Var(MTG)≤ 1/4

Published: 05/06/2020

As a life time supporter of 'funner', I'm not usually a linguistic perscriptivist. Ever since I read an article in Scientific American on the carpool to middle school about the inverse correlation between the probability a verb conjugates regularly and its frequency of use I've understood that language is more or less something we've agreed upon. My position somewhat softened when I read Tense Present at a friend's recommendation on my trip to GP Cleveland. And now I feel I must speak out against 'Variance' as it is used in the Magic the Gathering community. The platitude "Magic is a high variance game" is pervasive. Like other MtGisms, it is borrowed from Poker; tilt being the canonical example. But unlike poker, Magic is not a high variance game! Magic is a game with a binary outcome. A match is won or lost. I've never lost game 1 so hard that I automatically lost the match. And I've never lost a match so hard that I've been jettisoned from a tournament. Hands of Poker are fundamentally different in that they have a huge spread of outcomes ranging from a small blind changing hands to whole stacks being lost.

Variance and Expected value (EV) have precise mathematical definitions. In some respects they are nice mathematical definitions because they have nice mathematical properties, like linearity across independent random variables and linearity across all random variables respectively. But the real reason they are great mathematical definitions is because they help answer real questions about the world like "what price should stocks be" and "should I make this bet".

Let me anticipate some criticism here. Maybe the reader thinks to themselves "well sure you mathematicians have some technical and precise meaning of the word variance but gamblers got there first. Everyone knows what I mean when I say variance. In particular my notion doesn't have an implicit concept of a random variable which is being described. Do you also pedantically complain about how people use words like: line, curve etc.". To which I respond: when you say magic is a high variance game after someone gets unlucky why don't you just say "How unlucky"? The whole point of invoking variance is to sound like you're saying something technical and important when you're really complaining about luck.

A random variable is just some variable which depends (or not. A random variable can be constant) on a random event. You don't know what the event will be but you do know what it could be and the possibility's probabilities.

Example 1: The card you draw during a draw step is a random variable valued in the set of magic card, which I will denote, \(M_{tg}\). Your opening hand is a random variable valued in \(M_{tg}^7\), the set of seven magic cards.

If we want to compute the expected value of our random variables we'll need them to be numbers.

Example 2: The outcome of a die roll is a random variable which is 1-6. The match points from a game of MtG is random variable either 0, 1 or 3. For simplicity let's just say its 0 or 1.

The expected value of a random variable, X, is given by,

\[\mathbb{E}[X] = \sum_{x} \operatorname{P}(X = x) x.\]

Where the sum is over all \(x\) that our \(X\) might equal.

Example 2 continued: In a die roll all six outcomes are equally likely so the expected value is \((1+2+3+4+5+6)/6 = 3.5\). If your win probability in a match of mtg is \(p\) then your EV is just \(p\). This is a general property of indicator random variables. That is random variables that are \(1\) when some event happens otherwise \(0\).

The variance of the random variable is given by,

\[\operatorname{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2].\]

That is the variance of a random variable is the variable's average squared distance from its average. If your random variable is a constant, than its variance is \(0\).

Example 2 continued: If a random variable is 1 with probability \(p\), otherwise \(0\) then the variance is, \[p(1-p)^2 + (1-p)p^2 = p(1-p)(1-p+p) = p(1-p).\] So we get a quadratic with maximum at \(p = 1/2\) and roots at \(0\) and \(1\). That makes sense. The more weighted your coin the more confidently you can predict how it will land so the lower any good notion of variance should be. Notice that the variance is a simple function of win probability. So there is no mathematical difference from saying a match of mtg has high variance and saying a player can not achieve a win percentage far away from 1/2.

A hand of magic has a binary outcome but poker, where variance was complained about first, is different. In poker everyone can lose or win your whole stack every hand. But most hands just a few multiples of the big blind exchange hands.

I play casually with my friends sometimes. We're pretty weak players (except one player who dazzles us by accusing us of donk betting from time to time, whatever that means). The client we play with keeps track of the number of hands we've won with a little trophy count above our stacks. And something that's apparent on most nights is an inverse relation appearing between the number of hands someone wins and their stack. This isn't a surprising phenomena. The players who play tightest are the most likely to win the hand when they pay to see a flop. It doesn't cost a lot of money to sit at a table waiting for good cards. A corollary of this is over just a few hands the player who is up money is probably not the strongest player. In our weak group I think >90% of the money is exchanged in <10% of the hands.

One might complain here that its unfair to compare big blinds and match wins. Really people care about cashing tournaments anyway. Units are important. If you think one chess match win is worth a thousand big blinds than wouldn't that mean mtg was higher variance than poker?

There are some important corollaries of mtg being low variance in my sense though. In mtg you don't have to play that many hands to know your win rate. People say the opposite of this all the time. That no one has the sample size to say anything. But the problem is rarely that samples are small but that they are unrepresentative. No number of wins against my friend can tell me what my win rate on the MTGO Leagues will be. No number of games on the leagues will tell me what my GP win rate should be. The first 10 rounds of the GP don't tell me my odds of winning round 11 because there is a strong selection effect on that opponent.

In MTG if you win 10 games in a row you can be reasonably confident your true win rate is >1/2. And if you lose 10 games in a row you can be reasonably confident your true win rate is <1/2 (In the context of the environment you did your winning and losing in). Poker is different. Having 10s, Jacks, Queens or Kings is pretty rare. An opponent having a higher pocket pair in the same hand is even rarer. But what happens in those rare hands can be so catastrophic that it negates your winnings in all other hands. That doesn't happen in MTG. If I don't mulligan 1 landers that can only reduce my win rate by the probability that I draw a 1 lander. If there's one spot in poker where I am confident I have the strongest hand but I don't, the result can be arbitrarily bad.