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By: Shaterri, Steven Stadnicki
Oct 26 2009 10:05am
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With Zendikar newly released on MTGO, draft queues are packed to the brim again, and players are once more being tempted by the shiny rares they're offered.  Obviously if you crack a foil Lotus Cobra you're going to take it and easily pay for your whole draft, but what about that Goblin Guide?  What if you see a mid-pack Emeria, or your third pack offers a Chandra Ablaze to go with the 25 blue and white cards you've already drafted?  Just how valuable does a rare have to be before it's worth taking over a better card for your deck, even over an outright bomb?  And for that matter, which queue should you be joining in the first place, assuming your aim is to maximize your return on the draft investment?

Of course, there's no one right answer to these questions. Every draft is different, every opponent is different, and even the time of day can have a huge impact on the quality of your opposition.  But a few simplifying assumptions and a little mathematics will at least offer some straightforward guidelines. Since these assumptions are the biggest constraint on the process I'll make them explicit whenever I can, but they have a way of sneaking in unnoticed, so don't take what I say as gospel!

To answer these questions we'll need some basic probability theory.  The math we'll use centers on two core notions: event probability and expected value.  Event probability is just what it says it is, the likelihood of something happening; for convenience's sake, it's usually represented as a number between 0 (impossible) and 1 (guaranteed).  A probability of .5 means that something has a 50/50 chance of happening, a probability of .9 means it'll happen 9 times out of 10, and so on.  With this convention, probabilities satisfy three simple mathematical properties:

  • First, if the probability of something happening is X then the probability of it not happening is 1-X: if my probability of drawing a blue card is .2, then my probability of drawing a non-blue card is .8.
  • Second, and most importantly, if the probability of one thing happening is X, and the probability of something else happening is Y, and the two don't influence each other, then the probability of both happening is XY: if the probability I draw a creature card is .5, and the probability that my opponent has Essence Scatter is .3, then the chance that I draw a creature card and my opponent has Essence Scatter is .15.
  • Finally, if the probability of one thing happening is X and the probability of another thing happening is Y and it's impossible for both to happen, then the probability of one or the other happening is X+Y (more generally, it's X plus Y minus the probability that both happen): if the probability that I draw a blue card is .2, and the probability that I draw a red card is .1 (and I have no multicolor cards in my deck), then the probability that I draw either a blue or a red card is .3.

Expected Value is a little harder to define, but fortunately it's easy to calculate.  For our purposes, it's the average value of a draft payoff; if your EV for a draft is 1.5 packs, then over the course of a hundred of those drafts you can expect on average to win 150 packs.  To calculate the EV for a set of possible outcomes, multiply the value of any given outcome by its probability and then add the results together.  For instance, if we figure our odds in a 4-3-2-2 give us a 10 percent chance of winning 4 boosters, a 25 percent chance of winning 3 boosters and a 40 percent chance of winning two boosters (with an implicit 25 percent chance of winning nothing), then our EV for that draft would be 4*.1 + 3*.25 + 2*.4 = 1.95 boosters.  One important fact here is that EVs for independent events are additive: if you play two drafts, one with an EV of 1.25 boosters and one with an EV of 1.5 boosters, then your total EV for the two drafts is 2.75 boosters.

Because we're talking about rare-drafting and how it changes your EV, it makes sense to start off with game probabilities.  This is where the first explicit assumption comes in: namely, that the probability of winning game 2 (and if need be, game 3) is the same as of winning game 1.  This means we're ignoring the influence of sideboarding, and also means that we're ignoring the importance of having the play/draw choice.  Factoring in those elements would substantially complicate the formula; since this is an approximation anyway, I'm happily glossing over them.

Since our expected values for the different draft queues will depend on the probability of winning matches but a rare-draft decision changes probabilities on a game-to-game basis, the first thing we need is a formula to convert from game probabilities to match probabilities.  If the probability of winning a single game is X, then the chance that we win the match outright in two games is X*X; the probability that we win the first and third games (and lose the second) is X*(1-X)*X; and the probability that we lose the first, but win the second and third, is (1-X)*X*X.  Since it's impossible for more than one of these to happen, the overall probability of winning the match is M(X) = X*X + 2*X*X*(1-X).  Here's that match win probability, tabulated against game win probability:

X M(X)
0 0
0.05 0.00725
0.1 0.028
0.15 0.06075
0.2 0.104
0.25 0.15625
0.3 0.216
0.35 0.28175
0.4 0.352
0.45 0.42525
0.5 0.5
0.55 0.57475
0.6 0.648
0.65 0.71825
0.7 0.784
0.75 0.84375
0.8 0.896
0.85 0.93925
0.9 0.972
0.95 0.99275
1 1

Note that if X is 0, or .5, or 1, then so is M(X); if we can't win a game we can't possibly win the match, if we can't lose a game then we can't possibly lose the match, and if a game is a coinflip then so is the match, since then the opponent's chances of winning two out of three are exactly equal to ours.  If our odds of winning are better than 50-50, then our odds of winning the match are better than our odds of winning a game, and if they're worse than 50-50, then our match odds are worse than our game odds; this is part of why matches are 2-out-of-3 in the first place, to give the better player a better chance.  For clarity's sake, I'm going to abbreviate the match win probability as just 'M', with the dependency on the game probability X omitted except where it's explicitly relevant.

