Statistics in Magic: the Gathering
“Statistics show that of those who contract the habit of eating, very few survive.” – George Bernard Shaw
Following on from last week’s article advocating a more theoretically driven approach to Magic: the Gathering, in respect to both writing and play, I thought I’d take a shot at a statistics article. This is a topic which many of us find alienating and unengaging, myself included to an extent – we may not be comfortable with the maths, the subject matter is quite dry, and often statistics tend to disenchant our more hopeful notions about the game. Of these issues I will do my best to accommodate the first two. I am studying for a PhD in Sociology, which requires knowledge of statistics for data analysis. However, I was never especially comfortable with the subject, so I am sympathetic to the idea that the topic is both dense and daunting. The last issue is not one which I’ll sugar coat, though. The best way to learn is to set goals for yourself, understand how to achieve them and practice.
I’m not going to try and teach the whole subject in a single article, it’s a huge topic. Instead, I’m going to discuss some areas of the game in which people use statistics in either an unhelpful or unrealistic way, or simply fail to think of the topic in terms of numbers, and discuss how this might be approached differently.
“I’m better than the people I play against, but I can’t win a tournament – I’m super unlucky!”
The truth of it is that there is a good chance you’re not that much better than your opponents, and this is worth considering. How could you improve? Did you play your last match well? Did you keep a bad hand instead of choosing to mulligan? These sorts of thoughts are way more productive than being angry and entitled. There is reason to suggest that I am better than most of my opponents, but I didn’t win a PPTQ last season, or the one before it, and I played quite a few. I think I’ve messed up in about 5 top 8s in a row now, for various reasons…
…but let’s assume you are better, and that you’re also not making mistakes which cost you the game (two separate things – you can be better than them, but still mess up, and lose the game). This is a quincunx:
The idea is that balls are dropped in from the top, and each ball has a 50/50 chance of going left or right at each peg (dot) as it falls. If you drop a single ball, there is a good chance that it will end in the middle, because that’s where they’ll end on average, but it might go slightly – or sometimes extremely – to the left or right. If you drop 10 balls, it would be strange if they all went right, or all went left. 100 balls, very unlikely, 1000, more so, etc. with a large sample (which can in this context be taken to mean a more representative, or accurate, sample) the highest number of balls will fall towards the middle, and decrease in numbers as they diverge from the middle point. They will form a normal distribution curve, or “bell curve”.
If you think of the balls as players in Magic tournaments, the pegs as rounds, and the curve as the results of those rounds, this will give you some idea of how unlikely it is for someone who goes 50/50 with their opponents, on average, to win a long tournament. This will change based on the number of pegs (rounds), and (importantly, because some people do consistently win big tournaments!) the expected chance of winning a given round a given player has.
Imagine now that the area to the right of the bell curve represents a higher win rate, and the area to the left represents a lower win rate. Currently, assuming nothing else about our hypothetical Magic player, they have an equal chance of doing very well or very badly when the player enters the tournament hall (or when the ball enters the quincunx) and progresses through matches (falls through pegs).
If our player is favoured to do better or worse than average, their success can be represented by a Non-Normal Distribution as so:
If you’re better than your opponents, your chance of winning each match increases, and so does your overall result at the event. This results in the distribution above, skewed towards the right. How much better than the field are you, though? It would be unusual for you to be in an environment in which you were 90% to win a game against the average opponent. If this were the case, it is likely that you would already be qualified/winning enough that you weren’t frustrated. When I started writing this article, I put a 65% win rate into a 15 round event (the \swiss of a GP), in this simulator:
Currently, at 800 balls, there has been one 15-0. This model is imperfect as it can’t handle draws, but it does handle taking a loss at some point with a little imagination. For example, if you run the simulation for the Swiss only, you can count balls that only went 13-2 or better as having made top 8, then account for winning three rounds at 65%. However, you would be unlikely to win a GP if your win rate was 65%.
I’m using a GP as an example, but this tool can be used to look at your chances of winning a PPTQ or similar event by adjusting the numbers accordingly. The long and the short of this is that you’re unlikely to win events if you’re not *considerably* better than your opponents. The solution to that is to work out how to *become* better than them by such a margin that you’re in good shape for accomplishing your goals. Long term it will pay off the think about the game in terms of self-improvement, not results, because latter will come with the former.
“You should play >deck X< because it has the highest win percentage online, so it’s the best deck.”
Ross Jenkins mentioned an article to me a few months ago which gave me the tools to best explain why the above statement is problematic, which is fortunate as it is a recurring issue with many people who I discuss cards with. This article (which Ross tells me Richard Garfield wrote) discussed the intersection between random variables and skill using chess as an example.
Imagine that a player wins a game of chess. A die is rolled, and if it lands on a 6, the other player wins instead. The two players would achieve equal benefit equally from the dice element. However if Garry Kasparov is playing against a novice, then the dice is a problem because the novice, who will more often be the loser, will get to roll the dice more often, and will correspondingly win more as a result.
You needn’t be Kasparov, or some Magic pro equivalent (let’s say Sam Black, for name dropping’s sake) for this to be an issue. If you think you’re 65% to win a game against most people in the room, all things being equal, there is a problem with adding variable cards into your deck in this way, *even if the decks which have them win more*, because the win percentage is calculated based on the Population mean (everyone who plays the deck) not the Sample mean (you).
Let’s be super clear here. This isn’t an argument in favour of “playing what works for you” in the sense of preferences and individual strengths and weaknesses. I am a big advocate of playing all the decks and archetypes during testing so that you don’t end up with glaring weaknesses in your game, leaving you high and dry when one of the ones you haven’t bothered playing is actually a great deck that season. What I’m saying is that if you have an edge in the game, all things being equal, keep everything the way it is; don’t start having your opponent roll you out of wins!
Where this becomes an issue is with cards like [card]Collected Company[/card] which is clearly a very powerful Magic card, but can also just miss. I didn’t play this deck at all during the PPTQ season, and got a fair bit of flak about it from my peers. I didn’t feel that the deck was sufficiently powerful that the difference in win rate it had was greater than the detriment it added in terms of random variables (e.g. not hitting off the Collected Company, only hitting one creature, the creature not being good enough, not the mention the limitations the card places on deck building in terms of creature density).
Whether this was correct about that deck at that time, and about my expectations against the average players aside, the reasoning for this is perfectly defensible, and the idea that playing the deck with the best percentage is always correct is simply false. There have been times in the past, and there might well be in the future, when this is different, and the percentage the deck has is so great that even the best players are compelled to hope they don’t get rolled out of the event by worse opponents. Good examples include Mirrodin block, where Affinity was incredible, and a brief window in which Eldrazi was completely dominating Modern.
That’s it for this week, I’ll try and think of some more applied examples about statistics in Magic to write about in the future and make a mini-series out of them. Let me know if you have a pertinent question in this respect.
All the best,
P.S. my Quincunx program is at 1400 with four at 15-0 now, one in three hundred and fifty to x-0 a Grand Prix…