Magic Online has a variety of tournaments you can participate in, including regular 8-person draft queues, premier sealed and constructed tournaments, 8-person constructed queues, and 4-week sealed deck leagues. Each offers a different set of prizes and a different number of opponents. This article will show you that not all tournaments are created equally. Some Magic Online tournaments offer a much better return on your investment than others, so a regular player would play longer for less money by playing the high-paying tournaments and avoiding the others. If all you want is the conclusions and don't care about the math I used to get them, you can skip the math and just read the sentence after Conclusion in each section.
Introduction to Expected Value
First, I want to explain the idea of "expected value," or EV. The expected value of a random variable is the sum over each possible outcome of the value of that outcome times the probability that outcome would happen. For example, when you roll a 6-sided die, there are 6 possible outcomes, each with probability 1/6. However, each outcome has a different value (1-6). So, the expected value would be
Now, suppose you were offered two different games to play. In one game, we roll an 8-sided die and give you money equal to the value rolled. In the other, we roll two 4-sided dice and give you money equal to the value rolled. Which game would you rather play?
The EV for the first game is 4.5, while the EV for the second game is
So the second game would be a better deal for you, since over the course of many games, you would make more money. You can think of expected value as the average value you would get from repeated plays of the same game.
Conclusion: All else being equal, a game with higher expected value will produce better results in the long run.
DCI Ratings
At this point, you might be asking yourself, "Does this really apply to Magic? After all, Magic is a game of skill. Rolling dice is totally random!" However, we all know that Magic has a random component as well. Two equally skilled players with equal decks should split their matches over time. In fact, the central assumption of the DCI rating system is that a player's performance is a random variable. If you have a higher DCI rating than your opponent, you won't necessarily win, but you have a better chance of winning since you have proven to be a better player over time. The formula the DCI uses to determine the probability of winning a given match is
where r is your DCI rating and r' is your opponent's rating. For example, if you have a rating of 1700 and your opponent is rated 1550, you would have a 70.3% chance of winning that match. So, over time if you only played this same opponent, you should beat him 70% of the time.
MTGO Draft Queues
To apply the idea of EV to the various options for play available on MTGO, let's first look at the two options for 8-person drafts. There are two types of draft queues online with two different prize structures. The most commonly played type is known as a 4-3-2-2 draft, which awards four packs to the first place player, three packs to the second place player, and two packs each to the third and fourth place players. Players that lose in the first round get no packs. Assuming you have a probability of p to win each round, your EV for the draft would be
Suppose you were equally ranked with all your opponents. Then you would have a 50-50 chance of winning each round, so p = 0.5 and your expected number of packs would be 1.375.
The second type of draft is an 8-4 draft. These are considered more difficult and tend to have better players. They award eight packs for first place and four packs for second place. No one else gets a prize. The formula for the expected value of this draft is
which comes to 1.5 expected packs if you are equally matched with your opponents.
This implies that one should always play in the 8-4 drafts to maximize your winnings, since it has the higher EV. However, better players tend to play in the 8-4 drafts because of the higher payout if they do well. Therefore, the probability of winning any given round in an 8-4 draft is likely to be lower than in the 4-3-2-2 drafts. We can get an estimate the probability of winning a round in each draft using the formula I gave above for the probability of winning a match based on the two participants' DCI ratings.
For example, if you think that the average rating of players in the 4-3-2-2 draft is 1600, while the 8-4 draft has players with average rating 1700, then you could calculate the EV of each draft based on your own DCI rating by applying the formula for p above substituting r' = 1600 and r = your DCI rating. Then use this p in the formula for EV in each draft. This will tell you whether, based on your rating, you should be playing in 4-3-2-2 or 8-4 drafts to maximize your winnings.
I created the following chart to demonstrate the cutoff point for switching to 8-4 from 4-3-2-2 based on these assumptions.
As you can see, the 4-3-2-2 drafts are a better deal than the 8-4 drafts until you reach a DCI rating of around 1780. If the difference in average rating between 4-3-2-2 and 8-4 is larger than 100, than you will need a higher rating to do well there.
The other feature of this chart is the green line labeled "8-4 (split final)." This line is based on the assumption that the top 2 players in the 8-4 draft will always split the final and get 6 packs each. Notice that you are actually better off playing in the 8-4. If you are good enough to get to the final, you are also probably good enough to beat your opponent. However, this is based on the assumption that your opponent won't be better than your previous opponents, which is not likely to be the case in real life, so take this conclusion with a grain of salt.
Conclusion: There is a distinct cutoff point at which 8-4 drafts have a better payoff than 4-3-2-2. However, figuring out this point is complex. If you are usually higher rated than the players in 8-4 by about 80 points, you probably should play 8-4. Constructed Queues
The constructed 8-person tournaments award five packs for first place, four for second place, and two each for players who only won one round. The EV for this tournament is given by
If we assume all the participants in this tourney are equally skilled, then you would expect to win 1.625 packs per tournament.
