Expected Results: Which Set Should I Be Drafting?

Sebastian (aka neuhier) is a statistician and has been a passionate Magic player since the time of Urza’s Saga. He still can’t decide if his biggest success was winning a draft at FNM or opening a Gaea’s Cradle.

I’m not a good Magic player. That’s what my credit card bill says. Personally it’s no problem for me to spend some money to have fun on MODO, but I really don’t like spending too much money. Recently an episode of the Limited Resources podcast reminded me to be more self-critical on the topic of how to get the maximum number of drafts out of my limited money.

There are basically three factors influencing the number of drafts you are able to play for a given amount of money:

1. Entry Fees: This is the price for the booster packs plus the 2-tix additional fee.
2. Win Rate/Prices: On MODO this is the amount of packs you win, depending on your skill and the payout-structure of the tournament.
3. Card Value: This is the value of the cards you pick up while drafting.

This article will provide some hints on the first two points above, but focuses on the third. I will present a card-price evaluation of the sets Gatecrash, Return to Ravnica, and Avacyn Restored. Based on the expected prices of the cards in these sets and the prices of each booster pack, I will point out which sets have the best expected value with respect to card prices.

Entry Fees

For a long time I simply bought draft packages in the MODO store, consisting of three booster packs and two event tickets. For all of you currently doing the same thing, THIS IS WRONG! The price at the MODO-shop is $14 (14 US dollars). Now compare this to buying the boosters from, for instance, MTGO Academy’s bots. Their price for GTC boosters at the time of my authoring this piece is at $3.1/pack. This means that you are paying $9.3 + $2 = $11.3 for one draft. So you are able to save $2.7 per draft if you buy the packs from a secondary seller!

Payout Structure

MODO offers three kinds of tournaments: 8-4, 4-3-2-2 and swiss. There is one way to do it wrong! I have to thank Limited Resources again for opening my eyes, namely that 8 + 4 = 12 total prize packs, while 4 + 3 + 2 + 2 = 11 total prize packs. It seems to be obvious, but before, I had never wasted a thought on that. The payout of swiss tournaments is 12 packs, too. It is a huge difference in the long run if you are playing tournaments for 12 packs compared to just 11.

Expected Value and Card Prices

I decided to play swiss, but 8-4 would be ok, too. Moreover, I’m buying my packs from bots trading in the Classifieds portion of Magic Online. Now I wanted to know, “Are there more ways to easily increase the number of drafts I can play per dollar spent?” Therefore I thought about the third part of my list: As I’m not much of a constructed player, I tend to sell the cards I draft from time to time. This directly led me, being a statistician, to the expected value of the card prices for different sets. I wanted to know if there were sets that were more valuable than others.

Thus, I checked the card prices on MTGO Academy for the rares and mythic rares of some recents sets. All the below prices are from the 13th of April, 2013.

These so-called boxplots represent the prices. In this case the boxes in the bottom of the graph contain 75% of all rares and mythics of one set. They are the low-price rares. The more interesting cards are represented by their names in the top of the graph. You can see that Sphinx’s Revelation is the most expensive card, followed by Bonfire of the Damned. I guess most of you already realized that the red cards are of special interest, since they are rares (not mythics), e.g., Restoration Angel is priced at $17.90. This is important because rares show up when drafting much more frequently than mythics.

So do we draft RTR because of Sphinx’s Revelation? Or do we draft AVR because of the Restoration Angel, which isn’t worth that much but will occur more often since it is a rare instead of a mythic rare? Which set is better to draft from a financial perspective? We’ll have to figure it out, taking into account all the cards of one set and the corresponding probabilities of opening them. This is exactly the idea of expected value.

Let’s start with AVR as an example. Of course we are most interested in getting a Bonfire of the Damned. Luckily the probability of getting one is known: There is a mythic rare in 1 out of 8 packs. If we are opening a mythic rare, then it is one of the different mythic rares in that set. Thus we get the Bonfire in 1 of 15 cases of opening a mythic (since there are 15 different mythic rares in AVR). The final probability of getting a Bonfire out of a random AVR pack is 1/8 * 1/15 = 1/120 = 0.00833.

The expected value of a AVR pack is calculated by taking the price of the card, multiplying it with the probability of getting that specific card, and summing these values for all the cards in one set.

So the expected value of one AVR pack is $1.80. The following table compares the different sets:

Personally I was surprised by the huge difference between GTC and those other sets. Are we now able to make a final decision on which set to draft? Sadly no, but there is only one more step to go. Until now we did not consider the different pack prices. This may be done using the following table:

Depressing, isn’t it? Negative expectations aren’t good in most games. In the long run we will lose $8.45 for each AVR draft, $8.54 for each RTR draft, and $8.15 for each GTC draft we do. Based on this analysis, GTC drafts are the cheapest. The much larger EV of AVR and RTR does not compensate for the higher pack-prices, assuming that you won’t be winning any packs in the draft.

Conclusion

I was able to show that it is cheapest to draft GTC if you are not planning on winning any packs. Of course, you could remove the price for the packs from the calculation in the above table (the second row), if you are winning three packs in each draft in the long run. If you plan on winning an average of three boosters per draft, it is best to draft AVR, as its EV is the highest and you don’t need to consider the price of the packs in this scenario. But I guess this is pretty unreasonable, because you minimally would need to win 25% of your 8-4 drafts and place second in 25% of your 8-4 drafts to fulfill this condition. Whether or not this is attainable for you is not something I can calculate because each player drafts and plays differently, with different levels of seriousness and different skill levels. But you could certainly, upon estimating your winnings, recalculate using the above table to draw a conclusion — assuming you want to maximize the number of drafts you can do per dollar — regarding which set you should draft.

