I’ve heard it all.

“I chose more of a high-variance deck because Top 8 is all that matters to me for this tournament, so I need to gamble.”

“This deck is inconsistent, so while it can spike a 4-0 at an MTGO Daily, it will never hold up over 15 rounds at a GP. Don’t play it if you hope to go 13-2 or better.”

“To maximize my chances of winning the tournament, I chose a deck whose win percentage against the metagame has a low variance.”

“I’m not sure how good this deck is and haven’t tested it enough, but I’m playing it on the off-chance that it is broken.”

“It’s better take a 50% matchup to 60% than to take a 10% matchup to 20% because to go undefeated, I’ll have to dodge the bad matchup anyway.”

Is there any truth or logic to these statements? And when we are talking about variance in a Magic context, which random variable might we even be considering? It can be a confusing topic, so in this article, I aim to shed some light on it.

Variance in the Power of an Opening Hand

In my experience, most of the time someone calls a deck “high-variance,” they mean that their deck can either have an amazing draw or a terrible one. In other words, there is a lot of variance in the strength of hands that a deck can get. For example, a Modern reanimator deck could win on turn one with an opening hand like Simian Spirit Guide, Simian Spirit Guide, Faithless Looting, Griselbrand, Goryo’s Vengeance, Fury of the Horde, and Swamp, but it could also fail to assemble its pieces and lose without doing anything at all. Another example is a Standard U/W Heroic deck that can have unbeatable opening hands containing a perfect combination of lands, a heroic creature, pump spells, and a protection spell, but that will also frequently offer horrendous opening hands with only creatures or only pump spells. In contrast, a midrange deck like Jund in Modern has very few incredible or horrendous opening hands—most of its draws contain a reasonable amount of answers and threats, and there is little variance in the overall power level of its opening hands.

We may formalize this random variable for a certain deck as the probability that an arbitrary opening hand will win the game. We’ll abbreviate it as P. For a combo deck, its P might have the value 0.8 with a 40% chance and the value 0.3 with 60% chance. For a Jund deck, its P might have the value 0.5 with 100% chance. Clearly, the variance of P is higher for the combo deck than for the Jund deck, even if their expected values are the same.

 

In all fairness, ascribing a win probability of 0.3 to this opening hand may be overly generous, but that’s what I get for using illustrative numbers.

Now, how should this particular variance affect your deck choice? If all you care about is achieving a very high finish (say, winning the tournament) then should you gamble it all on a deck with a high-variance P? And if all you care about is achieving a reasonable finish (say, a money finish) then should you go for consistency and pick a deck with a low-variance P?

The answer is: It doesn’t matter.

Ultimately, the probability of winning an arbitrary match is unaffected by the variance in the deck’s P—in fact, it is completely defined by the expected value of P. Moreover, the outcome of a certain match does not depend on the results in previous rounds. So, if their P’s are the same, then the probability of going 2-0, 13-2, or any other record is the same for the Jund deck and the combo deck. Mathematically, comparing the probability of going 2-0 with Jund or combo, you can see that 0.5² = 0.4²∙0.8² + 0.6²∙0.3² + 0.4∙0.6∙0.8∙0.3 + 0.6∙0.4∙0.3∙0.8.

If you find it tough to understand this intuitively, then allow me to explain with an analogy. Consider two options for playing roulette in a casino. In the first option, a croupier spins a ball across the wheel and it will fall in a red pocket with 50% chance and in a black pocket with 50% chance. This is the “usual” option, and it is analogous to the Jund deck. In the second option, the croupier first throws a 10-sided die and chooses between two roulette tables based on the outcome of that roll: If it’s a 1, 2, 3, or 4, then the croupier will spin a ball on a table with 40 red pockets and 10 black pockets, and if the dice came up on a 5, 6, 7, 8, 9, or 10, then the croupier will move to a table with 15 red pockets and 35 black pockets. This is a more convoluted option, and it is analogous to the combo deck.

 

Spin the wheel!

If in the casino you can only bet on black or red, then it doesn’t matter which of the two roulette options you pick: Both give you a 50% chance of red and a 50% chance of black. The history doesn’t matter, and the variance doesn’t matter. Even if the previous three spins featured a die roll of 1-4 followed by a red pocket, then the next spin is neither more likely to land on red once again nor due for the table with more black pockets. It’s the same with Magic decks.

