Luke ‘Desultory’ Hatfield looks at variance in poker and why you need to understand it – and learn how to deal with it
Variance in poker may seem like a relatively obvious concept, yet is one that you need to know in depth if you want to be a long-term winner. I am often faced with students who ‘tilt’ or seriously worry about their short-term losses, and this suggests a lack of understanding of this apparently simple concept.
If you tilt one buy-in away, that is normally 100 big blinds. Given that the best players only make maximum of 8-10BB per 100 hands on average, you are now going to have to play a lot of hands to make up for your one instance of stupidity. If you fully understand variance you can mentally prepare for it.
Preparing allows you to take a step away from tilting and towards a better win-rate. Many players attribute their short-term wins to skill and their losses to luck. This is called ‘self-serving bias’. Have you met any player who thinks they are bad at poker? Decent players don’t attempt to improve because they already think they are the best and recreational players don’t stop playing for a while until they reach a certain loss, at which point they decide the game is rigged.
This is why variance is your friend. Many players fail at poker and it is all because they don’t want to take the time to fully understand the implications of variance.
So what is variance?
Variance is a measure of how far a set of numbers are spread out from the mean, or expected value. In poker, this basically equates to ‘luck’. In terms of winnings, the mean profit per hand in poker is the Expected Value (EV) from your preflop, flop, turn and river bets, calls and/or reraises all added together (EV = 0 for a fold preflop). Note that I also include losses in the term ‘profit’ and therefore it can be a negative number. The EV and mean is what you ‘should’ have won or lost on average. It is not what you have actually won or lost in real life.
EV is calculated from your equity%, and your equity% is the number of times you would win on average if the same situation was replayed 100 times. In real life, the situation only gets played once, so you either win or lose one time (100%), with the occasional split pot. For example, say you have A-A and you go all-in preflop for $100 in a $0.50/$1 game, and get called by K-K. The board comes K-3-2-8-J. Your real-life profit for this hand is -$100, but your expected or mean profit for this hand is 81.9% x $201.5 + 18.1% x -$100 = $65.11.
You could also total the entire mean $/hand profit values to get your overall mean profit. Win-rate in BB/100 (big blinds per 100 hands) is a measure of profit that poker players use. The mean win-rate can be calculated by the following equation:
{[(Mean total profit)/ (big blind)]/(number of hands played)} x 100. (The result is sometimes called ‘True win-rate’).
You could compare this to your real-life actual win-rate, but in practice no one bothers to do this just because of the sheer volume of calculations needed. The difference between real-life win-rate and expected win-rate is due to variance.
The impact of variance
I think pretty much everyone grossly underestimates how much effect variance has. Have a look at the graphs below. The first one shows the impact of variance on a mean win-rate of 8BB/100 over 100,000 hands, for a standard deviation (SD) of 100BB/100. (Standard deviation is just the square root of variance.) The SD will be smaller the ‘tighter’ you are and larger the ‘looser’ you are, but it is also related to the game played.
I am probably on the looser side of players and my SD is around 140 for heads-up cash (but I think this can go up to 160ish for some players), 100BB/100 for six-max, and 75BB/100 for full ring. The black dotted line shows the mean profit (expected profit) and the colours each show 1,000 possible profit lines caused by 140BB/100 standard deviations over increasing hands. If each colour was a player of exactly the same ability (all of them with mean win-rate of 8BB/100), you can see that after 100,000 hands the player with the best possible run would have earned about 17,000 big blinds and the player with the worst possible run would have lost about 1,000 big blinds.
Let me state again that these ‘players’ have exactly the same ability, but due to variance they have an 18,000 big blind profit difference after 100,000 hands. The second graph shows just the best and worst runs to make it clearer. As you can see there is still a swing of about 17,000BB profit from best run to worst run over 100,000 hands even after decreasing the standard deviation to 100BB/100.
Purely because of ‘luck’ the ‘best run’ player would not only end up with a great deal more money than the ‘worst run’ player, but it is quite likely the ‘worst run’ player will end up tilting or playing differently. This new playing strategy may decrease his mean win-rate and he may end up quitting poker soon after if ‘luck’ doesn’t turn for him.
On the other hand, the ‘best run’ player may get the glory, she may get a TV appearance, she may get invited to join a coaching site and she may become a guest pro at a large poker site. Poker certainly isn’t fair. You can work hard, be the best and still fail. It works the other way round too. Players who have a mean win-rate of -1BB/100 can still end up with a higher real life win-rate than quite a few of the players with a mean 8BB/100 win-rate. On average this won’t be the case, but on a large scale (real life) there will certainly be some unprofitable players with decent real-life win-rates in the mix after 100,000 hands.
These players may even get chosen as a coach instead of the mean 8BB/100 win-rate player, and they may start spouting their irrational decision-making process to the world. The worrying thing is, other players may start to listen.
The truth will out
The only way you will truly know if a player is good or not is if you have seen them rack up about a million hands at a respectable win-rate. A million hands is about the point where you can start to discount variance in poker. Compare that figure with the number of hands you play in a home game, or consider how many hands the ‘big names’ play in the high-profile live games. 30 hands per hour maximum? The worst player in the world can triple his buy-in in a few hours at these tables and then the commentators will glorify his ‘great play’.
Please note that there are also other unmeasured forms of variance that are not taken into account by these calculations. For instance, if you get A-A every hand for five hands then you are likely to do better than if you get 7-2o every hand for five hands.
Another unmeasured form of variance is not just how often you get a ‘good hand’ but how often both you and your opponents get a ‘good hand’ when yours is better. If this outcome occurs in your favour more than it should on average then you are going to have a much larger profit. You can sit all day at a table with poor players, but if they never flop top pair when you fl op something better it’s going to be hard to stack them before they leave.
This is especially true if the other regular players at the table are lucky enough to get into more showdown situations versus these recreational players. I wonder myself whether this form of variance is negated after a million hands. I suspect not. Remember though, at lower stakes there is never going to be a shortage of games, so it’s not a big deal if you run bad now, as luck will even itself out over a million hands.
