’s a lot of room for error here — the chance you will have exactly a $13/hour winrate going forward is remote. But you’d probably start at the $13/hour figure and tinker from there.

Maybe you’d tack on another $10/hour to that because you remember all the hands you got sucked out on and figure that probably won’t happen as much in the future.

And maybe you read somewhere that a good winrate is around $30/hour so you round up to that.

Ok, ok, that’s going a bit far. But nearly everyone would start at $13/hour and go from there. Maybe round up a bit for bad luck. How about $15/hour? Or round down a bit to regress to the mean. Maybe $10/hour is the right idea?

But that is completely wrong.

The reason this way of estimating winrate is wrong is that it ignores everything you would have used to help estimate your winrate before you played for 200 hours. What do I mean by this?

Well, let’s say instead of winning $2,600 in your 200 hours, you happened to win $30,000. I know. Pretty amazing to win that much at $2-$5, right?

But not inconceivable. That works out to $150/hour over that time.

So now I ask you to estimate your winrate. Do you start at the $150/hour? Does that seem like a reasonable estimate of how much you’ll win in the future?

Of course not. Why not? Because you have prior knowledge about what’s reasonable to win at $2-$5 and what’s not reasonable, and you know that $150/hour is simply not reasonable. No one wins that much. So you’d estimate a lower number. (Though whatever number you chose would still probably be too high.)

The thing is, even though $13/hour would be a reasonable answer, the process where you just divide dollars won by time and start there simply isn’t the right process.

The correct process uses a technique called Bayesian inference. The first step is to come up with what you would have estimated for your winrate before you played for 200 hours. This is not just a single number estimate—it’s a probability distribution. There’s some chance your future winrate is $13/hour. And some chance its $0/hour. And some chance it’s -$13/hour. And so on.

A reasonable place to start for this estimate would be to guess at what’s possible for all players in the $2-$5 player pool. In other words, you may not know what your winrate is, but you know it’s likely somewhere along the lines of what other players tend to make at $2-$5.

The average winrate at $2-$5 is just the amount of rake per player per hour. So if you guess that you tend to play 10-handed games with $120 in rake per hour from the table, then the average winrate would be -$12/hour.

So your estimate of the winrate distribution should have an average around -$12/hour and then spread out maybe like a bell curve from that amount.

Or maybe you might not want to assume a bell curve. You might just pick a flat distribution centered around -$12/hour. If you did this, you’d be assuming that you have an equal chance to have any winrate from perhaps -$62/hour to $38/hour. Or maybe even -$82/hour to $58/hour. Something like that.

The type of guess you make matters, but the longer you gather data (i.e., the more hours you play and results you rack up), the more your estimates will end up the same no matter what you started with.

The Right Process

The nitty-gritty of the math is way beyond the scope of this article, but if you’re interested please search for the term “Bayesian inference” and watch a few videos about it. But here’s an overview of the right process to estimate a winrate.

First, come up with a prior estimate for winrates based on no results whatsoever. This estimate should include lots of uncertainty — after all there are people who win a lot at this game and people who lose a lot, and even though you may have a suspicion where you fall on the spectrum, it’s not good to narrow it down too far.

Then every time you get a result, you update the estimate for your winrate. Again, the actual math of how you do this is too complex to summarize, but you can read more about it if you’re interested. But the basic idea is that any given result contains relatively little information and therefore your updated estimate for your winrate will change only a bit.

So if you play for 10 hours and win $1,000 then when you use this datapoint to update your estimate, the chance that you’re a $50/hour loser will go down a bit, and the chance that you’re a $50/hour winner will go up a bit. But only a bit for each. This will move the average estimate up from -$12/hour a little bit higher. But only a little bit higher.

And then you move on to the next result. The process keeps going. Have a prior estimate, get a result, update the estimate. And on and on.

This is why it’s wrong to start with the results and adjust from there. You should start with a prior estimate and adjust using the results. Because then you’re giving each individual result the correct weight to get the most accurate possible estimate.

Final Thoughts

Why go through all this? Well, because these ideas go far beyond just estimating a poker winrate. The same process can be used to figure out how good your local sports team is after a few games. Also the same process can give you the best estimate for how likely a medical treatment is to work. And so on.

Learning to think about winrates the right way will help you far beyond the poker table. ♠

Ed’s newest book, The Course: Serious Hold ‘Em Strategy For Smart Players is available now at his website edmillerpoker.com. You can also find original articles and instructional videos by Ed at the training site redchippoker.com.

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