So, provided they have identical expected Ps, a deck with high-variance opening hand strength is just as likely to finish at any record as a deck with low-variance opening hand strength. The only thing that matters is the expected match win percentage against the metagame. But that’s a random variable whose variance does matter.

Variance in the Match Win Percentage Against the Metagame

In many tournaments, prize payouts are top-heavy. This means that an infrequent first-place finish followed by a string of last-place finishes is far better than a stable diet of mediocre money finishes.

To make things concrete, suppose that you enter a three-round tournament that only gives out prizes to players who finish 3-0. Suppose that you can choose between two decks. For the first deck, the expected match win percentage against the metagame (expected MWP) is 55%. For the second deck, the expected MWP is 60%. Common sense would dictate that you’d take the second one. But this need not be the correct answer.

The thing is, the MWP is a random variable and its variance matters, too. There are two key types of uncertainty at work. I’ll illustrate both with an example.

The first type of uncertainty is that you don’t really know the true match win percentage against any opposing deck. You can specify some a priori estimates and you may have some results from a playtest session, but unless you’ve tested billions of games, you won’t know the true value.

To make an analogy, suppose someone gives you a weighted coin. Nature knows that it will land on heads 70% of the time, but we don’t know that yet. You may be able to hold it in your hand and, based on how the weight feels, give a probabilistic estimate of how often it would land on heads. You may also be able to toss it a number of times and record how often it lands on heads. But if you estimate that the heads probability is distributed uniformly between 60% and 80% and subsequently record 68 heads and 32 tails, then you still won’t be sure about the true probability of hitting heads. The probability of winning a certain matchup is akin to this probability of landing on heads: There is a true number out there in nature, but we don’t know what it is and can only get close via estimates and experimentation.

Let’s look at a numerical example. For simplicity, suppose that we are absolutely certain that every single person at the tournament will play the exact same deck, so you know what you’re up against. You consider Deck A and Deck B—both are new concoctions unlike anything you’ve ever seen, and you’re not sure how good they’re going to be. They could be incredibly good or incredibly bad. For concreteness, suppose that for either deck, you use expert judgment of the deck list to estimate an a priori probability distribution (rather than a density on [0,1] to keep things simple) that is described as follows:

• There is a 40% chance that the MWP is 0.2
• There is a 20% chance that the MWP is 0.6
• There is a 40% chance that the MWP is 0.8

With Deck A, you test only 20 matches, winning 11 and losing 9. With Deck B, you have enough time on your hands to test 1000 matches, winning 600 matches and losing 400 matches. With this new data, I’ll use Bayesian updating to get an updated distribution. In the following tables, the probability of getting a certain record, given an MWP, is obtained naturally from the Binomial distribution. The updated probability is proportional to the product of the second and third column, scaled by a constant factor so that the probabilities sum to one. Hope I did it correctly…

mwp table 1 mwptable2

Given the updated distribution, the weighted probability of going 3-0 for Deck A is 0.008 ∙ 0.5% + 0.216 ∙ 91.1% + 0.512 ∙ 8.4% = 24.0%. By contrast, for Deck B, it’s 21.6%. So even though you only won 55% of your test-games with Deck A compared to 60% of your test-games with Deck B, it is still better to go with Deck A!

The intuitive explanation is that, because you only tested a limited amount of games, there is still a chance that the true MWP for Deck A is 0.8 (in which case the probability of going 3-0 is very high) whereas this possibility is effectively ruled out for Deck B because you tested so many games with it. Now, a big part of this effect comes down to the particular values chosen, but the general insight is that if you haven’t tested sufficient games to rule out the possibility that the deck is insanely good, then it could be wise to play that deck.

 

I deserve a peanut after making these tables.

So far we tackled a situation where the metagame consisted of one opposing deck and the uncertainty lied in not knowing the true match win percentage against that deck. Recall, however, that we were initially interested in the random variable describing the expected match win percentage against the metagame.

The second type of uncertainty underlying this random variable is that you don’t really know the true metagame distribution. The metagame distribution describes the fraction of decks that you expect to play against at an event, for example 50% Abzan, 30% Sultai, and 20% Jeskai. However, once again these numbers are randomly determined and possibly correlated. For instance, the aforementioned metagame expectation might arise if you guess that there is a probability of 1/3 that the metagame will be 90% Abzan, 10% Sultai, and 0% Jeskai and a probability of 2/3 that the metagame will be 30% Abzan, 40% Sultai, and 30% Jeskai. In this scenario, you are very uncertain about the metagame distribution that you will face, and then it can be beneficial to choose a deck with lopsided matchups.

