Play or Draw?

Images courtesy of Alexander Shearer.

Have you ever wondered if being on the play is indeed advantageous, or how advantageous? Unfortunately, there are no statistics about winning rates on the play/draw publicly available. Wizards could probably generate them from MTGO at any time, but not having access to their tools, I took the liberty to watch a thousand games and then some. Don’t worry, I have not watched them all through to the end.

I’ve included the math for your perusal—but for the less mathematically inclined, I compiled the relevant results into aptly named results paragraphs at the end of each section.

Hypothesis (a.k.a. “So what question are we really asking?”

To do this properly, we need a hypothesis. An easy hypothesis would be: “The player playing first has a higher chance to win the game than the player on the draw.” The null hypothesis would then be: “Being on the play does not influence a player’s chances of winning.”


Which Games Do We Look At?

Although this hypothesis seems nice, there is a pitfall. Which games do we mean here? In game one it is random who has the choice of playing first. In the other games of the match it is not as random, though.

While we assume that the player who won the die roll has better odds to win, there is another factor we must not overlook. The player that is more skillful will on average win more games than his less skillful opponent. Thus, the less skilled player is more likely to be on the play in game two.

Now we have a bias to our data already. The less skilled player is of course better off being on the play than on the draw, but if we watch a random sample of games, the player on the draw might actually win more often in game two than the player on the play, because the player on the draw is more likely to be the more skillful player. Thus if we don’t take care, it might happen that our hypothesis is true, but our data will indicate that it is false.

In this case there is an easy workaround. If we choose to look at game one only, the bias doesn’t exist, because the chance to be on the play doesn’t depend on a player’s skill. It is solely up to Fortuna who wins the die roll.

Consequentially, our hypothesis will be: “The player that plays first in game one is a favorite to win that game.” Of course, the assumption that our hypothesis is true is the reason why everybody opts to play first all the time. The null-hypothesis would be: “Playing first in game one provides no measurable advantage.”

Method

I collected a bunch of data to look over. To be more precise, I looked at 200 games in each of the following categories:

• Game 1 of all matches that ended 2-0 in rounds 1, 3, and 4
• Game 1 of all matches that ended 2-1 in rounds 1, 3, and 4
• Game 3 of all matches that had a game three, in rounds 1, 3, and 4

I also counted the proportion of games that finished 2-1 to games that finished 2-0. As I will demonstrate later, this gives us all the data about game 2.

To test our hypothesis, the 600 game 3s and inferred game 2s are totally irrelevant, but we will get back to that later.

The Specifics (because they matter)

Before I present you any data, I would like to explain what data I took, how I took it, and why. This is important as the wrong method in collecting data can lead to systematic errors, which can invalidate the result. Only with this information can you be the judge whether my method was sound.

I watched only Standard matches. Thus every finding will only apply to Standard.

I watched only Magic Online matches. While I presume it is reasonable to expect that watching PTQ games will lead to results in the same ballpark, we cannot take that for granted. A lot of factors might make MTGO different from paper Magic. Players choose different decks for MTGO than paper Magic (for various reasons). The population is different and might be more or less competitive than a real life population you choose to observe (PT or FNM?) and it also might be more or less diverse than any other population. We cannot even fully rule out that the MTGO chess clock influences whether playing is advantageous.

I watched only matches of MTGO Daily Events. Premier Events or 8-man queues might have different populations of players and that might influence the results.

I watched only matches that were available for watching on Magic Online during European daytime, roughly 8:00-20:00 CET. Of course, this means that the observed populations may have some geographical bias, because not everybody is equally likely to be up and playing in that time frame. Geographical differences might correspond to differences in play style, but I believe that all things considered this should not be statistically relevant.

Matches that did not end 2-1 or 2-0 were ignored. On Magic Online there are only very few matches that end in a different score, but of course we might incur a small bias here. I doubt it is big enough to call an otherwise sound result into question, though.

I only looked at matches in round 1, 3, and 4. I assumed later rounds in the tournament might yield different results than earlier rounds, thus it was interesting to have the data for an early and a late round. Round 1 and 3 would have served for that. Collecting the data for round 4 as well was just my curiosity. Splits might warp this data, but that’s precisely what I wanted to see, even if that didn’t amount to much afterwards.

I assumed a player with the chance to play first did in fact always opt to play first. (See below.)

How to Sample Hundreds of Games

One major thing that merits further discussion is the way I came by the data. Of course, I didn’t really watch more than a thousand games. That would have taken weeks. I actually just inferred the data from the game result and the knowledge of who played first.

