Would changing who chooses to play first shorten Magic: the Gathering tournaments?
I need to come clean at the start of this article: I am not a regular Magic: the Gathering player. I’ve played Magic— the high-water mark of my competitive career was playing Birthing Pod in a Grand Prix or two in the early 2010's — and I tune into enough of today’s coverage streams to recognize metagame staples like Temur Reclamation or black-red cat sacrifice decks. But I am not an expert on Magic or Magic tournaments.
That said, I have spent my fair share of time competing in Android: Netrunner, running the gamut from eight-person tournaments to win a couple promos all the way up to the largest World Championships ever put together, where attendance soared to over 500 players. This isn’t a flex — the numbers certainly won’t stun anyone used to attendance at the largest Magic: the Gathering events. It’s just letting you know that I truly understand it when I say:
Long tournaments suck.
Playing nine rounds of a card game in a single day, against top-level competition, with the intention to win as many games as possible, is exhausting. It taxes your mind, it enervates your body, and it might even sap your spirit, depending on how your topdecks go. And that’s assuming the best-case scenario, where you’re keeping your energy up with healthy snacks and properly hydrating the whole time.
A tournament system should aim to fairly determine the best player. But if you have multiple tournament systems that approximate that goal about as well as each other, except that one does not run as long — and therefore involves less suffering? I’m all for it.
With that issue in mind, let’s consider a potential rule change to tournament Magic.
Who’s up first?
For those who are reading this article without a detailed background in Magic: the Gathering tournament structure¹, tournaments are divided into a number of rounds. During each round, each player will play a match, which is a set of one or more games. Matches are won on a “first to 2 game wins” basis, meaning that (with the exception of some edge cases) matches consist of either two games (e.g. player A wins game 1, then wins game 2) or three games (e.g. player A wins game 1, player B wins game 2, then player B wins game 3).
Another important piece of information for those new to Magic: the game is non-simultaneous. That is, player A will take a turn, then player B will take a turn, then player A will take a turn, and so on until someone wins. This probably isn’t surprising — most games have some sort of turn structure — but it does imply that one of the players will get to go first.
In a game where going first had no effect on the chances to win, this would make no difference at all. Historically, this has not been the case in Magic: the Gathering.
The last piece of the puzzle, then is to explain how the player who goes first is chosen. Before the first game of the match, the starting player is randomly chosen. After the first game, the player that lost the game chooses which player goes first in the second game. After the second game, assuming there is a need to play a third game, the player who lost the second game chooses which player goes first in the third game.
For the purpose of getting numbers to use for my analysis, I asked a friend who plays competitive Magic much more frequently than I do what he’d estimate for the probability that a player wins when they get to go first in Standard today, assuming equal play skill and no great disparity in deck quality due to matchups; his estimate was 53%. But let’s be clear: the number 53% isn’t important. The direction isn’t even important — 47% is the exact same, for the purposes of this analysis. The important thing is that choosing to play first or not play first offers some advantage. I’m willing to assume that players will know which decision is correct, even if it varies by matchups against specific decks — after all, I’m examining things in the lens of top-level competitive play. If there is some advantage to choosing which player plays first, then having the ability to make that choice increases your chances of winning.
The proposed rule change that I’m considering here is: instead of the player who lost the previous game choosing who plays first in the next game, should the player who won the previous game choose?
Looking for impact
As I mentioned above, there are two important things to figure out about any proposed rules change:
- Is it fair?
- Is it impactful?
I’m not going to tackle the first question at all, as I think it’s better handled by people who have a more thorough grasp of the niceties of competitive magic. My goal here is entirely tied to point 2: would making this rules change actually reduce the amount of time that a Magic: the Gathering tournament takes to run?
So many games, so much time
A round of tournament Magic lasts until all of the matches in that round are finished playing, or until a maximum time limit (usually 50 minutes) is reached. There’s no real way around that. The alternative would carry a host of challenges for both fairness and logistics. This does carry a serious implication for attempting to speed up a tournament, however, because it means that the following is true:
For a tournament round of Magic to finish in less than 50 minutes, every single match must finish in less than 50 minutes.
We can only expect to see a real reduction in the length of time a tournament runs if we see a notable change in the longest time it takes anyone to finish a round.
Experimental frenzy
I ran a few simulations to look at the proposed rules change’s effect on total round time.
Now, as any statistician knows, a key step in every simulation is making assumptions, and a key step of science is to be transparent about decisions you’ve made. In that spirit, here are the assumptions I made for this simulation:
- The length of time taken by an individual game of Magic roughly follows a Normal distribution. You’ve likely heard of the Normal distribution — it’s the classic bell-curve shape. This is a fairly strong assumption on the probability distribution that game lengths can fall under, so let me explain the implications.
