PV’s Playhouse – The Art of Reducing Variance

It’s the end of the round. You spot your friend coming down the hall.

“Hey, how’d it go?” you ask.

“I lost”.

“What was he playing?”

“Naya Pod.”

“But that’s your best matchup, what happened?”

Your friend gives a big shrug—“Variance.”

In retrospect, this “variance” thing sounds really bad—everyone keeps losing games to it. Is there a way we can stop it? No, there isn’t. Decrease it? Maybe.

The objective is not to minimize variance itself, but to minimize the impact variance has on you. Your opponent will have the Force of Will sometimes, and sometimes he won’t—you can’t predict it and you can’t do anything about it. What you can do, however, is make sure the outcome of the match is not decided by [card]Force of Will[/card] alone. The impact of variance is, therefore, diminished.

Variance is what changes from game to game, what could have gone one way, but went another. It’s drawing that spell when you needed a land, and vice-versa. If you’d played that same game 100 times, it would have gone your way in some of them. By itself, variance is not a bad thing—it can benefit your opponent, but it can also benefit you.

Why, then, should we try to reduce it? Because variance deviates the result from the “expected” one, and, if you’re a good player with a good deck, then the expected result is that you win. If you are so much better that you have a 65% chance to win, then variance is going to make you win roughly 65% of the games. We want to win more than that. After all, if we are so much better, so why shouldn’t we win all the games?

Take, for example, Chess. I wouldn’t say that the best player is going to win all the time, but if the difference in skill is significant, then you’d be surprised if the best player lost. If the difference is huge, it’s all but impossible. I’m not going to beat Kasparov, it’s just not going to happen.

With Magic, I could teach you to play today and promptly lose to you next week. Historically, the top players win 60-65% of their PT matches, and, though the number would obviously be bigger if they were playing at FNMs instead, it certainly wouldn’t approach 100%. This is directly caused by variance—or, in a way, luck.

This doesn’t mean Magic is an inferior game, it’s just different—in fact, to me it is a better game in part because of that. Variance makes sure every game is different, it makes sure you can beat someone who is better than you or with a deck that is supposed to beat yours, it gives players an escape mechanism for their losses—sometimes it’s frustrating to know that if you lose it HAD to be your fault, it’s good to know that maybe you can just blame luck.

The most important thing when it comes to reducing the impact of variance is increasing your sample size. The best way to decrease variance is, therefore, to just play more games. Say you have a 90% win rate against someone with a certain deck—that means you’re going to go 9-1, theoretically, in 10 games. That one loss might be the first game, and it might be the last. If it’s the first and you only play one game, then you actually have 100% losses—a huge impact. If only you’d played 9 more games, maybe you’d have gone 9-0 in them.

As a general rule, the bigger your sample size, the more accurate your result is going to be. So, you should play in as many tournaments as you can to balance things out. If you play a PTQ and do badly, that’s not an anomaly—it’s expected, even if you’re skilled with a good deck. If you play 15 PTQs and do badly in all of them, even though you’re skilled and have a good deck, well, now that’s an anomaly—now you can be upset, you can blame life and God and justice, but not before, because if you quit after one or two bad tournaments then you won’t be giving variance a chance to even out.

Take, for example, Bridge (yeah I like those references). The real beauty of the game is that it’s duplicated—everyone plays exactly the same boards, with the same cards, and the point of the game is to do better with the same cards than your opponent. If team A-B-C-D (four people) plays against team 1-2-3-4, then the match is going to be A-B versus 1-2 in a room, and C-D versus 3-4 in another. A-B will have a set of cards, 13 each; and in the other room, 3-4 will have those exact cards. By consequence, the cards that 1-2 held on the first table will be held by C-D in the other one.

Imagine you were playing Poker and every table in the casino had exactly the same hands and the same flops/turns/rivers, and your goal was to make more than the other people in your position—if you lose $100 and everyone else loses $200, you’re actually winning. As such, the “I have all Aces” type of luck does not exist—if I have all the Aces, then my opponents will also have all the Aces against my teammates at the other table, and the goal is not to win but to do better than he does over there.

This is not to say there isn’t luck in Bridge—there is. Sometimes, you make a play that has 70% chance of succeeding, and the cards are badly placed, so you fail. At the other table, the person makes the play that only has 30% chance of succeeding, but, since cards are badly placed, it goes well for them. They have beat you in that hand precisely because they played worse—that’s luck. For this reason, people say “in eight boards, anything can happen.” Maybe you luck out once or twice, and the other boards are all flat and then you end up winning.

