The win rate fallacy and 4-3-2-2s

Many people have pointed out that, from a pure expected value standpoint, drafting in the 4-3-2-2 queues on Magic Online is a misplay because it pays out 11 packs, compared to the 12 packs that the swiss and 8-4 queues pay out. A common response is that I win more matches in 4-3-2-2s than I do in 8-4s and so I’ll end up winning more packs in 4-3-2-2s and thus they are better value. I’ve heard this response a number of times, but I’ve never actually seen anyone include their win percentages, so I decided that to dig in and find out whether there are win percentages that make 4-3-2-2s a better choice.

Let’s start out by looking at some data. I don’t draft a great deal online, so I don’t have a nice MtGO dataset of my own. But I do have my real life limited data that we can use to approximate my expected MtGO win rates. I went to the Planeswalker Points website and aggregated all of my limited match records. I grouped these data into three categories – matches in prereleases and release events; matches in PTQs, GPs, and Pro Tours; and all of the rest of my matches. Let’s map these three data categories to swiss, 8-4, and 4-3-2-2 queues respectively. This seems like a decent first approximation, although I’d imagine that my win percentage in actual 8-4s would be better than my PTQ/GP/Pro Tour win percentage. This data includes both sealed and draft data as otherwise I’d have very few data points for the top and bottom categories. So, mapping my results to the queue payouts, here’s how I did:

Queue Record Win Percentage Expected Value
8-4 80 wins, 51 losses 61% 2.4 packs/draft
4-3-2-2 192 wins, 78 losses 71% 2.29 packs/draft
Swiss 61 wins, 18 losses 77% 2.32 packs/draft

As we would expect I have a higher win rate in swiss than I do in 4-3-2-2s and I have a higher win rate in 4-3-2-2s than I do in 8-4s. But even though my win rate is 10% lower at 8-4s than 4-3-2-2s, I still have a higher EV in the 8-4 queue. This is partially due to the structure of the tournament (8-4s reward higher win percentages more than 4-3-2-2s), but it is also because there is a pack missing from the prize pool (if it was a 5-3-2-2 my EV would be 2.65). You’ll notice that I also have a higher EV in swiss than in 4-3-2-2. When people make the “my win percentage is better in 4-3-2-2s than 8-4s” argument they frequently don’t take into account that they probably have an even higher win rate in swiss.

Let’s look at some more data. My friend Daniel Duterte kindly let me publish his MtGO draft numbers from the last year:

Queue Record Win Percentage Expected Value
8-4 85 wins, 72 losses 54.1% 1.8 packs/draft
4-3-2-2 13 wins, 7 losses 65% 2.0 packs/draft
Swiss 13 wins, 2 losses 86.7% 2.6 packs/draft

OK, small sample size on 4-3-2-2s and swiss queues, but this is an example of how just crushing swiss queues gives you a huge edge on 4-3-2-2s. The whole “my win percentage is higher in 4-3-2-2s than 8-4s” argument is a double edged sword.

One more set of data, Limited Resources listener Vis posted his win percentages in the comments of one of the latest LR podcasts:

Queue Number of events Win Percentage Expected Value
8-4 35 events 61.9% 2.48 packs/draft
4-3-2-2 127 events 63.35% 1.92 packs/draft
Swiss 82 events 67.9% 2.04 packs/draft

Well, we keep having pretty bad EV with 4-3-2-2s. We’ve got a decent sample size here and see that 4-3-2-2 is wildly less profitable than 8-4 when there’s a small difference in win percentages and we’re a solidly winning player. And, again, swiss manages to edge out 4-3-2-2s, even though the win percentages aren’t as dramatic as they were in Daniel’s case.

So, we’ve got some real world data sets in which 4-3-2-2s seem to be a losing proposition, in terms of EV opportunity cost. Let’s take a look at the actual EV numbers for particular win rates and see if we can draw some conclusions. Here’s the relevant info:

Win % 8-4 EV 4-3-2-2 EV Swiss EV
5% 0.01 0.10 0.15
10% 0.04 0.21 0.30
15% 0.10 0.33 0.45
20% 0.19 0.45 0.60
25% 0.31 0.58 0.75
30% 0.47 0.72 0.90
35% 0.66 0.87 1.05
40% 0.90 1.02 1.20
45% 1.17 1.19 1.35
50% 1.50 1.38 1.50
55% 1.88 1.57 1.65
60% 2.30 1.78 1.80
65% 2.79 2.00 1.95
70% 3.33 2.23 2.10
75% 3.94 2.48 2.25
80% 4.61 2.75 2.40
85% 5.35 3.04 2.55
90% 6.16 3.34 2.70
95% 7.04 3.66 2.85
100% 8.00 4.00 3.00

