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By: CZML, Cassie Mulholland-London
Sep 29 2014 12:00pm
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 Hello, and welcome back to The Perfect Game. Today's article is going to be a big one.


In your time playing the game, you have probably heard at least a little bit about Magic theory—use your life total as a resource, card advantage is important, et cetera. But do you know why those axioms are true?


Today, I'm going to break down the two most fundamental concepts in any strategy game—Expected Value (EV) and Opportunity Cost—and lay the groundwork for my next few articles, which will introduce and expand upon my own personal Unified Theory of Magic. It is innovative in that to my knowledge no Magic writer or theorist has put things in exactly the same terms, and I am confident that you will gain something from it. At the very least, this series of articles will cause you to seriously reconsider your preconceived notions about how Magic: the Gathering really works.


Before we get into the definitions, I want to explicitly state a few assumptions about competitive Magic (and most games in general). This may seem redundant, but I want to clarify the parameters under which I am operating.


This article series assumes that we make every decision with the intent to win the game, and that we believe—to the largest extent possible—that the decision we end up making gives us the best chance to win the game.


Now that that's been established, let's talk about the first fundamental gaming concept: Expected Value. If you play poker, have been playing Magic for a while, or remember basic probability from high school math, you probably have a solid grasp of what Expected Value is, but I'm going to provide an explanation for those of you who aren't familiar with the concept.


Expected Value is the average value you will get out of an event given an infinite number of iterations. Essentially, if the same thing happens the same way an infinite number of times, the average result of that thing is your Expected Value. For example, say every day you flip a coin. If the coin comes up heads, you get $10. If the coin comes up tails, you lose $10. Your EV on any given coin flip is exactly $0, because half the time your value will be +10 and half the time it will be -10.


The way you calculate Expected Value is simple. Just take the chance of each outcome occurring (as a decimal out of 1) and multiply it by the value you get if that outcome occurs. For example, say you're trying to figure out whether to choose between a guaranteed $1,000,000 and a 50% chance at $5,000,000. In this case, you want to calculate the EV of each choice and compare them. The first choice is 100% likely, so its EV is 1 x 1,000,000, or a straight 1,000,000. The second choice is 50% likely if you choose it, so its EV is 0.5 x 5,000,000, which is 2,500,000. The second choice has higher EV.




“But Casper,” you say, “why the math lesson? How does this relate to Magic: the Gathering?” Well, Expected Value is inherently a mathematical concept, so a small bit of probability is necessary to explain it. Then again, Magic is a very mathematical game, and a strong grasp of the fundamentals of probability is helpful in many ways. But Expected Value is particularly necessary to understand when you're comparing two plays. If you have multiple lines of play—one that loses to what you think is a 2-of in your opponent's deck, and one that doesn't lose outright but puts you at a significant disadvantage to a 4-of, Expected Value lets you find the optimal play. In addition, Expected Value shines when making mulligan decisions if you have a relatively strong grasp of your chances to win with your current hand versus your chances to win with a variety of 6 or 5 card hands. There are obviously a lot of variables in these scenarios, but EV usually gives you a relatively solid idea nonetheless.


Most Magic players make judgments based on estimated Expected Value without having to do the math, but if you're fast enough at arithmetic or your opponent is also going into the tank for a bit, doing the calculations can be very beneficial. In addition, being able to calculate EV (or at least having a strong grasp of it) will help you with other games and even some financial decisions.


Something I should address is that Expected Value is different from Actual Value. If you make a play in Magic that you estimate has an Expected Value of .8 (80% chance to win the game), what that really means is that approximately 80% of the time that play has an EV of 1 (a win) and approximately 20% of the time that play has an EV of 0 (a loss). One in five times, you are still going to lose that game, and that's okay. It's the same thing with taking a mulligan; you should never look at your top cards, because it doesn't matter if you actually would have gotten there or not. What matters is that the Expected Value pointed towards you having a better chance if you mulligan.


One of the most important things to remember with Magic (and any game of variance or imperfect information) is that the right play does not always win you the game. That does not make it any less correct, however.


Opportunity Cost is fairly self-explanatory, and closely related to Expected Value. In fact, one way of thinking about Opportunity Cost is that it is the interaction between the Expected Values of two decisions. Simply put, the Opportunity Cost of making a given play is that you don't get to make some number of other plays you were considering. If the play you end up choosing has a higher Expected Value, then the Opportunity Cost is worth it. If you miss something and miscalculate the Expected Value, or if things change in a way you didn't expect, then the Opportunity Cost can end up punishing you fairly badly.


A very straightforward example of Opportunity Cost is in modal Magic cards like all of the Charms and the Command cycle form Lorwyn. If you use one (or two) modes, you lose access to all of the others. This is completely fine if you choose the optimal configuration, but if not then having the options can be detrimental.


Opportunity Cost is usually a good thing, because it means you have options, and the more options you have the more likely you are to have the best possible line (more on that in my next article). But sometimes, it can keep you from committing to the correct line because you're worried about things changing or being wrong. This is where it helps to remember what I mentioned above: focusing on Expected Value is better than dwelling on Actual Value. As Jon Finkel put it, “the right play is the right play regardless of outcomes.”


So how can you use Opportunity Cost to your advantage? Well, if you keep in mind that committing to a single line has a price, you will automatically start committing as late as possible. In many situations, it's important to wait to play your spells and even lands until your postcombat main phase, both to possibly bluff a combat trick but also to see how combat plays out so you know which land and spell to play. Holding an instant until the last possible moment is another good illustration of this.


Thanks for reading! This might seem a bit dry to some of you and a bit basic to others, but don't worry; next week we'll get into some more interesting and innovative stuff. I've found something that I think is extremely important and I can't wait to share it with all of you.