For any game, if you start watching the replay, you can immediately see which player had the choice of whether to play first. Your personal experience will confirm that more than 99% of the players in Standard choose to play first. Thus, I generally assumed that the player having the option to play first did indeed play first. This might lead to a very minor error as that is not necessarily the case all the time.

For matches that ended 2-0, I watched who won the die roll in the first game. The player that won the match obviously won both games, thus we need to know whether he won the first on the play or on the draw, and know that he has won the second on the draw.

For matches that ended 2-1, I watched the third and first game. The player that won the match obviously won the third game. Again the second game doesn’t need to be watched, as it was won on the play by the player that lost game one. Otherwise, the game would have ended 2-0. By taking this logic one iteration further, we can assume that the player, who was on the play in game three, won game one. Combining that knowledge with watching the die roll allowed me to determine whether the player on the play had won game one.

Finally, the observed games come from various tournaments. In some cases I didn’t watch all games of a single tournament, because I collected data until I had 200 games of every category and then stopped. This does reduce the value of the data because it means that the games are not fully representative even of all games played in MTGO Standard Dailies. To understand this, consider what would have happened if games would generally be won on the play all the time, only in one fringe matchup they are won on the draw all the time. This one fringe matchup would account for as much samples as the vast majority of the other matchups. As I counted the proportion of 2-0 matches to 2-1 matches, we can figure in that proportion if necessary.

Does Being on the Play Matter in Game One?

Now, I believe I bothered you long enough with statistical minutiae. Let us take a look at the data of all the game ones:


This data is very homogenous. Apparently preboard games are won on the play a bit more often than 56% of the time, and as far as we can tell this is not influenced by the round number. Also, this is not influenced by the way the match played out. If different match scores would have yielded different observations, we would have to care about the proportion of matches ending 2-1 to those ending 2-0, but as we can observe no significant differences anyway, we don’t need to worry.

RESULT 1: *In the actual Magic Online Standard metagame about 56.3% of the first games of a match are won on the play*

We cannot infer from this, that there are no matchups where you would be advantaged by choosing to draw first preboard. However, you don’t know what matchup you are playing, thus playing first seems to be the correct choice. While there still might be decks or even archetypes that are generally advantaged on the draw, you should have a lot of evidence for this being the case before starting to draw first in the dark.

Side note: I had expected that the round of the tournament has some influence on the chances of winning on the play. This might seem counterintuitive, but as the tournament progresses, for any given match, the difference in playing skill between the opponents will be reduced. When the skill differences are large, being on the play should probably not matter as much as the better player will win anyway. Apparently this is not correct, though. It seems to be more likely that while the better player will win more often against an inferior opponent, he is still accordingly more likely to win on the play than on the draw.

Does Being on the Play Matter in Game Two?

The data for game two I could obtain in a very efficient way. I just looked at the result of the match and knew whether game two was won on the play or on the draw. This might seem surprising if you never really thought about it, but the reason is really quite simple. If a match ends 2-0, one player won game one, and the same player won game two. Of course that same player was on the draw in game two. Thus all matches that end 2-0, have a win on the draw in game two. Conversely, if game two is won on the play, then there has to be a third game. We don’t know who won that one, but certainly the score will be 2-1 in the end.

What interesting thing might we find out in game two that we didn’t already find out in game one? Maybe we could find out whether sideboarding has some impact on win percentages? While this would be a very interesting finding indeed, there should be one effect that we can observe. It is not only the player who won the die roll that is more likely to win game one. The player that is more skilled is also more likely to win. The chance to win game one should be ranked like this:

Better player otP > Worse player otP ? Better player otD > Worse player otD

If a better player is more likely to win game one, then he is also more likely to be on the draw in game two. This means that game two should be won on the draw more often than game one as the higher average skill of the player on the draw should somewhat compensate his disadvantage of being on the draw.

Unfortunately, this is not the only effect that can influence the win percentages in game two. While the effect just described is almost certainly present, other factors might figure in as well.

Sideboarding might have an effect on the chances to win on the play/draw. For example, if players generally prefer cards that interfere with their opponents’ game plan, then the games might be less about tempo and more about grinding the opponent out. This is just speculation, but having such an effect is definitely a possibility.

Another factor might be knowledge of the matchup. For example, a pilot of an interactive deck like UWR might desire very specific hands when on the draw against aggressive decks. On the other hand, an aggressive, but not hyperaggressive deck like Gruul might need [card]Tormod’s Crypt[/card] on the draw against Human Reanimator to have a chance at all. In both cases the knowledge of the matchup might be almost irrelevant on the play, as improved tempo will carry the player all the way, but on the draw that knowledge might lead to improved mulligan decisions. If this were the case, post-board game win percentages would improve on the draw. Again, this is just an effect that might be there and it might not be, but it illustrates that many things might influence the chances to succeed on the play/draw.

