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Poker Strategy | Advanced Poker

Which Game Has Lower Variance?

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I recently participated in a forum discussion on the following question: "Which game has more variance, limit or no-limit?" I thought the answer was obvious, but as I soon learned, I was obviously mistaken in this thought. The answer isn't obvious, as lots of people disagreed. I did some research and found similar discussions on other forums, with a similar split between those who thought limit was more variance and those who thought no limit was more variance. Well, I'm here to tell you: no-limit is higher variance though it's also lower variance. Puzzled? It's all how you look at it.

My initial reaction in encountering the disagreement was that people just do not know what variance is. Many use the term loosely, but as it turns out, even settling on the definition fails to fully clarify things. Nevertheless, getting on the same page in terms of understanding variance is quite helpful. Variance is a measure of the distribution of a data set around the mean or average. Usually in relation to a poker game variance is a measure of distribution around the win rate. More precisely, variance is the average value of the squared differences from the win rate. Less precisely, the set of values that one uses to calculate variance could be by hand or by session or really by any number of different measures. An example will help clarify this. Let's say you have played five sessions of 100 hands each with the following results:

  1. $5
  2. -$2
  3. $10
  4. $2
  5. -$5

Your win rate for these five sessions is $2/100 hands. If this were a game with a $1 big blind amount, then the win rate expressed as big bets (BB) would be 2BB/100. To figure the variance, we look at each session determining how far from the win rate the results of that session are. This gives the following values:

  1. 3
  2. -4
  3. 8
  4. 0
  5. -7

Now to determine variance we square all these values and then take their average (add the squared values up and divide by five). That gives us a variance of $27.6/100. The figure that statisticians usually use, however, is standard deviation (SD), which is just the square root of variance. The SD in our example is $5.25/100.

Poker Tracker and other tracking programs calculate win rate and SD for you. Determining which game, limit or no limit, has more variance is as simple as comparing SD values for each game. The average limit player has a SD of 15BB/100 or so. The average no limit player has a SD of 40BB/100 or so. Which game has more variance? Pretty clearly it is no limit (40 is much higher than 15).

"But wait," some of you may say, "you can't compare BB/100. You need to compare games with equal win rates." This sort of concern appears to be one in which the player wants to achieve a particular win rate expressed in dollars per 100 hands or per hour or perhaps per month. Essentially, the concern is about bankroll fluctuation. Which game offers me less risk given that I need or want to earn a certain amount of money? Let's work with another example in which the player wants to earn $10/100 hands. Now let's say that this player must play a 5/10 limit game to earn that win rate but only a $1 big blind no limit game. If we use the typical SD numbers (15BB/100 for limit and 40BB/100 for no limit) then in dollars the limit game value is $150/100 hands but only $80/100 hands for the no limit game. When expressed in $/100 no limit appears to be the lower variance game. What has happened?

By switching our measure of SD to $/100 we get the exact opposite as when SD was expressed in BB/100. It isn't improper to express SD in either formula, so there isn't a correct or more correct way. Each offers a different kind of comparison. One must be careful with each. Expressing SD in terms of $/100 is best for comparing the $ win rate and $ swings of two games. If you are a winning player at a $1 big blind no limit game and decide to move up to the $2 big blind game you will see larger dollar swings to your bankroll. Is that because variance has increased? Well yes and no. In dollar terms yes it has, but in terms of BB, no it's probably about the same. In fact, you would get some strange looks if you asked someone which no limit game was more variance, the 25c big blind game or the 50c one. What people are often more interested in, however, is the "variance" that one will see in one's bankroll; that is, people want to know what will happen to their bankroll (expressed as dollars or pounds or whatever currency you prefer) given different games of varying blind amounts. If this is one's concern then it may appear that no limit has more variance because in dollar terms the swings appear larger. In terms of big bets, however, limit has less variance.

As the dollar win rate that one wants to achieve increases, no limit offers a less risky place for one's money. How is that possible? A strong win rate in limit hold'em is 2BB/100; while a strong win rate in no limit is 5BB/100 (one might prefer a higher value for no limit, but 5BB/100 will suffice for now). Essentially this means that you can make more at lower blind levels in no limit than you can in higher blind limit games.

With these win rates, for example, you make $5/100 hands in a 50c big blind no limit game while only making $4/100 hands in a 1/2 limit game. If we use the same typical SD values as I used above (15BB/100 for limit and 40BB/100 for no limit) then the dollar equivalents of those values are $30/100 and $40/100. At these lower stakes, limit appears the less swingy of the two for one's bankroll.

Typically, as one moves up in stakes, one can expect a drop in one's win rate (expressed as BB/100). Thus, while one may make 2BB/100 at ½, a move up to 2/4 may only generate a 1.5BB/100 rate. Similarly, a 5BB/100 rate at a 50c big blind no limit game may drop to 4.5BB/100 at a $1 game.

Variance numbers, however, stay fairly close, so we can use the same SD values for different levels.

At 4.5BB/100 a $1no limit game will produce $9/100 hands. The limit player in the current example can only make $6/100 hands. Assuming 1.5BB/100 is possible at the next level, our player would need to play a 3/6 limit game to achieve $9/100 hands. Now, however, look at the SD values. The limit player at 15BB/100 for standard deviation is $90/100 in dollar terms whereas the no limit player at 40BB/100 is only $80/100. So as you move up in games, the swings to your bankroll in dollar terms are less in no limit games than in limit games relative to equal win rates.

So the answer to the question is: it depends on the expression of standard deviation. In terms of BB/100 no limit is clearly higher variance. In terms of $/100, at a certain point as one moves up in stakes, no limit will offer lower variance.

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