Let's use this knowledge to compare the profitability of drafts versus the constructed tournaments. We already saw that the 8-4 draft has a better EV than the 4-3-2-2 draft, so we'll compare it to this constructed tournament. The entry fee for a draft is 3 booster packs plus 2 tickets. The entry fee for a constructed tournament is 6 tickets. If we set the price of a booster pack at 3.5 tickets, then we are spending 12.5 tickets on the draft and 6 tickets on the constructed tournament. We expect to win 5.25 tickets worth of product from the draft, and 5.6875 tickets worth of product from the constructed tournament. This amounts to an expected loss of 0.3125 tickets in the constructed tournament, and a huge loss of 7.25 tickets in the draft. Obviously, this doesn't count the product you open and get to keep from the draft, but still the constructed tournament is much more profitable than any draft.
Now you can't take just any deck into the constructed queue and expect to have a 50-50 chance of winning your matches. You need a high-level deck even to compete, so the initial investment is high.
Conclusion: Constructed queues are more profitable -- if you have a competitive deck.
Premier Events
What about the premier events? Starting with the sealed deck tournaments, the standard prize structure is
PlaceBoosters
1st 12
2nd 9
3rd-4th 6
5th-8th 2
Plus each player in the top 8 gets 3 packs to play the top 8 draft. Since these tournaments allow 64 players, the EV given a full tournament is
where p is the probability you have of winning a Top 8 draft match, and q is the probability of making it to the top 8 in the first place. P depends on the average skill of players in the draft just as in the draft queue section. However, q depends on a number of factors, including your card pool, the relative skill of the other players in the sealed deck, but also the number of players.
Let's try to model the probability q that you will make it to the top 8. The number of players is easy to model. All else being equal, if there are fewer players in the tournament, you have a better chance of making Top 8. We also already know how to account for the relative skill of the other players. The hard thing to quantify is the quality of your card pool. How much of an effect does card pool have in an individual match? This requires a significant amount of study and is out of the scope of this article. For our purposes, we'll just assume an average card pool. That is, your card pool will be better than half your opponents' and worse than the other half. This will let us ignore the effect of card pool for now.
If there are N players in the tournament, there are always log_2 N rounds. Generally, you are guaranteed a top 8 slot if you win your first log_2 N - 2 rounds since you can just draw your last two matches and get into the Top 8. Generally, players that lose one match but win the rest also get into the Top 8, as do players that lose one, win the rest, and draw the last match. Thus, we can compute the probability q as
If we use this formula for q, then the expected number of packs won in a premier event depends on the average rating of players in that tournament compared to your own DCI rating and the number of players. If we assume every tournament is full (64 players), and the average player skill is the same as in the 8-4 drafts we get the following graph
As you can see, at just the same point where a player should switch from 4-3-2-2 drafts to 8-4 drafts, sealed deck tournaments become more profitable than both drafts. However, remember the assumptions we based this analysis on. We assumed that the players in the Top 8 would be average for the sealed deck as a whole. Also, remember that we are not taking into account the increased cost in product necessary to join a sealed deck tournament.
If a premier event isn't at the maximum number of players, then it becomes an even better deal.
Now what about those big 2x and 4x tournaments? We can analyze them just like the regular tournaments, except they also pay out prizes to players outside the top 8. The payout for a 2x tournament is
PlaceBoosters
1st 24
2nd 18
3rd-4th 12
5th-8th 6
The payout for a 4x tournament with a maximum of 384 players is
While it was fairly easily to analyze the chance of getting Top 8 in regular tournaments, there are many ways in which a player could get into the top 64 of a 4x. To simplify this analysis, let's assume that if you don't get into the Top 8, you are assigned to one of the remaining slots with equal probability. This gives an EV of
The following graph compares the expected value of regular, 2x, and 4x tournaments given your DCI rating.
As you can see, 2x events are generally a better deal than 4x events unless you are an especially good player. However, they have an increased risk because many players will not win any packs at all.
Conclusion: If you are good enough to play in the 8-4 draft queues, you are probably also good enough to do well in the sealed deck tournaments. In general these provide a better payoff.
I hope this analysis will show you how to spend your time more profitably on MTGO. However, remember that MTGO isn't all about winning money. If that was your goal, you would probably be better off playing poker on the Internet or investing your money in stocks. Magic is a game that we play for fun!
To summarize, the most profitable tournament depends on the quality of players in them. If we assume the 4-3-2-2 queues have generally worse players than the 8-4 queues, then they are a better deal unless you are a good player yourself. Also, sealed deck premier events offer a good deal for a good player. Constructed queues and premiers are always more profitable, but they require a solid deck investment.