However, the player on the play in game two should win less often than the player on the play in game one. This is inferrable from the the fact that the more skilled player will be less often on the play in game two. Now if we don’t observe this, then we know that there must be a strong effect at work that we didn’t think of. The data:

We got what we expected for round 1 and 3, but the result for round 4 is very different. I believe that the round 4 result just means that some people split in that round, and then the match always ends 2-0 where it might have ended 2-1 otherwise. The other data is evidence to our hypothesis, but unfortunately we cannot discern from that which factors make it true to which degree. It just means that we did not find anything out of the ordinary, which in turn would make us want to investigate further. We may also assume that it is still advantageous to be on the play. While we don’t have proof, the data is at least evidence in that direction.

RESULT 2: *For game two there is an anomaly for matches in round 4. Assuming that this is cause by concessions, the win percentage on the play is about 52%. The reduced win percentage in comparison to game two is likely to be caused by the better player being more likely to be on the draw in game two.*

Side note 2: In round 1 and 3 the second game seems to be won on the play about 52% of the time, a.k.a. the match goes to game three. Only 46.7% of the matches in round four go to game three, though. A logical explanation for that would be that roughly 10% of the round four matches are determined by a concession/prize split.

What about Game Three?

Analyzing game three is way more difficult than analyzing game one and two. It is not only that a lot of things have happened already that led to a specific player being on the play in game three—things we don’t know anything about—but also now we don’t work with the full population any more. Almost every regular match of Magic has a first and second game after all, but not every match has third game. Thus we only observe matches where both parties have demonstrated the ability to prevail in a game. If that means anything at all it should mean that being on the play is rather more important now, because the matches where one side couldn’t win are filtered out. In these matches being on the play was rather unimportant, though.

Looking at this from another perspective, the winning percentage on the draw in game one should have been boosted by these matchups, and the winning percentage on the draw in game two should have been boosted even more. Why one side couldn’t win in these matchups we don’t know. It might be because one player was too much better than the other, or because the matchup was too favorable for one side. It doesn’t matter as in both cases who plays first shouldn’t be all that important.

Another factor for the third game is that the better player is now more likely to be on the play as the sequence (from his POV) win-loss is more likely than loss-win. Thus the player on the play is likely to have the double advantage of being on the play and being the better player. This should lead to further increased win percentages on the play.

So far we have identified two reasons why the win percentage on the play should be higher in game three than in game two. The data looks like this:

Now this is just not what we expected. There seems to be only a very slight advantage to being on the play at best. If the effects I mentioned have figured into this data there has to be a strong influence that we didn’t think of.

One part of the explanation might be that here we are only observing matches that have two post-board games. We cannot infer it from my data only, but post-board matchups might just be very different from the preboard matchups.

If this were true, the following might be going on: Game one player A has a deck, that is favored, and thus wins. The player that was disadvantaged in game one is afraid of that particular matchup and thus dedicated a lot more sideboard cards for that particular matchup than his opponent. Being on the play he usually wins game two easily. Despite being on the draw in game three he is still favored to win that game as his sideboard has improved the matchup so much. If this happens often, we will indeed observe a decline in the on-the-play win percentage in game three.

Another consequence would be that in spite of our data, being on the play might still be significantly advantageous. The player that is on the draw is just more likely to have the better deck for the matchup. Thus he can compensate his disadvantage of being on the draw rather well, but this doesn’t mean his opponent should decline to play first.

A small part that might influence the game three statistics is that if a match runs out of time it will be very likely to do so in game three. However, it should be essentially random which player runs out of time. This factor should pull the win percentages towards 50/50, but I don’t believe this factor is big enough to explain the rates I observed in game three.

There might be other explanations for the anomaly in game three. Right now I can think of none that would explain the effect as convincing as the first one I gave, but that might just be my lack of imagination. If you have alternative hypotheses that would explain the anomaly, I would be happy to read about it in the comments.

RESULT 3: *Game three is won on the play about 51% of the time. This is significantly less than was to be expected. The reason for this is unclear. A possible explanation would be that the player winning game two is likely to have a better sideboarding plan than his opponent and is thus more likely to win game three even when he is on the draw.*

Side note 3: If we recombine our game win percentages to make a whole match of them, we arrive at the conclusion that the player winning the die roll should win about 53% of the time. Thus for every nine games the player on the play wins, the player on the draw wins eight.

Florian Koch

Thanks to Alex Shearer for the images and headings.

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