That’s useful reference data, but it’s a bit hard to interpret. Let’s visualize this by looking at equivalent win percentages for 8-4s and swiss compared to the 4-3-2-2 queue:

4-3-2-2 Win percentage Swiss 8-4
5% 3% 15%
10% 7% 21%
15% 11% 26%
20% 15% 29%
25% 19% 32%
30% 24% 36%
35% 29% 39%
40% 34% 42%
45% 40% 45%
50% 46% 48%
55% 52% 51%
60% 59% 54%
65% 67% 57%
70% 74% 59%
75% 83% 62%
80% 92% 65%
85% N/A 67%
90% N/A 70%
95% N/A 73%
100% N/A 75%

Here’s how you read this chart: let’s say you have a 40% win percentage in 4-3-2-2s, so you look at the 40% line under “4-3-2-2 Win Percentage,” then look at the percentages for swiss and 8-4, which are 34% and 42%. This means that a 40% win rate in 4-3-2-2s is equivalent to a 34% win rate in swiss or 42% in 8-4.

Looking at the chart, it’s pretty obvious that if your win rate is 60% or less in 4-3-2-2s then you would do better by simply playing swiss queues, since you’ll have the same or better win percentages and a higher EV (actually the break-even point is at a 61.9% win rate). If you have a greater than 60% win rate in 4-3-2-2s there needs to be a pretty large difference in your win rates between 4-3-2-2 and 8-4 queues AND there has to be a small difference in win rates between 4-3-2-2 and swiss queues for 4-3-2-2s to be correct. The win rate differences between 4-3-2-2s and 8-4s are so big, however, that this isn’t really plausible. If you win 70% of your 4-3-2-2 matches then you should be able to win either 60% of your 8-4 matches or 75% of your swiss matches (or both!).

It turns out that the “I win more matches in 4-3-2-2s than I do in 8-4s, so I’ll win more packs by playing 4-3-2-2s” argument just doesn’t hold any water. First off, you have to consider swiss queues as well, and if you assume that your win rate is higher in 4-3-2-2s than 8-4s, you also need to assume that your swiss win percentage will be higher than either of the others. Given that, you now need to be an excellent drafter with a greater than 61.9% win rate for 4-3-2-2s to possibly be better than swiss queues. And you need to be doing much worse in 8-4s than you are in 4-3-2-2s, like 10% worse or more, to not move up to 8-4s. These circumstances just aren’t going to occur for the vast, vast majority of drafters.

The best way to figure out which queue to play in is to keep track of your results, determine your win percentage, and then figure out which queue offers the best value. If you want to challenge yourself and play against the best, or if you are a very winning player, you should draft 8-4s. If you’re just want to play some magic, or if you’re still learning and are not yet a strong drafter you should draft in the swiss queue. But please, do not ever draft in 4-3-2-2 queues – they’re a terrible deal that you always come out behind on.

Cheers,

Joe

Postscript:

For the mathy completionists out there, here are the equations I use to calculate pack EV:

  • Swiss: EV = 3 * Win%
  • 4-3-2-2: EV = Win% * (Win % * (4 * Win% + 3 * (1 – Win%)) + 2 * (1 – Win%))
  • 8-4:: EV = Win% * Win% * (8 * Win% + 4 * (1 – Win%))
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3 thoughts on “The win rate fallacy and 4-3-2-2s

  1. Great article. One aspect that wasn’t discussed was the ‘net EV’, or # of packs won minus all the entry fees (3.5 packs, if you’re paying retail), which would give you a feel for where you would need to be to ‘go infinite’ on MODO. That would be a win rate of 75% in 8-4, 95% in 4-3-2-2, and impossible to do in Swiss. 🙂

  2. While it’s true that I didn’t address net EV specifically, it actually doesn’t matter. Since the entry cost is the same for all three you maximize your net EV by maximizing your gross EV.

  3. Pingback: The draft queues conundrum | MTG Turtle

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