No-limit Hold'em & coin flip

[maltz]

This is a rather long article of my dig into the analogy between NL hold'em & coin flip. I hope you enjoy reading it. Comments & corrections are most welcome.

***

I've heard that people using the analogy that poker is like a coin flip. For an average player the coin is exactly 50-50. If you are highly skilled, your coin is slightly favored to your side. A 60-40 advantage would be huge. At that time I thought "60-40 is not huge. I can still lose 40% of the time. A 90-10 would be huge". Yeah in my dreams.

For a few days during my walk to work I've been thinking of a model to describe this coin flip analogy. Let's say each hand is a coin flip. How big an advantage does a good player have on his coin? How good is your coin?

First, let's define the problem a little. We probably agree that a good player earns, in average, about 10xBB per 100 hand. When I played NL2 I actually did better than that (around 50XBB per 100 hands) but we all know how laughable NL2 is.

Let's use 10XBB / 100 hands as an example.

Next we need to know the average pot size. I went to pokerstars.com and took a look at all available NL Hold'em tables at the time:

Table (BB=) Average Pot (XBB)
$6 11.4
$4 11.9
$2 12.6
$1 12.4
$0.5 12.2
$0.25 15.1
$0.1 12.3
$0.05 20.7
$0.02 25.5

You can see that at higher limit, the average pot size is around 12XBB, while this ratio grows at lower micro limits. Let's use 12 times big blind as the magic number for now.

So how does 10XBB / 100 hands look like in a coin flip?

In 100 hands, there will be a total pot of 100 x 12BB = 1200 BB.
You get a profit of 10 BB of it.

Let's say you are doing a coin flip of 60-40.
Per 100 average hands your net win is 20 pots, which is 20 x 12 BB = 240 BB.
This is far better than the reality, 10BB. 24 times better.

Your real coin is 24 times weaker than a 60-40 coin.
To be exact, it gives you an advantage of 5/6 (0.83) of the pot every 100 pots.
Your coin is a 50.417 - 49.583 coin.

That's not really an excellent coin flip you might say! But keep in mind that you are also fighting against the casino rake. From my experience, say if I have an net income of $1, the caisno has raked away $0.6 from it. This means that you are actually flipping something close to 50.7 - 49.3.

Also, most of the times you are not really participating in a pot. When you do you probably wins 60% of the time or even more. But that's exactly why you are earning money. This coin flips applies to every hand, playing or not. If you play more hands you can expect that edge to disappear or go to the red side.

Now if you look at my case. When I play NL2 I earn 50XBB / 100 hands.
In NL2 the average pot size is 25xBB.

My coin was a 51 - 49 coin at NL2. I am not really that much better than NL2 donks!

***

Another interesting question is to see how good the pros are doing. Last time I read an article saying that a pro is earning an hourly rate of $100 (just as an example) but his standard deviation is $600. Now how good is his coin flip?

First I checked pokerstars and worked out the average hands / hour. It is about 90. For simplicity we will say 100 hands / hour.

Let's first simulate what is going to happen with a 60-40 coin, flipping 100 times.

If everything goes averagely, the pro wins 60 pots and loses 40 pots, netting a 20 pots average (per hour).

His standard deviation would be 0.98 pots (thanks to MS Excel).
That's better than 20:1 ratio between profit and standard deviation -- too good to be true. He is actually getting 1:6 !

The pro is actually getting a coin that is 120 times worse than the 60:40 coin. It is 50.083 vs. 49.917. A tiny edge makes all the difference for someone to make a living out of it.

Now let's go back to my 51:49 coin. I would have an average earning of 2 pots / 100 hands, and a STDEV of 1.0 pots.

Now let's go back to your 50.4 - 49.6 coin flips.
You would earn 10BB/100 hands, which is roughly 5/6 pots per 100 hands.
Your STDEV is 1.0 pots. That's a 5:6 ratio.

You probably have seen noticed something - no matter how much your edge is, your standard deviation is always around 1.0 pots per 100 hands. You can call this "LUCK".

***

I still remember the first night I learned poker as a loose passive donk. After watching people bad beat each other like drinking water, I thought "poker is not really all about skill. I might say it is 3 skill -- 7 luck in any given night".

