Monday, May 27, 2013

Expected Value and Magic Online Drafts

I've been playing quite a bit more Magic online, and if you're playing a lot, the more you're losing per event is a lot more impactful on your pocketbook. I had read a few sources online regarding expected value (EV) and various MTGO event formats, as well as what they've said on the subject on Limited Resources. The general consensus seemed to be that 8-4s were the way to go, but I really wanted to see for myself, so I coded up some simulations in Java and let 'em rip.

I was also curious about the EV for Phantom Sealed events, so I included that in the simulations. Let's look at the first graph, which doesn't take into consideration the cost of playing an event or the resale value of cards:

I start at 50%, because if you're not winning at least that much, you're gonna have a bad time anyway. As you can see, 8-4 is indeed the most profitable payout structure, no matter what your win percentage. It has a steeper curve than the others, which means that it also scales better as you improve (smaller improvements in your skill lead to proportionally bigger earnings).

The much-maligned 4-3-2-2 isn't really that much worse than Swiss below about a 70% win percentage, and after that it is actually better than Swiss. But obviously, no matter what your win percentage, if you care about maximizing your earnings, you should be playing 8-4s. 

Phantom Sealed looks the worst here, but we're not considering the cost of playing the event, which is a pretty big factor. Let's look at the data when we take cost into consideration:

Here the payout is adjusted to tickets, and I've also made some assumptions that may or may not be valid. First of all, for the three draft formats, the cost is 14 tickets (assuming each pack is worth 4 tickets and the entry fee is an additional 2 tickets). I'm also assuming a pretty generous resale value of 5 tickets for the drafted cards (or about 1.67 tickets per pack). Your tendency to money rare draft will affect this estimate, but it seemed like a reasonable baseline for comparison.

I think the data gets a lot more interesting when we adjust for cost and resale value. The first thing I noticed is that 4-3-2-2 and Swiss are both unprofitable until you reach around 73-74% win rate, which is quite high. So you have to be playing at near the highest levels in order to just break even in those structures, although again, below that, the difference between them is not that large.

The next thing to notice is that you only need about a 60% win rate in order to be profitable at both 8-4 and Phantom Sealed (which cost 4 tickets, but don't let you keep any cards for resale). Assuming you were able to resell your packs at or near face value from Phantom Sealed, you could "go infinite" with a 60% or better win rate. Same with 8-4s, though since the payout is proportionally larger, it would be quite a bit easier to go infinite in 8-4s. Again, this is assuming a card resale value of 5 tickets per event, which may be right on, or may be much too high. 

I would love to see some actual stats for various players regarding win percentages with fairly large sample sizes. There are some obvious reasons why some top players may or may not want to share this information. If any of you out there keep records, by all means share if you don't mind. From the records I've kept, here are some of my stats:

Live play at my local game store:

88 events played
Match win percentage: 67.37% 

Some of that is a result of intentional draws either into or in finals, so it should actually be a little higher. My win percentage by set/format is as follows:







For records I kept drafting last year online, most of which were Swiss events, my average win percentage was 64.20%. This year, so far it's 64.58%, mostly playing 8-4s. So I seem to do a little worse online than in person. Part of that may be issues with the MTGO interface, slightly tougher competition online, or other variables. 

For a little more context, Paulo Vitor Damo da Rosa at one point had a 65% match win percentage at Pro Tour Events, and this was considered outrageously good, considering the level of competition. Another article notes that the all time winning percentages of pros in the Hall of Fame, starting with the highest all time are:

Jon Finkel, 63.39 %
Kai Budde, 62.86%
Olivier Ruel, 61.97%
Zvi Moshowitz, 58.58%
Tsuyoshi Fujita, 58.53%
Antoine Ruel, 57.72%

Again, this is a pro-level environment. We'd expect these players to have significantly higher win percentages in softer environments (though how much more...who really knows?). My intuition is that maintaining any win percentage above 75%, even for top players, is very difficult. Though again, I'd love to see some actual data.

One more note, since I'm interested in making a run for the Magic Online Championship this season (which runs from the middle of each month to the next). You need 15 qualifier points (QPs) to qualify for the prelims, and 35 to go straight into the main event. According to my rollout simulations, with a 65% win percentage, the distribution of event outcomes is as follows:

Events: 1,000,000
3-0: 273,396
2-1: 147,883
1-1: 228,048
0-1: 350,673

Some interesting things to note: Even with a respectable win percentage, you'll still need to play over 50 events, on average, to get to 15 QPs. You'd have to play about 130 events to get 35 QPs. Luckily, if you're playing at this win percentage, you're profitable. Still, that's a bit of a grind (you'd need to be playing about 2 events per day on average).

Another thing to note is that you will have more 1st-round losses than any other kind of outcome, so you shouldn't get too bummed when you are 1-and-out. This will happen the most often. If you didn't know this was a statistical certainty, you might get discouraged that you were going 0-1 more than 3-0, but even with a respectable win percentage, this is the norm. Along those same lines, this means you will not win any packs in well over half your drafts (58%). But that's okay. You'll have short term variance that may eat into your bankroll, but if you are a reasonably good player, you'll regain those losses if you keep at it.

Unfortunately, you're not going to buy a summer home with your winnings, but you will actually eke out a small profit while playing a game that you presumably really like. Seems like a decent deal to me.

If you have any questions/criticisms of my methodology, please let me know in the comments. Also, if you have any data to share, go for it.