I’ll explain with a numerical illustration. Take the above-described metagame guess as a given. Suppose that you can choose between a risky Deck A and a solid-all-around Deck B. Deck A has a match win probability of 0.90 against Abzan and 0.20 against Sultai and Jeskai, making for an expected win probability of 0.55 against the expected metagame. Deck B has a match win probability of 0.60 against Abzan, Sultai, and Jeskai, making for an expected win probability of, well, 0.60 against the expected metagame.

Yet, if all you care about is going 3-0, then you have to look beyond the expected metagame—you need to take into account the uncertainty about the metagame distribution, too. Intuitively, if you choose Deck A, then you might luck out and find yourself in a field with 90% Abzan, in which case your chances of going 3-0 are super high. Numerically, the calculation (assuming that the metagame distribution stays the same across every round, which need not be true in reality, but the incorporation of that effect in a more complete model lies far beyond the scope of this article) is as follows:

• With Deck A: The probability of going 3-0 is 1/3 ∙ (0.9 ∙ 0.9 + 0.2 ∙ 0.1 + 0.2 ∙ 0.0)3 + 2/3 ∙ (0.9 ∙ 0.3 + 0.2 ∙ 0.4 + 0.2 ∙ 0.3)³ = 0.237.
• With Deck B: The probability of going 3-0 is 1/3 ∙ (0.6 ∙ 0.9 + 0.6 ∙ 0.1 + 0.6 ∙ 0.0)3 + 2/3 ∙ (0.6 ∙ 0.3 + 0.6 ∙ 0.4 + 0.6 ∙ 0.3)³ = 0.216.

Indeed, Deck A is better! It can pay off to take the deck whose MWP has lower expectation, yet higher variance. Once again, this is based on the specific numbers chosen, but it’s interesting to see that this can happen.

Conclusion

I’m aware that things can get horribly confusing when venturing into probability distributions over probabilities, but those things were needed to describe my view on variance pertaining to deck choices in Magic. To wrap it up, let me revisit the questions I posed in the beginning and provide my answers to them.

• I chose more of a high-variance deck because Top 8 is all that matters to me for this tournament, so I need to gamble.

If you mean a deck with more variance in its opening hand strength, then that’s completely irrelevant. You’re spinning a nonsensical narrative.

• This deck is inconsistent, so while it can spike a 4-0 at an MTGO daily, it will never hold up over 15 rounds at a GP. Don’t play it if you hope to go 13-2 or better.

If this inconsistency would pertain to a deck with more variance in its probability that an arbitrary opening hand will win the game, then once again that doesn’t affect your chances of getting a certain record—if the metagame and matchup percentages are known and fixed, then only the expected probability that an arbitrary opening hand will win the game can affect this. If this inconsistency would pertain to lopsided matchups against the metagame coupled with a large degree of uncertainty regarding the metagame from tournament to tournament, then there might be some truth to it, but please define your terms less ambiguously.

• To maximize my chances of winning the tournament, I chose a deck whose win percentage against the metagame has a low variance.

Generally, it will be the other way around: a high variance in your win percentage against the metagame, either because you don’t know the true matchup percentages or the metagame distribution, often leads to more lopsided records.

• I’m not sure how good this deck is and haven’t tested it enough, but I’m playing it on the off-chance that it is broken.

This could make sense. Just don’t sacrifice too much of your expected win percentage against the metagame by picking the riskier deck. Also, consider the feeling you got from your playtest session—the numbers are nice, but they don’t say everything.

It’s better take a 50% matchup to 60% than to take a 10% matchup to 20% because to go undefeated, I’ll have to dodge the bad matchup anyway.

I didn’t really analyze this particular case, but if you suppose that the metagame and matchup percentages are known and fixed and that the 50% matchup is just as popular as the 10% matchup, then it doesn’t matter because both yield the same increase in the expected win percentage against the metagame in every round. I usually don’t buy the narrative regarding giving up a bad matchup. I admit that the dodge strategy might make sense if you expect that deck to have dissipated at the top tables, but then we venture into more complicated situations where the metagame distribution doesn’t remain constant over records.

Thanks for reading, and Happy New Year!