In eight boards, the Brazilian junior team could beat the world champions. In fact, at the junior tournament, we beat the extremely superior USA team in a 16 board game (which, I’m not going to lie, was highly satisfying. I don’t think they were very happy about it). The way the game evens that out, though, is by having a lot of rounds—in this case 19. The US team still qualified, we did not.

In the knockout stage, you play a lot more boards—the finals can be, for example, 64 boards rather than 16. We’d never beat the world champions in 64 boards, because the ones we luck out at will be diluted among the ones they beat us because they’re better. In Magic, it’s not so different—you have a lot of rounds, you can afford to lose some, and finals are sometimes best out of 5. Since they aren’t about to make finals best of 63, the only thing you can do in this regard is play more tournaments.

A big part of reducing variance comes from deck selection, and whether I have control over what is going on is a very important aspect in my deck choosing process. Take, for example, a deck like Belcher—playing Belcher at the tournament is the equivalent of flipping a coin every match (a less than 50% coin if you ask me, since I think the deck is very bad, but let’s assume for the sake of this article that it’s a 55% coin). Can I kill him turns one-two? Does he have [card]Force of Will[/card]? That’s all there is to it. Going back to what I mentioned in the introduction, half the games he will have Force of Will, half he won’t—this is pure variance, and this is what is going to define whether this deck wins or loses. With every other deck, he’s still going to have [card]Force of Will[/card] half the games—the variance is still present—but the impact of it is greatly diminished.

You can also take the Miracles deck that won the Pro Tour—one person won the tournament with it, everyone else had a horrible record. Surprising? No, not really. It’s the kind of deck that does that. If you’re lucky you’ll be unbeatable, and if you’re not you won’t beat anything. I like to stay away from those decks, and miracle cards in general (though some of them are very good even if you don’t miracle, such as [card bonfire of the damned]Bonfire[/card]—so they’re worth playing—but I hope I never have to play [card]Revenge of the Hunted[/card] in my life).

Increasing your sample size works not only for playing many matches, but also for playing many turns inside a match. A deck that bases its game plan around turns 1-3, for example, is a lot more susceptible to slight variation. Imagine two Limited decks: one is very aggressive, one controllish. The aggressive deck aims to “close” or define the game by, say, turn 5. By turn five, you’ll have seen 12 cards. It is not likely that those are 6 lands 6 spells, but it is certainly very possible—it’ll happen, say, 30% of the games (completely made up number for clarification), but those 30% will be a huge blow to your game plan. Instead of killing your opponent you are playing lands you don’t need. Things might balance out eventually—maybe you’ll draw four spells in a row—but that’s not going to help you, because those spells are not good in the late game, they are 2/2s for two, they’ll all get blanked by your opponent’s bigger guys.

Now, take the control deck… if it draws 6-6 early, that’s still bad—but it’s not nearly as bad, because the game is supposed to go longer anyway. The game will get to turn 17, and by then, well, maybe you haven’t drawn 12-12—maybe there was a sequence of spells on the top of your deck after all, and you can actually use them. If you draw 30 cards, it’ll almost certainly not be 15-15. With a slower deck, you can basically work with more scenarios—imagine a situation where the top of your deck is five lands and five spells. If they’re arranged in that order, then control might live to see the five spells, but aggro might not, it needs a better distribution—say, a spell, two lands, two spells.

By virtue of playing a control deck, you do not magically remove the times where you flood out, but you make sure you get to a point where you can draw your spells once variance evens out and those spells still matter. That’s an advantage control decks have had over aggro decks since the dawn of time, and it’s not going to change anytime soon—in Constructed we can see the same thing with the all-in aggro decks such as Tempered Steel and Kuldotha Red, where if you flood out and draw seven lands, it doesn’t matter you’re drawing a sequence of [card]Signal Pest[/card], [card]Ornithopter[/card], [card]Memnite[/card], [card]Memnite[/card] later on, because later on those cards do nothing. In general, I try to stay away from those decks—I mean, look at the cards you’re playing, a 1/1 for zero.

You could, of course, argue that the control deck is also susceptible to early variance because, if it stalls, then it might not even survive to get to the late game. That is true to an extent, but control decks have a lot more leeway to stall as a general rule, because they have 20 life to work with; if you draw two extra 2/2s where I drew two extra lands, then I still have five turns to give luck a chance to even out before I die, you’re still not killing me that quickly.