- One consequence of the Normal distribution is that we have to assume an average value for the length of time of a game of Magic. I used a few different values for the mean (12 minutes, 12.5 minutes, 13 minutes, … 15 minutes) and looked at results for each of them.
- Another consequence of the Normal distribution is that we have to assume a standard deviation for the length of a time that a game takes. I chose 2.5 minutes. Because of the properties of the Normal distribution, this implies that a majority of games (68%) will finish in the 5-minute time window straddling the mean, and the vast majority (95%) will finish in the 10-minute time window straddling the mean. To give a concrete example: if we believe 15 minutes is the true average time that it takes a game of Magic to play, then a standard deviation of 2.5 minutes corresponds to almost every game taking somewhere between 10 and 20 minutes.
- I have to assume a probability that a player wins when they get to choose who goes first. I assume 55% in this simulation. The farther this percentage is from 50%, the higher the likelihood we’ll see an improvement (to see this logic, consider the extreme case of 100%, which results in every match lasting exactly 2 games under the rules change).
- I have to assume a number of matches in a single round. This assumption is a big one, as I’ll get into later. For the main analysis, I chose 85 matches per round, which loosely lines up with the size of recent in-person Star City Games invitationals.
- We make simplifying assumptions to ignore basically everything else. Player skill? We assume players are matched up with players of equal skill (which is relatively reasonable given high-level play and a tournament structure). Mana screw? We assume it washes out. Metagame dynamics like matchup? We assume that’s factored into that 55% in all the ways that matter. These simplifying assumptions are certainly not perfect, but explicitly modeling them in requires making a whole bunch of stronger assumptions that are metagame-dependent. I didn’t set out to do that in this analysis for a number of reasons, not least of which is that I’m far from an expert on the current Magic: the Gathering metagame.
A final note on these assumptions: I’ll include the code I used for this simulation at the end. If you don’t like my assumptions, not a problem! Feel free to change them and run the analysis yourself.
A rule by any other game
One important mathematical consequence that I’d like to spell out before we continue: win probability on the play implies a probability of two games vs. three games.
Under the old rules, to win in two games, the player has to win an unfavorable game two. That means P(win in 2 games) = P(win a favorable game 1)*P(win an unfavorable game 2) + P(win an unfavorable game 1)*P(win an unfavorable game 2) = 0.55*0.45 + 0.45*0.45 = 45%.
Under the new rules, the player gets to win a favorable game 2. That means P(win in 2 games) = P(win a favorable game 1)*P(win a favorable game 2) + P(win an unfavorable game 1)*P(win a favorable game 2) = 0.55*0.55 + 0.45*0.55 = 55%.
This is the only way that the rules change affects the mathematics of the simulation.
Unexpected results
Using this data setup, I simulated quite a few rounds (1000, for each data setting). To give you a sense of the distribution of match times, I generated some density plots:
For each simulated round, I checked whether at least one of the matches in the round crossed the 50-minute boundary. I then took the overall proportion of rounds where at least one match crossed the 50-minute boundary as an estimate of how likely a round is to go to time under a particular rule set. Here’s how the probability of hitting time varies based on the value we assume for the average length of a game:
As we can see here, the rules change does seem to offer an improvement in the probability that a round goes to time. In some cases, the change is notable — for instance, it’s near 10 percentage points when the average minutes per game is in the range of 13.5–14 minutes.
We note, however, that the assumed average number of minutes per game appears substantially more impactful. When the average length of a game of Magic is 12 minutes, rounds will virtually never go to time, regardless of rule set; when the average length is 15 minutes, they will virtually always go to time, regardless of rule set.
This suggests that we might want to delay any firm conclusions until we have a sharper estimate of how long a game of Magic lasts. If the true average time of a game of Magic is 12 minutes or 15 minutes, then the rules change is unlikely to move the needle at all. Conversely, if it’s 13.5 minutes, the rules change might offer enough time reduction to take seriously.
Extending the model: uneven games
Now, as any player who’s played competitive card games, including Magic: the Gathering can tell you, not all games are games. Because of the variance involved in shuffling up a large number of cards and then relying on those cards, along with other factors like extremely bad matchups or extreme imbalances in player skill, some games are blowouts. Blowouts tend to take much less time than the average game.
I ran an additional simulation where I incorporated blowouts in the following way:
- Any game has a 10% chance to be a blowout.
- If a game is a blowout, it follows a Gamma distribution with shape parameter 2.5 and scale parameter 2. This isn’t as easy to translate to English as it is with the Normal distribution, but a key consequence of this assumption is that we assume a blowout takes 5 minutes on average.
