Probability in MTG: What’s in a Win Rate? (Part 3)

The first installment of this series defined Win Rate as an unknown parameter, a number that corresponds to the expression: Pr(Deck A, Pilot B wins versus Deck C, Pilot D) or the probability that Deck A piloted by Player B wins versus Deck C piloted by Player D. It was shown that this unknown number can be estimated by the use of appropriate statistical tools, but that estimating even the simplest win rates accurately can be quite the cumbersome task.

In the second installment, we turned our attention to perhaps the most practical win rates for estimation, which are: Pr(Specific Deck, Any Player versus Any Deck, Any Player in a meta) and Pr(Specific Deck, You versus Any Deck, Any Player in a meta). While once again, estimating these numbers is not easy, it was shown that there were practicable ways to do it and that published win rates, while most likely biased, are still quite useful to the competitive MTG player.

In this third installment, I attempt to answer a question that is likely obvious to many, but is a constant source of irritation to some. The question: is luck all that matters in this game? A quick search in both the MTGA reddit sub and the MTGA forums reveals at least some people having significant amounts of vitriol on this issue.

https://forums.mtgarena.com/forums/threads/63306?page=1

https://forums.mtgarena.com/forums/threads/41735

https://old.reddit.com/r/MagicArena/comments/f535jk/luck_and_randomness_in_mtg/

https://old.reddit.com/r/MagicArena/comments/dccgl8/how_much_luck_and_skill_is_actually_involved_in_a/


Now, it is of course obvious that there is luck involved in every game of Magic. In fact, I can go so far as to say that there are some games where it truly does not matter how good a player you are in terms of determining such games’ outcomes (e.g. having to take a mulligan to 4 or never drawing a third land in 10 turns). However, as pointed out several times by level-headed people in the links provided above, Magic shares this quality of randomness with all other card games such as Poker, and so accounting for this factor when playing is, in fact, part of the game.

Instead of repeating the reasonable arguments made by those people, I propose the following argument that is framed simply around the concept of win rate. The argument is as follows: If luck was all that matters in Magic, then the probability of winning (i.e., the Win Rate) should be independent of who is playing. 

Mathematically, this means Pr(Deck A wins |Player B) = Pr(Deck A wins|Any Pilot) or the probability that Deck A wins given that Player B is piloting it is equal to the probability that Deck A wins given that any player is piloting it. Logically, if it can be shown that this equality does not hold, then it follows that luck is not all that matters. Incidentally, it is quite obvious that the two probabilities are not always equal. If a player only learned how to play Magic today and is asked to play any Tier 1 deck on the ladder, I do not think anyone would argue that this player would get the same win rate with this deck as another player on the same deck who has been playing Magic for years.

Still, the point of contention that those who continue to embrace the idea that only luck matters in the game is that the correct decisions in the game are so obvious and that only the luck of the draw matters once you know “the basics.” While this is somewhat true, and indeed, when pitting one pro player versus the other, it is more likely that luck rather than skill will decide the game (as neither is likely to make a mistake), many people who embrace this idea do so while grossly underestimating what it means to “know the basics.” Sadly, being preoccupied by this notion makes it that much more difficult to self-evaluate one’s plays properly, such that one would typically attribute losses to bad hands or mana drought, or whatever randomness happened to occur in the game without realizing that he could have won despite of these occurrences had he just played better. This is, again, not to say that there are games where a player would have lost regardless of what he did; there certainly are plenty of games like that, but there are also many games where a mistake or two did cost the player the game, but the player was too preoccupied with not having made his third land drop on turn 3 to realize it, fixating instead on this random element (being late with the third land drop) as the reason why he lost.

Another way to look at win rate as a means for showing that luck is not all that matters is by considering the special MTGA Events where the possible decks that can be selected are fixed. In particular, consider the MTGA Release After Party event from a few months back where players got to play with some of Magic’s best cards (Black Lotus! Moxen! Sol Ring!) by choosing one of two possible preconstructed decks, Elspeth or Ashiok. Consider the two win rates:

Pr(Ashiok deck wins |Pilot A)

Pr(Elspeth deck wins|Pilot A)

This situation is convenient because there are only two possible decks. Furthermore, the matches are fixed such that if you play the Elspeth deck, you need to be paired versus a player who chose the Ashiok deck (which led to relatively higher queue times for the Ashiok deck). This thus provides a uniquely tidy setting to observe if luck is all that matters or not. If luck is all that matters, then it would mean that the win rates of the two decks are independent of the pilot. Therefore, if we also assume that correct decisions in the game are so easy to make, then the only factor that one has control over which affects one’s win rate would be whether or not one makes the correct deck choice. A consequence of this, given that there are only two decks, is that the true win rate of each deck must sum up to one. Another way of looking at this is that if the pilot is not a factor, then what remains to be the factor is the deck. So, if the Ashiok deck is stronger then its true win rate will be higher than 0.5 regardless of pilot, but since the Ashiok deck is always matched against an Elspeth deck, it would mean that the Elspeth deck would have a win rate that is lower than 0.5. It is of course, also possible for the two decks to have equal win rates, in which case that would mean that both their win rates would be close to 0.5. MTGA was generous enough to actually provide win rate numbers for the event in a tweet, where they said that the Elspeth deck is slightly advantaged with a win rate of 51.6% (based on what can be safely assumed as a very large number of games). Thus, if indeed luck is all that matters, then for any player who plays enough games with the Elspeth deck, their win rate should converge to 51.6% (or 48.4% if they were playing the Ashiok deck).

Anecdotally, this is not the case in my experience with the event as shown in the tracker snippet. Unfortunately, I only played the Ashiok deck very few times, mainly because I was turned off by the long queue time, so the sample size is really not enough to make any meaningful inference. Nonetheless,  suppose these win rates are consistent (that is, if we assume that they would remain unchanged even after playing 100 more games), then it would seem that at least for this event, where the decks were both preconstructed and limited to two options, the choice of deck actually does not matter. My win rate is the same regardless of what deck I choose, and for each deck, my win rate is very different from what the true win rates of each of the decks were supposed to be.

Finally, while Magic is certainly a game whereas lot of luck is involved, it is also a zero-sum game. That is, in every game of magic, a win for one player means a loss for another. Considering all player-controllable factors (deck choice, technical play, sideboarding, etc.), if none of these mattered then everyone would have a true win rate of 50%. Those who keep screaming from the top of their lungs that only luck matters need only look at their dismal tracker data to realize that this is not the case.

May the shuffler be with you.