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Variance in Poker

Good Variance vs Bad Risk

Poker RiskThe "Chen Coin Flip" article explains the basics of what variance is: the statistical measure of how your results will be dispersed. Despite how some players use the term, variance does not mean "bankroll swings". Variance comes into play in poker when observing the difference between the expected value of an action and the actual results.

Just in a raw universe sense, if you are playing a nine-handed No Limit Hold'em game with a stack of $500, your results for a single hand could range from -$500 (losing your whole stack) to +$4000 (winning when all eight opponents cover your $500 bet). If you are a longterm winning player, winning $30 an hour, while playing 30 hands an hour, you could conclude your EV (expected value) for each hand you play is $1. So the event of a single hand could have an expected result of $1, but the variance of the actual results will fall between -$500 and +$4000.

That's a pretty big spread for a pretty mundane EV, but that is the reality of poker.

To be able to win that $1 every hand, you have to be able to stand the far bigger fluctuations -- stand it both in terms of bankroll and mental attitude.

The $1 a hand concept isn't a very helpful way to look at things though. First, your EV for hands played first under the gun will be different than your EV on the button. And your EV in the blinds will be quite a bit worse (especially in Limit games) than from the other positions. Likewise, in a ring game your EV second under the gun with 7♠2♥ will be different than your EV when holding A♠A♥.

But there isn't just a difference in EV between holding 7♠2♥ second under the gun and holding A♠A♥ in the same position. When holding 7♠2♥, you should fold every time, your EV should be $0, and your variance should be zero. On the other hand, when holding A♠A♥ you should have a positive expected value while also having fairly extreme variance. Certainly some of the times you hold A♠A♥ you will lose your $500 stack. And, some times you should have wins exceeding $1000.

The bottom line in this is: variance is your friend. If you don't like variance, then hope for 7♠2♥ every hand. If you hope to get good cards, then you need to accept a goodly amount of variance. Big hands, like A♠A♥ or the nut flush on turn, should have large variance. They shouldn't be folded because we lose a big pot when we do lose.

At the same time, it is important to not solicit pointless, unprofitable variance... just like we don't want to solicit extra random luck. Here is an example of bad variance:

Suppose John is such a bad player that to keep him in the game Mary offers to pay John $1 every hand. No risk, no gambling to it, just a $1 per hand payment... or, she offers John a gamble-y alternative prop bet: if the flop comes two or three red cards, Mary will pay John $102, but if the flop comes two or more black cards, John has to pay Mary $100.

The EV of these two situations is exactly the same: John is EV+$1 every hand. But, if he takes the 102/100 option, John is turning down a sure thing profitable situation for a higher variance one where he could actually lose money.

Suppose after 1000 hands it has come black for Mary 525 times and red for John 475 times. This works out to John being behind -$4050, even though he had a EV edge of +$1000. Mary could then decide that she wants to go home, and quits the bet. Via pure luck, Mary turned her -1000 EV into +4050.

In contrast, John has no similar good quitting strategy to employ, since every bet is EV+$1 for him, he should never quit. Taking on the extra variance has no advantage for John.

More mundanely, Mary might quit after the 21st red/black flop, when she has won 11 and John has won 10. Unlike the above 525/475 scenario, it is not remarkable at all that early on in the red/black game that Mary would be ahead at some point, whether it be 11/10, or 2 of the first 3, or even Mary winning the first flop... and promptly quitting and pocketing her $100 win.

When you have the best of it, inviting pointless variance is a catastrophic way to play a game with significant short-term luck like poker. The close equivalent of the 102/100 situation in ring game poker is when a terrible player wins a pot from you via a miracle river card and then promptly quits the game. Losing to bad players is not the problem. That's part of variance. The door swings both ways. You have to lose pots to win pots.

However, there are ways to play hands that while they have the approximate same EV, one way might have far more variance than the other. There is no reason to choose the higher variance way.

Don't be afraid of variance. Winning players like variance. They like to play big pots with the nuts where their opponent may have a one-outer to beat them, and when they hit it, the results variance table will look all extreme and goofy. But don't be a sucker like the 102/100 bet. Don't give your opponents extra chances to win via luck when you don't need to. For example, a lot of players like to slowplay aces in No Limit. They want to let their opponent bet-bet-bet and hang themselves. Totally fine tactic... unless you don't have the courage of following through because suddenly the variance of a river bet intimidates you. If you just call with aces preflop, on the flop and on the turn, then fold to a large river bet when it comes a blank card... what the heck was that? If you are going to play a hand in a high variance way, then play it. Don't bail out early because suddenly your high variance play causes you to chicken out.

Likewise, if you have bet-bet-bet your nut hand to set up a $300 river bet that you calculate is the perfect EV bet against this opponent, why would you suddenly bet $600? If you are called 50% of the time the EV works out the same, but why would you want $600 half the time and $0 the other half instead of $300 every time? Suppose your opponent doesn't call the $600 this time, and then never plays another hand of poker again. Yes, it could work out via luck that your opponent calls the $600 this time and then never plays poker again, but that is leaving it to luck, like the 102/100 example versus the $1 sure thing. Just take your EV. Don't give your opponent the chance to luck into a way to give you less than your EV.

Play with an appropriate bankroll. Play to maximize your EV, regardless of variance. When you find the best EV+ way to play, don't add unnecessary, extra, lucky variance.

Also: Poker Expected Value, Changing the Poker Math and Poker Odds and Gambler's Ruin