For this 3 skill - 7 luck (3:7 ratio) or better to happen, say 100 hands are dealt during the night, you need to at least have a 50.2 vs. 49.8 coin. Are you this good? Hopefully.

***

By the way, Casino games usually feature odds much better than that. For example, a single 0 roulette gives the house an edge of 1/37. This is like a 0.514 vs. 0.486 coin. Once I read that a good pro's edge at a casino NL hold'em table is as high as 5% (52.5 vs. 47.5 coin). I wonder whether that only belongs to the past as average people play hold'em much better than before.

***

Thanks for reading. Now if you want to learn 3 things from your last 5 minutes I wish it to be:

(1) In pokers, you are usually flipping a coin (a lot) worse than 51:49 per hand.
(2) Small advantages build up to give you a profit, in the long run.
(3) Playing against the casino is worse than playing against the best poker pros.

 

[zachvac]

Good post, one comment. You said playing against a poker pro is better than against the casino. That is semi-true. One hand against a poker pro is better than one casino event. 10 hands against a poker pro is better than 10 casino events. I can bet one event in a casino easily. How many people sit down and play one hand against a poker pro? Mostly you're sitting for a session. If he had a 52.5-47.5 edge, you're playing a few hundred hands against him, that edge grows and grows. If you flip a 51-49 coin a million times, the 51% side will come up more than the 49 side like 99.999999% of the time. Same with playing an entire session against a pro. Good post though, interesting to read even if you didn't outline your methods completely (mainly the standard deviation part)

 

[maltz]

Ops I think I've made a mistake in calculation. Let's go back a little bit.

** Let's review how a coin flip is done.

Say you bet $1, and your opponents (in the poker sense, you are playing against all of your opponents as one entity) bet $1.
You flip the coin. If you win, you get $2 (pot size = $2). Your net win is $1.
If you lose, you get nothing. Your net loss is $1.

Next we can look at a situation of 60-40.
When you invest $1 for 100 times, in average --
Your result is 60 x 2 + 40 x 0 = 120.
You win $20 per 100 hands, with an average pot size of $2.

We know the average pot size is 12 times of BB. BB in this case = 2/12 = 1/6.

Therefore, your winning per 100 hand of a 60-40 coin is:
20 / (1/6) = 120 (BB). You are expected to win 120BB per 100 hands.

Now this is 12 times better than your average earning of 10BB / 100 hands! (in my original post I said 24 which is incorrect, as I failed to consider the pot size is actually twice as large as your bet size.)
In reality, your coin is 12 times worse than a 60-40 coin.
Your coin is actually a 50.083 vs. 49.017 coin.

So on my above post my numbers should be multipled by 2.

***

Just a quick correction of all relative numbers:

Your coin with 10BB earning: 50.83 vs. 49.17
My coin with a 50BB earning (pot size 25BB): 52 vs. 48

***

That's all the correction.

Now let me revisit the method for calculating the standard deviation.
The formula of standard deviation is given as:

(The sum of variance / sample size)^(1/2)
where variance = (individual sample - average)^2

This is what I wrote in the Excel spreadsheet.
Let coin flip win rate = A%
lose rate = (100-A)%

Hand number = N
Average = AVG = (A*2+(1-A)*0)/100
Standard Deviation (STDEV) = (((2-AVG)^2*A+(0-AVG)^2*(1-A))/N)^0.5

For 100 hands STDEV is more or less fixed at 1
(For 10 hands it is about 3.2, for 10000 hands it is about 0.1)

Poker Pro's coin with hourly rate : standard deviation = 1:6
So we are aiming at a winning rate per 100 hands (in about an hour) of 1/6.
This gives us the 50.083 vs. 49.917 coin flip.

***

Well most people do play a lot of sessions in the casino, and when people play online they do have the option to gamble just a few hands with a pro. Anyways, you know what I mean -- the casino always knows how to run its business!

 

[maltz]

more models coming

After working out the Pro's weak coin, I was wondering about one thing obvious - how can we all beat the pro by that much? There must be something to be improved in the model.

When we use our own examples in the coin model, we start from our BB/100 hand. However, when we apply the coin model to the pro, we start from his standard deviation. Now, if our coin model underestimates the variance, we are going to give the pro a poorer coin than he deserves.

