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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Is Elixir of Immortality right for my (limited) deck?

Marshall_LR is quick to dismiss cards like Elixir of Immortality. And with good reason – the effect Elixir has on the game is not worth the card you expend. The co-host of Limited Resources advocates a straightforward, value-based approach to card evaluation well suited to his audience and rejects cards that aren’t playable in most decks. The real life Marshall, however, is definitely interested in digging deeper and understanding when the normal case doesn’t hold. A couple of days ago, at our weekly poker game, Marshall posed the question “what kind of deck should run Elixir of Immortality?”

It’s an interesting question.

Let’s talk about what Elixir of Immortality does for you:

  • It gains you five life
  • It improves your future draw steps
  • It lets you recycle specific cards
  • It gives you inevitability
  • It protects you from decking

It looks like Elixir of Immortality does a lot of things! Unfortunately, the things it does are irrelevant or marginal for most decks. Specifically, recycling specific cards, gaining inevitability, and gaining (some) protection from decking are simply irrelevant to most decks. Recycling specific cards is only relevant if you are subsequently tutoring for those cards – if your plan is to go long with Diabolic Revelation then Elixir makes sense as a way to cast your removal spells and Archaeomancers over and over again, but it’s only relevant if that specifically is what you are doing. There aren’t many decks that want Elixir to provide inevitability, simply because there aren’t that many decks that can beat every card in their opponent’s deck but don’t have a way to actually win the game. And unless you have a ridiculous amount of card draw AND plan to go super long you simply don’t need a card in the main to protect yourself from decking. There are definitely decks that want Elixir for these purposes, Brian DeMars’s GP Boston winning deck was a UR Control list that needed Elixir to balance out his tremendous card draw and slow threats, for example, but the decks that want Elixir for one of these three effects are both very rare and easy to identify.

So what about all the other decks out there – do any of them want Elixir of Immortality?

In the rest of our decks Elixir of Immortality reads “Gain 5 life. Improve your future draw steps.” Is this effect worth a card to us? Well, that depends on how much we’re improving our future draw steps. I wouldn’t play a three mana spell that gained me five life and drew me half a card. There are some decks that might play that life gain spell if it drew a full card, but Renewed Faith was never exciting and I expect this wouldn’t be either. I would be excited to play Elixir if it drew me two cards, but does that ever happen? Let’s find out:

Let’s say it is turn eight and we have 25 cards left in our library. We’ve drawn all the lands we need for the game, so any future lands we draw are dead cards. Our deck started with 17 lands and 23 spells, all of our spells are reasonable draws, and we’ve drawn spells and lands in proportion to our deck, so our deck now contains 15 spells and 10 lands.  If we shuffle eight cards into our library with Elixir we’ll increase our spell percentage from 60% to 70%, which means we’ll draw one more spell than expectation after ten turns. That’s a long time to wait to get our one card back from the Elixir. If we want to get a card back from Elixir after only five turns of waiting we’ll need to shuffle 25 cards back into our library, more spells than were in our deck to start the game!

In order to actually be up a whole card on the deal (i.e. replace Elixir and draw an additional card) we have to jump through some pretty big hoops. If we wait until there are only 15 cards left in our library and we shuffle 15 spells from our graveyard into our library we’ll be up a card after ten additional draw steps.

Limited games just don’t go this long with any routine frequency. Most limited games are over in less than ten turns, so expecting to live ten turns after you use Elixir, and getting a large number of spells into your graveyard before using Elixir is just something that realistically won’t happen. There are some decks that can use Elixir of Immortality, but those decks want it because of specific properties with recycling cards or needing to not deck yourself. Elixir simply is not a viable value card – you’ll never get a full card out of it by using it fairly.

Cheers,

Joe

Post Script:

If you’re interested, here’s the formula I used to calculate how many turns it takes to “draw” a single card after using Elixir:

You can calculate how long it takes to draw X cards by changing the 1 in the numerator to X for whatever value of X you want.

Also, here are some sweet quotes from the LR guys about Elixir of Immortality:

  • “Everyone thinks it’s a freeroll, and it’s a complete blank.”
  • “You’re increasing your chance of drawing a spell by a small percentage, which is not card advantage.”
  • “This will be the most played F.”