# Variance is not a kid's fundament afterall!!

• Bronze
Joined: 02.12.2010
I have seen so many threads about variance recently and unfortunately every poker player is bound to go thru variance.
I just came by a decent article on variance at another site. Some people might have already read it but just thought i should share it and maybe enlighten others

Webster's Dictionary defines luck as "the events or circumstances that operate for or against an individual; favoring chance." In poker, luck, determines when a hand holds or doesn’t. We as poker players call this variance. There has always been a bit of mystery in the world of poker around exactly what variance means. How does variance control our success or failure as players?

There are two major types of variance in poker. The first, card variance, is isolated hand by hand. It is a measure of how the cards fall. Card variance is measured in a vacuum and represented in chips…the dollar value of the chips in the pot is irrelevant. No matter how big a favorite or underdog you are in a specific hand, there are no guarantees on which side of those odds you’ll end up. The odds of sucking out with KK v. AA are the same in a \$1 MTT as in the biggest NLHE cash games. In the short term, chance determines when your hands hold up and when they don’t. As players we know the results even out over time, with enough samples. Mathematicians rely on the central limit theorem to prove this.

We all experience stretches when dominant hands seem to lose time after time. These periods are much easier to handle if you remember no hand is affected by a previous one. Cards don’t know you’re due to have hands hold up because you’re running awful. They can’t know that you need your hand to hold up, just this “one time” because winning puts you in a great spot. It’s really not personal, even though it may seem as if it is.

The hardest loss I've ever taken was a true flip. There were 17 players left in a large field with a great structure. I have a loose aggressive player a few to my right who has made a very small raise almost every hand for the last hour. He seems comfortable playing small ball. To succeed, I need to take this opponent out of his comfort zone. I look down at KJs on the button; my opponent again puts in a small raise. Taking a stand, with a lot of fold equity, I shove. My opponent snap calls with a pair of deuces. I’m about 50/50 to be in a great position.

The flop, turn, and river bring nothing but heartache and I finish the tournament in 13th place. This run was one I am very proud of; the tournament was the Sunday 750K on Full Tilt Poker. If I had won that flip it would have certainly put me in a great spot to take home life changing money. My bankroll would not support playing this tournament, I had “taken a shot” by playing a satellite for my entry. I certainly could not afford to play it enough to beat “card variance,” allowing me to reasonably approach the expected win rate of my hand, about 50%. By placing such a large amount of potential winnings on one hand I allowed card variance to become tournament variance.

“Tournament variance” is determined by results, when you win or lose hands in relation to the tournament’s payout and chip structure. Unlike card variance, tournament variance is only measured in real dollar amounts. If you have a dominating hand hold up at the final table of a large field tournament your likely to see a sharp increase in your real dollar expected value, often representing a large increase in your bankroll. If you were to win that same hand in the first few levels of that same tournament it is much less likely to directly represent the same type of bankroll swing. The odds of winning the hand may be the same, yet one brings greater profit. Tournament variance attempts to measure the ups and downs of results.

Tournament variance is calculated based on the size of the field, payout structure, and your expected return. Card variance is completely ignored in the calculation. We are able to justify doing this with a statistical theory known as the "central limit theorem”: over enough trials actual results will equal expected results. The math is tedious, but in layman’s terms, it’s the reason we know we win about the number of times we’re supposed to. The central limit theory relies on having a very large number of trials. This is a primary reason we use bankroll management.

Tournament results, like all measures of variance, are subject to skew. Skew is the measure of how far your results might vary up or down from expectation. Unlike with cash games, tournaments have a very positive skew. If you play a tournament, particularly a large field tournament, your worst negative result is normally one buy-in. Your best positive result can be several hundred buy-ins!

This can cause a player to run well above expectation and maintain an inflated ROI for thousands of games. As the number of tournaments you play increases your skew decreases, slowly approaching one. When enough samples are observed, you will have be able to calculate what is known as your True ROI, your ROI will be as close to accurate as its ever going to get.

A player can run above expectation for thousands of games and believe their ROI is actually higher than it is. If you’re basing your bankroll requirements on an overestimated ROI you will end up with a false since of safety, and a very real chance of going broke.

In playing out of my bankroll I ignored the central limit theorem and exposed myself to tournament variance. Every time you play in a game that you aren't positive you can support playing regularly, you open yourself up to lady luck as well. If you want to limit her role in your poker career, consider evening out the average field size and buy-in you play. You would be amazed and how much less a bad beat hurts when you know you will see that same situation a ten more times before the end of the day.

It’s human nature to allow ego to alter our perception’s of our own abilities. Ego isn't all bad though as it can be part of attaining confidence and being confident is necessary for poker players. However, letting your ego overestimate your true expectation is very dangerous. Believing you are beating a game by more than you actually are puts your entire bankroll at risk. This doesn’t affect everyone equally; understanding why could make a huge difference in your long term success. In part two we’ll talk about why and what steps you need to take to protect your hard earned bankroll.