Imagine a hand of bear, bear, 5 lands (in an aggro deck) versus a hand of 6 lands, [card]Shock[/card] (in a control deck). Even though the bear hand has more action and is better (though both are very bad), I’d certainly bet on the control player in this game. By the same reasoning, card draw will help reduce the impact of variance by giving you a bigger sample size. If you drew seven lands and a spell, and that spell is [card]Harmonize[/card], well, you’ll probably find more spells. If you drew only two lands but one of your spells was [card]Sign in Blood[/card], you can find more lands.

Variance does not only come in terms of lands and spells, of course. If half of your deck does something and the other half does something else, then you have to draw the right half at the right time. Playing a deck with four [card]Duress[/card]es and four [card]Mutilate[/card]s is asking to be beaten by variance from time to time, because you will draw [card]Duress[/card]es against aggro and [card]Mutilate[/card]s against control.

If those cards are [card]Kird Ape[/card]s and [card]Loam Lion[/card]s instead, well, I will draw [card]Kird Ape[/card] sometimes and [card]Loam Lion[/card] the others, but big deal, they serve the same purpose. In this regard, aggro has a big advantage, because it is more united towards a single goal whereas control has to play disparate answers.

Being consistent does not, however, necessarily mean you reduce variance, because you might increase it for your opponent, which in turn increases it for you. Take Dredge—Dredge is an incredibly consistent deck, it does the same thing every game. Manaless Dredge is even more so. But, when you play Dredge, you make the game about their sideboard, much like Belcher makes the game about [card]Force of Will[/card]—you don’t have any variance yourself, you’re unaffected by it on your side, but you are making the game all about their variance. They will draw [card]Leyline of the Void[/card] half the games, and in those, you will simply lose. If you are a good player, I don’t see why you’d want to subject yourself to that—you’d much rather be in a position where you can handle any card they draw and the game is not defined by luck alone.

Another way to fight variance blowouts is card selection, and cards like [card]Ponder[/card] and [card]Preordain[/card] make it so that you draw what you want, rather than what is on the top of your deck—you’re effectively cheating variance, and reducing those times where you’ll draw the stuff you don’t want. There were three lands on top when I [card]Ponder[/card]ed? How about two [card]Gut Shot[/card]s and a land when I have no target? Well, that was unlucky, but they’re gone now. If you have three lands on top and you can’t [card]Ponder[/card], you’re stuck drawing those three lands.

Incidentally, one of the reasons Delver is so good is that it has those variance-cheating elements, while retaining many aspects of an aggro deck. Sure, [card]Wild Nacatl[/card] decks are powerful, but their powerful plays of a three-power guy on turn one are somewhat countered by the times where they flood out and then draw Nacatl on turn ten, when it’s no longer useful.

With Delver, you get the same [card]Wild Nacatl[/card] draws, but way less of the flood draws—after all, if you don’t want a Delver on turn ten, you have the tools to not draw it. You draw a second [card]Mana Leak[/card] early on in the form of [card]Snapcaster Mage[/card], but you’re not stuck drawing it later on, because that useless “[card]Mana Leak[/card]” becomes a [card]Vapor Snag[/card] instead. You also naturally flood less since you can afford to play less lands, due to all the filtering and cantripping.

In the end, you should strive to find balance—reduce variance and go for consistency, but only to the point where your win % is benefited by it. When I asked Jon Finkel in an interview what he thought was the most important thing in a deck, he replied:

“The answer is, has a good win %. The rest is all very deck- and metagame-dependent. I’ll play a deck with a poor sideboard if it wins a lot. Don’t mind auto-losses if it mostly wins. It’s whatever combination of the factors you mention produce the best win rate. I know it’s a kind of non-answer, but by just factoring one or two things to the exclusion of others you’re really costing yourself. That’s how you end up playing the same deck all the time.”

I agree with that. I think being consistent and reducing variance will, in general, lead to the most win %—but sometimes it won’t and then you should just accept it. Take, for example, a deck that is 25 Swamps, 35 [card]Relentless Rats[/card]—it’s very consistent, it’s going to be affected by variance very little, but it’s going to be consistently bad, and that’s useless. I’ve played decks like Tempered Steel, which are incredibly affected by luck, because overall the deck simply won more than other decks, despite being affected by luck so much. I don’t want to go into a tournament flipping a 55% coin, because I think I can win more than that if I just play. But give me an 80% coin and I’ll certainly take it, even if it disgusts me.