Actually, it is very likely that the coin model is underestimating the variance of NL hold'em. This is because:

- People don't play every hand. By playing fewer hands there are more variance. Our coin model assumes that you do play every hand.

- The frequency of all-ins (either doubling up or lose everything) is not negligible, and when that happens the pots are usually HUGE.

The next closest thing that comes up in my mind is DICE. A standard dice has 6 faces, numbering from 1 to 6. When you roll the dice you can expect 1-6 to show up with equal chances.

A standard dice would have a higher variance than the coin model. By rolling the dice 100 times, you are expecting an average of (1+6)/2 = 3.5 (or, 0 gain). Your standard deviation would be 1.71. That's 70% higher than flipping a coin.

Yet the good thing about dice is that we can change its numbers on the faces. Let's now exaggerate the dice a bit to reflect:

(1) Most of the times you don't play a hand, hence the gain/loss is minimal
(2) Sometimes when you do get heavily involved, your either win a lot or lose a lot

Let's paint the dice with:

5
1
0
0
-1
-5

This is the amount you are going to win/lose.

The next interesting concept is the average pot size.
For one approach we can average the dice numbers. (5+1+1+5)/6=2
For another approach, in reality both you and your opponents (as one entity) still each contribute $1 to roll, so the pot is still $2.


Actually, I purposely designed the dice to be this way, so the two approaches are equivalent. Now we don't have to worry about which approach is correct. (In fact I have no idea which one is correct. )

Now let's roll this 5,1,0,0,-1,-5 dice for 100 times. For an average player:

Expected win: $0
Standard deviation: $2.94

Now we have successfully elevated the variance of our model, from $1 (coin) to $1.71 (standard dice) to $2.94 (our customized poker dice).

That's for an average player. How about a good player? The dice is now biased towards the positive side. We can imagine the positive side of the dice (0, 1, 5) as one side of the coin, and the negative side (0, -1, -5) as the other side of the coin. Therefore, a 60-40 dice has 20% (60/3) chance to hit 0, 1 or 5, and 13.33% (40/3) chance to hit 0, -1, -5.

Now let's revisit our examples:

[Case 1] Professional with his hourly rate:standard deviation = 1:6

Let's still assume he plays 100 hands per hour.
It turns out that the Pro is using a poker dice of 50.122 vs. 49.878.
(Compare to our previous coin model of 50.083 vs. 49.917.)

You might think there isn't a lot of difference at all - indeed there isn't! Our pro, using our new poker dice model, is just having a coin (dice) that is 50% better.

[Case 2] You earn 10BB / 100 hands.

Everything remains the same. Your poker dice is 50.417 vs 49.583.

Your poker dice is still much better than the Pro! But if you pay a closer attention, the difference between the Pro and you have reduced a bit, due to the increased variance of our new model.

***

It seems to me the poker dice is doing ok but not that great. I further improved the model with an imaginery dice of 20 faces.

Face 1 -- 15
Face 2 -- 4
Face 3 -- 1
Face 4-17 -- 0
Face 18 -- -1
Face 19 -- -4
Face 20 -- -15

Now that sounds like poker even more! When I run the same calculation, the standard deviation becomes 4.92. This is the turbo version of our poker dice (stdev = 2.94).

So what kind of Turbo dice is our pro using? I will save you the trouble of reading and just tell you the result: 50.205 vs. 49.795. Our pro's true edge is getting more obvious now.

Yes, this is still about twice as weak as your dice (10BB/100 hands). We are doing better probably because our opponents are much weaker at low/micro limits.

***

To show that you didn't waste your past 5 minutes, here comes the summary:

(1) You can design multiple models to describe Poker (or anything). Some models are closer to reality.

(2) Even though we approach the ideal model of poker, your edge is still tiny (way less than 51:49 per hand) no matter how good you are.

(3) Multi-tabling micro-stake tables may be more profitable than simply moving up the limit and play single table. Your edge is greater down there, and your variance is reduced by playing more hands.

p.s. My BB here means Big Blind. I just read that people use BB (big bet) as 2x big blind. So in the 10xBB example, you are actually earning 20 big blinds. Your dice would be twice as good!