Understanding how the factors of tournament variance behave in relation to each other allows us to make a few very important observations. The most important relationship in tournament variance is its inverse relationship with expectation. When expectation increases, variance decreases. However a one point increase for a player with a 5% expectation reduces variance dramatically more than it would for a player with an expectation of 50%. Unfortunately, this theory also works in reverse. Players with smaller expectations due to playing at a higher level, or in smaller fields, must be much more attentive to their bankroll management systems. A loss of only one percent of expectation would be much more dangerous to them.

When a player's expectation drops from 50% to 40% the risk of ruin only falls about 1.2%. If we take that same 10% loss of expectation from a player with a 20% ROI the risk of ruin increases almost 10%. The same loss of expectation yet a dramatically different increase in risk of ruin with the only difference being the initial expectation. This may seem simple but it can sneak into your life and shake up your bankroll management without you really noticing.

Changing field sizes generally result in a fairly constant change in tournament variance. As field size increased tournament variance seemed to increase at a fairly steady rate. There were very slight differences that are fairly insignificant. No great mystery here; Increasing fields size also increases variance, its fairly logical so we wont waste a lot of time on it.

In part one we discussed two types of variance, how they effect each other and how some of the factors of tournament variance are not constant. It is important to understand tournament variance because of its relationship with risk of ruin. Risk of ruin is the calculation behind every decent bankroll management system. You'll find risk of ruin used in poker, blackjack, horse racing, and even by investors on wall street. The formula tells us how likely we are to go broke given a certain bankroll, number of trials, and variance.

Variance has a complementary relationship with risk of ruin, as it increases so does your risk of ruin. Variance is a factor of risk of ruin and expectation is a factor of variance so through the communicative property of math we also know that expectation has an inverse effect on risk of ruin. As expectation decreases risk of ruin increases.

This explains why a player with a lower expectation (all other things being equal) would be subject to greater swings and consequently why his\her bankroll requirements are likely to be much higher. As your average buy in increases we assume that competition becomes tougher, leaving less expectation to pass between a larger number of talented players, significantly decreasing your expected return in most cases.

Our first example involves a player that has made no changes in his play, game selection, or buyin level yet his bankroll has been exposed to a greater risk. I have a good friend that plays the \$16 18 mans on poker stars religiously. With one of the lowest rakes in all of online poker and they are filled with grinders. Due to field size and composition, expectation for even top players doesn't get very high. They also happen to be one of the best ways to rack up frequent player points. Toward the end of the year you will see a ton of very good players scrambling to make supernova firing them up. Even though this is his regular game, he may experience an increase in his risk of ruin without changing a thing. The influx of good players to this game means less profit is available driving his expectation down. This decrease in expectation means he will need to make a bankroll adjustment to maintain the same risk of ruin.

The next example involves the players who migrated to that game as they are likely multi table tournament specialist who are used to having an expectation that is simply unachievable in this new format. If they were to maintain the same bankroll management system they would be exposing their bankroll to considerable risk. You might naturally assume that the dramatic decrease in field size would be sufficient to offset the drop in expectation. It would be prudent to remember that as expectations get closer to zero each percentage point lost has a greater effect on risk of ruin, so assuming your covered could be a grave mistake.

My final example comes when players attempt to move up to a new buy in level. Players will often assume that if they are playing the 2.20 180 man multi table sit and go’s at Poker Stars with 150 buy-in’s they could safely move to the next level without adjusting their bankroll management. Unfortunately players often fail to consider that increasing quality of competition is likely going to cut into expectation. With a new lower expectation the player would need to increase his minimum required buy-ins to maintain the same risk of ruin.

We often fail to make adjustments at times as it can be hard to identify which aspect of bankroll management has changed. The best way to better protect ourselves is by gaining an in depth understanding of all aspects affecting variance and bankroll management. There really is no acceptable shortcut, yellow brick road, or easy way out. In a world where we casually accept giving up a percentage of our expectation in order to increase our hourly rate, how often do we consider how that changes our exposure to variance? Or even how playing a particular number of tables, a new tournament, or different level might effect our bankroll management? Variance can change without warning, protecting your bankroll requires constant attention, an objective mindset, and often painful and honest self assessments.
• 14 replies
• Bronze
Joined: 08.12.2009
tl;dr

it would be nice if you could summarize the thread for ppl like me i.e everyoone else on the forum
• Bronze
Joined: 02.12.2010
lol i know its a bit long,but it was totally worth the time, atleast for me.
• Bronze
Joined: 08.12.2009
• Bronze
Joined: 09.09.2010
nice one
• Bronze
Joined: 22.11.2009
Nice article, and I suspect his conclusions are correct, but:

i) That's not what the central limit theorem says. What it actually says is that after enough independent trials everything becomes normally distributed.
ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.
• Bronze
Joined: 02.12.2010
Originally posted by clawindsouza
Glad u read it. . Personally i liked the parts where he gave sum real life examples and shockingly i hv been thru them like moving up from the \$2.2 180 man sng's for example, etc. and i didnt know about them.Now i understand a lot that u not only need skill for different buy-ins but also the correct bankroll and mindsetup to beat them. One more gud point he gave was hw many times we hv to repeat a same situation again n again to be +ev.(kj vs 22)