You can also reduce variance inside the games, by not making all-in plays or plays that will make the outcome of the match hedge on whether your opponent has drawn a card or not, or on whether you’re going to draw a card or not. If you think you’re ahead, then avoid making those plays – you don’t need them.

Yesterday, I was watching a SCG match where a player had three 1/1 infect guys, and the other a 3/3 [card]Phantasmal Image[/card] and two open mana. The infect guy had two alternatives—he could attack and [card]Titanic Growth[/card] an unblocked guy, killing his opponent if he didn’t do anything, or he could [card]Titanic Growth[/card] the Image and attack for only three, giving his opponent an extra turn but protecting himself from [card]Mana Leak[/card] and [card]Doom Blade[/card].

He chose the second option, and his opponent did in fact have [card]Doom Blade[/card], so he’d have been in a horrible position with the first play. Here, the person had the choice to make the game about a card (the [card]Doom Blade[/card]), or not. He (correctly) chose not to, because he doesn’t have to. Even if he doesn’t think the opponent is all that likely to have the [card]Doom Blade[/card], there is no reason for him to risk the entire game based on that, when he can actually beat the card (and the lack of it), just by making the safer play.

Who has never been extremely frustrated that their opponents drew that third [card]Vapor Snag[/card]? Sometimes you couldn’t have done anything, but in other situations you actually put yourself in a position where the game result depended on that draw step. Sure, third [card]Vapor Snag[/card] is unlikely, but if you can prevent that from killing you by keeping one of your guys back, why not do it? Why can’t you wait three more minutes to kill your opponent, are you in such a hurry to eat the horrible concession stand food? Again, you’re not stopping luck here—he will draw the third [card]Vapor Snag[/card] sometimes and there is nothing you can do about it—but you can reduce the effect this has on the game. When you attack and leave only one blocker, YOU are choosing that the result of the game is going to depend on that one draw step, YOU are increasing the impact luck is going to have on the game.

“But, PV… what if I am not good? What if the expected result is that I’m going to lose?”

You’ve been thinking about this since the opening paragraph, haven’t you? Don’t worry, it’s OK. If you want to reduce variance when you’re the favorite, then it follows logically that you want to increase it when you’re not. The problem with this approach is that most steps people take to increase variance are actually harmful in the long run by way more than they help your win percentage—again, it’s a trade-off.

Say you’re playing against LSV—you’re not a favorite to win, he is. You could decide to side out 10 lands for 10 spells to increase the luck factor in the game, and you would be doing exactly that, but, even though you’d have a small gain on the amount of luck involved, you’d have a huge net loss on the fact that your deck is now almost unplayable. The whole “I’ll keep this one-land-[card emrakul, the aeons torn]Emrakul[/card]-hand-because-I-can’t-win-with-a-normal-draw,” is also a big mistake, not to mention the “he’ll never expect thisplay!”—if he’ll never expect it it’s for a reason. Maybe you do something horrible and you didn’t even have a bad matchup to begin with!

The moment you should increase variance is not when you’re playing against someone better, because it’s very hard to do that and have it be profitable—the great majority of the time you’ll just make things even better for them. No, the time to do that is when you’re losing on board, and luck is simply your best chance to win, not before the game even started.

Let’s go back to the infect scenario. Imagine that he knows his opponent is going to untap and play [card gideon jura]Gideon[/card]—now, suddenly the game is not looking so good for you. As opposed to shying away from the result being decided on the presence of a single card, you actually welcome it. If that doesn’t happen then you will lose. In this case, your options are: a) whether he has this card or not will decide the outcome of the game and b) I lose. Obviously option A is better.

Increasing variance can be beneficial when you, for example, splash something in Limited. Obviously you’re making luck play a greater role—you’ve just added a spell to your deck that is going to be uncastable a significant portion of the time, while also adding lands that might make every other spell in your deck uncastable.

When should you do it? When you think that you need to. If your deck is extremely good, then there is no reason for you to give luck this much power in the game—just strive for consistency.

Well, that is about it. To sum it up:

• If you’re better, then you want to diminish the impact that luck and variance have on the match. There are some ways to do that:

Play a lot of matches to dilute luck.
Don’t play decks that naturally make the outcome of a match depend on one card.
Try to avoid completely situational cards.
Try to have some card selection and/or card drawing.
Don’t make plays that place the outcome of the match on a single factor.

If you feel you do not have the edge, then you can make risky plays. But make sure you actually don’t have the edge, and do not make crazy decisions in the hopes of randomly increasing variance, because it’ll generally do more harm than good.

I hope you’ve enjoyed this, see you at GenCon!


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