Reducing Variance With Simultaneous Play

Variance in poker outcomes is often frustrating, even when you know you're playing well. You can reduce your variance in different ways: by playing more conservatively, by seeking out more sedate tables, and sometimes by sacrificing some narrowly profitable situations. Online poker offers another option, one that wouldn't be possible in live (brick-and-mortar casino) play. You can potentially reduce your variance by playing in multiple small games instead of one larger game. This opens up a whole spectrum of approaches to managing the volatility of your bankroll.

Suppose your regular game is $15-$30 hold'em, a game in which you expect to win one big bet (BB) per hour online. If you could maintain that same BB per hour rate in three simultaneous $5-$10 games, you would yield the same overall expectation but a much lower hourly standard deviation (SD). How much lower?

Your standard deviation for a series of outcomes is the standard deviation for a single outcome multiplied by the square root of the total number of outcomes. So, if your hourly standard deviation is 10 big bets, and you play 100 hours, the standard deviation on your total outcome will be 100 big bets. If you play 300 hours (or 100 hours at three tables simultaneously), your standard deviation will be 173.2 big bets. If your SD in big bets is the same for both the $15-$30 and the $5-$10 game, your standard deviation in dollars will be $3,000 for 100 hours of $15-$30, but only $1,732 for 300 hours of $5-$10. So, all else being equal, your variance over a fixed interval of time will drop dramatically if you divide your play among smaller tables. Or, you could play at two $10-$20 tables, with a standard deviation over 100 hours (times two tables) of $2,828 – which is slightly less than at $15-$30, with a potentially higher earning rate.

Overall hourly standard deviation is a reasonable measure of how stressful your poker life will be. But reduced variance can also have a dramatic effect on your bankroll requirements. Bankroll requirements don't depend on how many hands you play per hour; they depend on just your variance and earning rate. So, for bankroll purposes, playing three $5-$10 games is just like playing one. Restricting your play to $5-$10, even if you're playing three tables simultaneously, means you can get by with one-third the bankroll you'd need for $15-$30, all else being equal.

Of course, all else is never equal. If you're really playing multiple tables simultaneously, your win rate may suffer from playing multiple games. You may also play each individual game with higher variance (in bets). Conceivably, the smaller game could have weaker players, which could offset the cost of playing multiple games, but it isn't guaranteed.

Since there are a few variables at work here, it's worth laying out some of the possibilities in a table. The bankroll requirements below are derived from a simple formula (Mason Malmuth first applied it to poker in his landmark Gambling Theory and Other Topics). To keep the numbers intuitive, I've included dollar amounts for hypothetical $5-$10 games (and one $15-$30 game for reference). The hourly standard deviation (SD) and expected value (EV) are for a single table.

The first line of the table is the hypothetical original game, from which you might consider switching.

A few things are immediately obvious. First and most obviously, if your hourly SD and EV are roughly the same at both limits (measured in bets), you need only one-third the bankroll size to play the smaller game. Even if you're able to muster only two-thirds of your full earning rate, your bankroll requirements are still cut in half. So, if you have a smaller bankroll, playing multiple smaller games might be a good way to trade some of your expectation to reduce your risk of ruin.

As the hourly SD slides up, you need to be more cautious. With an hourly SD of 18 bets, you need to maintain about two-thirds of your original earning rate to realize a substantial reduction in required bankroll. Anything less and you're sacrificing EV for no benefit. At an SD of 21, you'd need to maintain five-sixths of your original earning rate, and with an SD of 26 bets (which is roughly 15 times the square root of 3), there's no overall variance reduction. Three simultaneous $5-$10 games would give you the same hourly SD as one $15-$30 game, so it would be pointless to drop down unless you could realize an improved earning rate.

Of course, there are other options for dividing up your play, and there's no reason the bet sizes have to come out even. As I mentioned earlier, two $10-$20 games could conceivably give you increased EV and reduced variance, the holy grail of poker bankroll management. The important thing is that you consider the full range of options afforded by online play, and arrive at a scheme that meets your needs.

Is playing multiple small tables really a good idea? Some experienced players believe it's too difficult to play well in multiple games simultaneously. If they're right, your win rate suffers much more than it would appear from the table above. I personally believe that many winning players, especially those who are generally able to make quick decisions in their regular game, should also be able to beat multiple games. And they should be able to realize winning rates comparable to what they earn in larger games. Of course, I wouldn't recommend it with stud, due to the burden of remembering boardcards. And I'd be reluctant to try multiple big-bet games. But for limit hold'em, and perhaps Omaha, I believe the game is often mechanical enough to afford opportunities for quick-thinking players. Although there will certainly be some errors due to managing multiple tables, and you'll likely miss things as you divide your attention, in some cases this can be offset by the reduced quality of competition at the lower-limit tables.

Finally, one of the nice things about reducing variance in this way is that it doesn't undermine the beneficial effects of variability (namely, that it gives losing players a legitimate chance to win). When you play multiple tables, your variability at a given table isn't necessarily going down – it's probably going up. You're just getting into the long run faster by playing more hands. Losing players may actually benefit, as long as widespread adoption of this approach doesn't overpopulate low-limit games with solid players.

Daniel Kimberg is the author of "Serious Poker" and maintains a web site for serious poker players at www.seriouspoker.com.