@jbpatzer

ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

if u square the risk of ruin, then hw does it become 1% from 10%??
• Bronze
Joined: 22.11.2009
Originally posted by shehanshah

ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

if u square the risk of ruin, then hw does it become 1% from 10%??
10% = 0.1

0.1*0.1 = 0.01 = 1%

• Bronze
Joined: 02.12.2010
Originally posted by jbpatzer
Originally posted by shehanshah

ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

if u square the risk of ruin, then hw does it become 1% from 10%??
10% = 0.1

0.1*0.1 = 0.01 = 1%

ahh now i get it. This risk of ruin thingy is a bit hard to learn
• Bronze
Joined: 31.01.2009
ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

That aint true. Decreasing your true wr does affect your variance. Better winrates=lower variance
• Coach
Coach
Joined: 03.08.2009
Originally posted by Alverine
ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

That aint true. Decreasing your true wr does affect your variance. Better winrates=lower variance
lol

low variance style = lower variance

high variance style = higher variance

low win-rate = higher risk of ruin given a certain style

Better winrates may even be higher variance depending on your style.
• Bronze
Joined: 22.11.2009
Originally posted by Alverine
ii) Decreasing your win rate doesn't affect your variance, but it does affect your risk of ruin. If you double your win rate, you square your risk of ruin, e.g. 10% becomes 1%.

That aint true. Decreasing your true wr does affect your variance. Better winrates=lower variance
Risk of ruin is exp(-2 mu/sigma^2), where mu is the win rate and sigma^2 the variance. Are you saying sigma^2 is lower for higher mu? That would be interesting. HEM can tell you your standard deviation (=sigma) and obviously your win rate. Maybe we should start a thread where everyone posts their win rate and standard deviation and see whether sigma does decrease with mu in some way. Interesting. Or maybe someone's already looked at this. Is it in the Maths of Poker? I left my copy at work.

• Coach
Coach
Joined: 03.08.2009
Originally posted by jbpatzer

Risk of ruin is exp(-2 mu/sigma^2), where mu is the win rate and sigma^2 the variance. Are you saying sigma^2 is lower for higher mu?
I do not think these are proportional to each other. It is possible to play a low variance style with a poor winrate and a high variance style with a good winrate.

However....you may notice various trends. For example, in order to capitalize on ever-decreasing edges at the higher stakes it is likely necessary to adopt a higher variance style to maintain a decent winrate. Therefore, you may notice a correlation between mu and sigma^2 if you were to analyse specific players/games, but theoretically I do not believe these are inherently proportional - even if they are not totally unrelated in practice.

People do have a tendancy to suggest higher-winrates are lower variance, but this is likely a perception issue. It feels lower variance when you have a high winrate - you are more likely to be be up in \$ even when variance is kicking your ass. It's certainly "lower variance" from a risk of ruin point of view. Theoretically though, your variance depends on your play style not your winrate.

Anyways, just my 2 cents, flame away.
• Bronze
Joined: 22.11.2009
Originally posted by w34z3l
Originally posted by jbpatzer

Risk of ruin is exp(-2 mu/sigma^2), where mu is the win rate and sigma^2 the variance. Are you saying sigma^2 is lower for higher mu?
I do not think these are proportional to each other. It is possible to play a low variance style with a poor winrate and a high variance style with a good winrate.

However....you may notice various trends. For example, in order to capitalize on ever-decreasing edges at the higher stakes it is likely necessary to adopt a higher variance style to maintain a decent winrate. Therefore, you may notice a correlation between mu and sigma^2 if you were to analyse specific players/games, but theoretically I do not believe these are inherently proportional - even if they are not totally unrelated in practice.

People do have a tendancy to suggest higher-winrates are lower variance, but this is likely a perception issue. It feels lower variance when you have a high winrate - you are more likely to be be up in \$ even when variance is kicking your ass. It's certainly "lower variance" from a risk of ruin point of view. Theoretically though, your variance depends on your play style not your winrate.

Anyways, just my 2 cents, flame away.
That all sounds very reasonable to me.
• Gold
Joined: 21.01.2010
Originally posted by w34z3l
Originally posted by jbpatzer

Risk of ruin is exp(-2 mu/sigma^2), where mu is the win rate and sigma^2 the variance. Are you saying sigma^2 is lower for higher mu?
I do not think these are proportional to each other. It is possible to play a low variance style with a poor winrate and a high variance style with a good winrate.

However....you may notice various trends. For example, in order to capitalize on ever-decreasing edges at the higher stakes it is likely necessary to adopt a higher variance style to maintain a decent winrate. Therefore, you may notice a correlation between mu and sigma^2 if you were to analyse specific players/games, but theoretically I do not believe these are inherently proportional - even if they are not totally unrelated in practice.

People do have a tendancy to suggest higher-winrates are lower variance, but this is likely a perception issue. It feels lower variance when you have a high winrate - you are more likely to be be up in \$ even when variance is kicking your ass. It's certainly "lower variance" from a risk of ruin point of view. Theoretically though, your variance depends on your play style not your winrate.

Anyways, just my 2 cents, flame away.
+1 sounds good to me