- If a game is not a blowout, it follows a Normal distribution as described above.
- All other aspects of the simulation remain unchanged.
The distribution of match lengths changes when we add in blowouts. Here’s what it looks like:
Let’s translate that into the probability of going to time:
The conclusions here are similar: while the proposed rule changes clearly offer an improvement in the probability of going to time, that improvement should be taken with a grain of salt given how important it is to know the average length of time that a game of Magic lasts. In the best case, it could be a substantial improvement to the tune of nearly 10%; in the worst case, it would not improve things in a way that anyone would be likely to notice.
Extending the model: large tournaments
Let’s revisit one of the assumptions we made to start with: the number of matches in a single round. Not all tournaments are created equally sized, even among high-level competitive events. High-level competitive events might have as few as 16 players or as many as 700.
As I mentioned above, this is pretty impactful. The reason goes back to the same core logic we’ve been dealing with this whole time: for a round to finish before time, every single match has to finish before time.
A quick thought experiment: let’s say that, under a specific rule set, every match has a 5% chance of going to time. If one match is played in a round, then there’s a 5% chance the match will go to time. If two matches are played, then either game can go to time, which makes the overall probability of going to time 1-(95%*95%)=9.75%. If three matches are played, then the chances are 1-(95%*95%*95%)=14.3%. By the time we have 50 matches, there is a whopping 92% chance that a match will go to time.
Obviously, our situation is more complex than this setting, and the number 5% is almost certainly wrong. But the number 5% and the uniform probability of going to time aren’t really important. The point is that similar logic will apply: the more matches that are played in a round, the higher the chance that one of them stretches super long and goes to time. And one match is all it takes.
So, what do things look like if we have a very large tournament — say, 300 matches in a single round?
Overall, in a large tournament, there are many more realistic scenarios where the rules change doesn’t have a chance of decreasing time per round.
What the model misses
The model that underlies these simulations is not perfect. Here are a few things it misses:
- Some decks are designed to play long games. Netrunner players who were around in 2017 likely remember the slog-filled reign of Jinteki: Potential Unleashed, which prompted at least one top player to say “it’s good that the clock exists, because it’s how this deck loses.” If this is the case in your current metagame, you’re less likely to see any improvement in time length from a rule change.
- Many Magic: the Gathering tournaments do not operate with the timeline they “should” have on paper. Deck checks, judge calls, and the logistics of internet-streamed games can lead to time extensions — essentially a flat amount of time that’s added to the clock for a given match. This means a round can last 50+ minutes even if every game theoretically does not go to time — sometimes even by a substantial margin.
- Competitive card gamers who are trying to win adapt to new rules. If the rules change happens, it’s possible that people’s play habits, as they translate into the amount of times their games take, will not change. But if the correct strategic decision becomes to lengthen game 1 and look for every possible line because game 1 is even more impactful than it is today, we can trust top players to make that decision. This, in turn, will erode any time gains that a rules change carves out. This is something I can’t predict, and it’s completely missing from the model.
- I’ve only considered overall time in the tournament. Obviously, that’s not the only way that a tournament creates wear and tear: there’s also active playing time to consider. As we showed earlier, the rules change would likely lead to a reduction in the number of game threes played, which in turn would lead to a reduction in average active playing time. That might be a benefit big enough to justify cost-benefit analysis, independent of the rule’s impact on the overall tournament time.
Final thoughts
I want to be clear that I’m not offering any definitive answers today.
For one thing, this is all simulated data based on some assumptions that seemed reasonable to me but might not actually be borne out by data from the real world.
For another thing, everything I presented ignores the entire question of whether the rules change is fair, which is ultimately more important.
I’m not advocating for changing the rules, and I’m not advocating for rejecting this idea. But I hope I’ve demonstrated that a little mathematical reasoning can go a substantial way to shedding insight in situations like this.
My hope is that someone (or a group of someones) who actually has insight into the data on a level I don’t might see this post, think “huh, that was kinda cool,” and incorporate this sort of thinking. And my hope is that when rules changes like this are talked about, debated, and implemented, people want to see them backed up with math.
Wait, you said something about code?
You can find all the code I used for this analysis at https://github.com/mtlawson/mtg-rules-change-medium. It’s open source, and you’re free to use it as you see fit. It’s certainly not my finest work, in terms of software quality — there’s a lot of copy-pasting where there really should be function-writing — but such is life. Caveat emptor.
- Note that there are other types of Magic: the Gathering tournament structures, especially with the introduction of Arena. I am limiting the scope of my analysis to first-to-two-wins tournaments, with a special emphasis on the Swiss rounds (since those affect the largest number of players) in this article.