 
  1. The GTC boxplot is super depressing.

    Also, if you plan on winning 1.5 packs per draft (the average), the EVs are AVR -4.5, RTR -4.7, GTC -5.05. So for most players, AVR will be the cheapest. The only problem is that this calculation does not factor in the cost of having to draft AVR….

  2. Yeah, nice point with the average winning (playing swiss and winning 50% matches).

    Concerning the prices of GTC: I guess this might change a bit as Dragonsmaze comes out.I will be observing the price changes after the next set comes out. Always wanted to see if this has a relevant influence on the prices of the current set.

  3. I”m pretty sure this happens when each new set comes out – as less and less people are playing the older sets, the prices for their cards start to climb – especially those mythics and rares that prove their usefulness in standard. I used to dump all my cards right away when playing the pre-release and release events. Now I wish I still had the foil revelation I opened in one of those events.

  4. Yeah that’s what I think, too.

    But as a statistician i’ll have to check the numbers. :-) I will try to gather some data on this. I think it would be especially interesting if there is a real rise in prices and to be able to estimate how much the prices raise after a set gets older.

    It’s becoming even more complex, when you try to model the prices over larger periods, including the time when the cards wheel out of standard.

  5. Great article. I like the suggestion of tying in win percentage to the payout EV’s so we can see the result of getting prize in the form of AVR packs vs GTC packs – maybe something like expected EV per format vs win percentage? I’m super surprised by the result that GTC is actually the cheapest set to draft!

  6. Thanks for the article, its an interesting attempt to analyze mathematically the ticket costs of playing in a limited tournament.

    First off, everyone always analyzes the tournaments with the cost to enter, and packs won by all players as the primary indicators of expected value. But they do not take into consideration the chances to win prizes are actually quite dramatically different depending on which queue you enter. Since player-skill determines your win percentage (and it is almost never ~50%), if you are a decent magic player, but not at the top tier of players, you could consistently end up placing 3rd or 4th in an 8-4 draft, but rarely get the 1st or 2nd slot. Lowering the perceived value of this tournament dramatically (as your prize is, lets say – 90% of the time, 0 packs)

    secondly, I would argue that a format such as phantom draft or phantom sealed is of the highest value to someone who is just looking to play games of magic on a limited playing field, as opposed to constructed play. – the prizes might not be spectacular, but the cost of the event compared to the amount of actual playtime is higher. (plus you dont have to spend time to try and trade away the cards you opened up at a decent/profitable exchange) – so the value is given in time spent playing magic, rather than in the prizes earned.

  7. An important option to remember is doing the new “phantom” sealed queues – it’s 4$ to join, and if you win 2 matches you get a booster (win 3 matches and get 3 boosters for the full blowout). Thus, for something like 14$ you get to do a draft and 3 seald events, which I think is quite a lot of bang for buck. Granted you don’t get money for cards you open in the sealed phantom, but it’s best to think of it as paying *slightly* more in order to multiplay by 4 the amount of play you are getting. Highly worth it IMO.

    Also, about the swiss/ 8-4 consideration – I ran the number once, and it’s only smarter to do 8-4 queues if you are able to win more than 50% of your matches. While that may sound good (“I’m a good player – of course I can taken more than half of the games!”), remember that the meaning of the above statistic is that half of the people playing in the 8-4 queues are making a mistake with their money – think long and hard to make sure you are not one of them!

  8. Now you can re-do all the numbers to figure out the “new” 4-pack sealed – 4 pack of product and 2 tickets to join – payout of 5-3-1 for round wins – you keep the product.

    These are a whole new ballgame, and strategy for GTC 4-pack sealed is interesting!

  9. Another point to consider when playing swiss is that sometimes players drop, so some matches aren’t played out which lowers the overall prize payout.

    For example, if 2 players quit before round 3 starts, there will one match less played and one booster less paid out.

  10. @Original-pete, that’s actualy ireelevent. You shouldn’t care how many boosters are given away in total, you should care about how many boosters are given to YOU. and other players dropping can’t change that number, as long as you don’t drop either. And if your opponent drops? well, you get a guranteed booster…

  11. @Carrot

    Actually, that can be relevant depending on the number of players that drop. If two players drop and you are doing poorly, it’s possible you’ll get paired up against someone who has more wins than you, potentially causing you to lose a game you may have otherwise won.

  12. First of all: Thanks for all that feedback.

    I really didn’t think about phantom as an alternative. I always had a bad feeling when drafting phantom. Think it’s important for me to actually keep the cards. But I guess I’ll have to re-evaluate that aspect.

    Second point: Did it ever happen to you, that 2 guys dropped? I rarely see even one person drop.

  13. I’ve seen it happen a number of times, but it’s only affected me twice, one on either end.

    One time it happened in a phantom draft, and I was paired up against the leader (his would-be opponent was one who dropped). I beat him, and no one went 3-0 and the pack payout was almost nothing.

    The other time I was on the receiving end in a Swiss draft. I was doing relatively poorly, but was hoping to at least pick up a win in the final round to get a pack back as I currently sat 0-2. Two people dropped, pairing me up to someone who was 1-1, and he ended up beating me.

    Not a frequent occurrence, but I did want to point out other people dropping can potentially affect your outcome.

  14. @Original-pete, Carrot and Hercules.

    ATM players dropping out are irrelevant to prices. If a player drops out another player will get a automatic victory and the prize structure will be the same.

  15. Also. In swizz you play against those with similar points as you. You play 3 matches and you play opponents you have not played before. If you look at the structure you will see that an opponent is dropping will almost always be good for the other players.

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