Magic Math And Why I’m Against It

Just how useful are Magic Math articles, anyway? Andrew Elenbogen highlights its limitations and argues that there’s no substitute for old-fashioned playtesting.

In the wake of the London Mulligan introduction, there has been a substantial increase in the number of Magic articles that “run the numbers,” so to speak. These articles purport to answer questions like how reliably Dredge can find two lands, a Dredger, and a discard outlet; or how many spells are needed to play Augur of Bolas. I believe these articles, for the most part, provide easy answers to complex questions, allowing players to think they know things that they could not possibly know. As a result, I have long believed such calculations are of relatively little value to a ruthless competitive player looking only to maximize his or her win percentage.

What This Article Isn’t

Before I get into the meat of this topic, I want to be totally clear about something: this article is attacking an idea, not any person. I believe that many of us in the Magic community are math-oriented. A significant proportion of players I have interacted with work in some STEM-related field, and even some of those that don’t find math fun in its own right. As a result, many people in the Magic community find game-related calculations to be enjoyable thought experiments. I have personally found many articles on such topics enjoyable and am friends with people who have written them.

Nonetheless, I contend that these articles, while fun to think about, don’t serve to noticeably improve one’s results.

A Short Definition of Magic Math

I want to be very clear about the concept that I will refer to as “Magic Math.” Magic Math is content involving numbers or calculations using estimation or assumptions, whether explicit or implicit. We’ll see that, while perhaps reasonable at first, these pieces require substantial contextual justification that simply isn’t feasible.

Many players, when considering whether to mulligan, will estimate how likely their hand is to work out and how likely it is to win if it does. I think this is generally a fairly good process, and the reason for this is that said process is grounded in context. Players who do this know their particular deck, hand, and matchup, and should factor all of it into their decision. For instance, if you are playing an awful matchup for your deck, you should be much more willing to keep excellent hands that are unlikely to work out, since your average draw is not going to beat them anyway.

Everything Is Contextual

This brings us to the case of Augur of Bolas. Augur of Bolas will, as of the release of War of the Spark, be legal in Standard again. This card has seen play in the past and has a good chance of seeing play again. As a result, the question on every deckbuilder’s mind is this:

What spell count do I need to justify the inclusion of Augur?

The answer, as you may have guessed, is that it depends. Are you playing a broken combo deck that needs to assemble a specific combination of spells? Then maybe a very low rate of hitting is fine, as you mostly just want to make a chump blocker and dig three cards closer to your combo pieces. Are you playing a card-advantage-light control deck? Then you need to hit almost every time, as a single miss has a significant chance of losing you the entire game.

For the first deck, hitting on 50% or less of looks might be totally fine. For the second, finding a spell 85% of the time might not be enough. Those are drastically different rates that could never be reflected on a single chart. They are also far from the only possibilities. A tap-out control deck with an overwhelming late-game, for instance, would fall somewhere between these two extremes. The reality is that you need to understand your deck’s overall strategy and the role you want Augur to play in that strategy in order to know how often it needs to hit to be good.

Do Your Homework

Here, a Magic Math fan would reasonably suggest the following strategy: go out and play a bunch of games with your deck in order to gather the context required to interpret the formulas, and then apply them in order to translate your experiences into a required spell count or equivalent.

I agree that the primary way to gather context like this is playing the games. Much like any other playtesting, I would expect most people executing this method to try several different small changes over the course of a playtest session. As they do so, they will most likely alter whatever variable they had done their math about in the first place. But at that point, why look at the math at all? Why not simply draw conclusions based on how the various configurations performed in playtesting?

If your 27-spell Augur of Bolas deck performed better overall than the 26-spell version, then I think you have your answer. Playtesting will also provide other relevant contextual information, such as exactly how good or bad the specific 27th spell under consideration performs against the field. Even the staunchest Magic Math fan would agree that the correct decision changes depending on whether the 27th spell is Anticipate or Preordain.

In summary, math requires playtesting to be useful, but once you’re doing the playtesting, you no longer need the math.

Assumptions Using Math

The other common problem I see in Magic Math articles is with assumptions. For instance, consider calculations regarding Dredge hands under the London Mulligan rule. Writers in these sorts of articles generally determine the likelihood of finding a particular combination of cards assuming the Dredge player is willing to go to five or even four cards. Then, they use this information to deduce what hands the Dredge player should keep or mulligan. Perhaps they define a good Dredge hand as any hand with a Dredger, a discard outlet, and the lands to cast that discard outlet. Then they calculate the odds of achieving that hand if the Dredge player mulligans and use this information to gain insight into whether some borderline hand is a keep.

Methodologically, I think this process is basically sound, but its validity rests entirely on the validity of its assumptions. For example, maybe the Dredge deck should not keep hands which have no green mana, as they lock it out of casting Life from the Loam for the entire game. This is obviously a small difference, but such differences add up and often increase the overall complexity of the calculation beyond feasibility. To make matters worse, it’s impossible to know what assumptions are reasonable unless you’re already a Dredge expert. For all I know, the mulligan strategy I outlined above has large holes or does not apply to certain matchups.

The task of the Magic Math then becomes straightforward: play the deck in question enough to discern what sets of hands are really keepable so that can be included in the calculations. But, as you’ve probably guessed, if you’ve played the deck enough to have an informed opinion on these questions, you’ve most likely played it enough that you won’t need the calculations anyway. Simply mulligan according to what you’ve deduced. Staring at math will never allow you to determine if your assumptions are right or wrong, and that is the entire ball-game.

The Role for Magic Math

What I’m getting at here is this: competitive Magic is fundamentally an empirical process. But given this, I still think Magic Math can be useful in some limited sense. Primarily, it can help spark your intuition and help you articulate ideas that have been at the back of your mind throughout the testing process. This can in turn generate potential changes to try, whether a different mulligan strategy or swapping out that 60th card. Assumption-based math can also be used for estimation, although it is certainly no substitute for playtesting and the user must be aware of the inbuilt assumptions at every step.

As I alluded to earlier, math concepts are also fun to think about. At the end of the day, most of us play this game to enjoy ourselves and I see no reason why Magic content should be any different. There’s absolutely nothing wrong with writing a fun, entertaining Magic Math article if we’re all on the same page about what it is.

The Takeaway

Math done for Magic-related purposes is typically not very useful. The combination of context and assumptions frequently prevents it from being applicable. While statistics avoid the small sample problems inherent in using purely empirical data, the cost is too high. Context and making correct assumptions are critically important, and the best way to get a handle on them is to simply do your homework, eat your vegetables, and play the matches.

Drawing the correct conclusions in this way is never easy, but it is also the essence of the game and a big part of what makes Magic amazing. If your intuition is not yet up to the task, I recommend making that your next area for improvement. If you want to take your game to the next level, investing in your intuition will pay dividends for years to come.