Now that we have a match win percentage, we can start figuring out EVs for the different draft queues.  There's another fairly subtle assumption at play here, namely the idea that you want to maximize value on a draft-by-draft basis.  If your play model is to try to get in one draft a night, or one every three nights, or whatever, then this makes sense; on the other hand, if you're planning on spending all day just entering draft after draft after draft, then you might be more interested in knowing your hourly rate.   If your EV on a Swiss is slightly higher than your 4-3-2-2 EV, for instance, then the longer runtime of a Swiss draft means that it might be worth your while to play in the 4-3-2-2 just because it lets you get in more drafts over the course of a day.  This effect is hard to estimate, so I'm skipping it for this article.

The simplest queue to figure out EV for is Swiss: since you get one booster if you win your first match, one if you win your second match, and one if you win your third match, and since these are independent events, then the EV for the draft as a whole is EVSwiss(X) = 1*M + 1*M + 1*M = 3*M(X).  If you're paying close attention, you might have noticed that I snuck in a huge simplifying assumption there: I claim that these are independent events, but on average you're going to find a better caliber of opponent (and thus, have a lower win probability) if you're 2-0 in match three than the random opponent in your first match.  This is even more exacerbated in the elimination queues, where your only chance of winning the top prize is to beat someone who's also 2-0 and thus presumably skilled.  This is another factor that's hard to properly estimate so I've chosen to omit it here, but it's relatively straightforward to tweak these formulas to use, say, M(X-.05) instead of M(X) to represent the third round match probability if you want to assume that a finals opponent will have a 5% higher chance of winning a game than a random opponent.

With that issue nimbly elided, it's straightforward to calculate the EVs for the other two draft types: for a 4-3-2-2, your payout is 2 boosters if you win your first-round match and lose your second-round match (with the probability for this being M*(1-M) since the two are independent), 3 boosters if you win your first- and second-round matches and lose the third (probability: M*M*(1-M) ), and 4 boosters if you win all three matches (probability M*M*M); this means the total EV is EV4322(X) = 2*M*(1-M) + 3*M*M*(1-M) + 4*M*M*M.  Similarly, the EV for an 8-4 is EV84(X) = 4*M*M*(1-M) + 8*M*M*M..  I've written (and tabulated) all of these as functions of the game win percentage, since that's what we're going to need when it comes to estimating the impact of a raredraft.  Here are the EV tables for the different kinds of draft queues:

X   EVSwiss(X) EV4322(X) EV84(X)
0   0 0 0
0.05   0.02175 0.0146 0.0002
0.1   0.084 0.0568 0.0032
0.15   0.18225 0.1254 0.0157
0.2   0.312 0.2199 0.0478
0.25   0.46875 0.3407 0.1129
0.3   0.648 0.4887 0.2269
0.35   0.84525 0.6652 0.407
0.4   1.056 0.8715 0.6701
0.45   1.27575 1.1082 1.031
0.5   1.5 1.375 1.5
0.55   1.72425 1.6697 2.0808
0.6   1.944 1.988 2.768
0.65   2.15475 2.3229 3.5457
0.7   2.352 2.6645 4.3862
0.75   2.53125 3.0001 5.2504
0.8   2.688 3.3141 6.0886
0.85   2.81775 3.5893 6.8432
0.9   2.916 3.8071 7.4525
0.95   2.97825 3.9495 7.8558
1   3 4 8

Comparing EVs for the different queues reveals something interesting: the 4-3-2-2 queues are never the 'best' choice.  If your win percentage against the field is less than 50%, then the Swiss queues are your best bet; if it's better than 50%, then 8-4 becomes better than 4-3-2-2.  In fact, this still holds if we skew the win percentages to account for different opponent strengths: if you assume that your win percentage against the average Swiss opponent is 5% better than vs. the average 4-3-2-2 opponent, and 5% better against the average 4-3-2-2 than the average 8-4 opponent, then Swiss becomes your best bet up to about a 65% win percentage vs. the field (and thus a 55% chance against the 8-4 opponents), and above that 8-4 takes over again.  In fact, Swiss and 8-4 queues still dominate if you assume a 10% strength difference — all that changes is the crossover point.  4-3-2-2 only becomes the best option if you assume that opponent quality is similar between 4-3-2-2 and Swiss, but much harder in the 8-4 queues — and even then, the 4-3-2-2 queues aren't better by much and only in a very short win probability range.  That's not to say that you should never play in 4-3-2-2s; as I mentioned in passing above, it's possible that the (potentially) shorter duration of a 4-3-2-2 queue could give you a higher EV per hour than the Swiss drafts.  But in general, 4-3-2-2s are seldom your best option.

To figure out what the EV impact of a raredraft is, we first need to figure out its impact on game probability.  Imagine that you normally have a 60% chance against the field, and you're given a hypothetical 0-mana 'I win' sorcery.  How much would that improve your win probability?  Given that draft matches generally last about 8-10 turns, you have roughly a 40% chance of drawing the card. In the 40% of games where you draw it, you auto-win, but of the other 60% of games you play, you'll still win 60% of the time (your abstract chance against the field), so your total chances of winning are .4*1 + .6*.6 — in other words, about .76; you've increased your probability of winning by 16%.  If your normal win percentage against the field is 70%, then your chances with the card in your deck go to .4*1 + .6*.7, or .82; you've increased by 12%.  Of course, Wizards hasn't actually printed a 0-mana 'I win' spell yet (after all, even Skullclamp cost 1!), and even the best of bombs still run into circumstances where you just can't cast them, or they don't affect the board enough to win you the game.  10% seems like a reasonable upper limit on the how much a single card can improve your winning chances, and 2-5% would be more common for even very good cards.  Rare-drafting will mean giving up that increase to your winning chances, and so change the EV of your prizes for the draft by some amount; for instance, if you give up a card that would have improved your winning probability by .05, then the overall change in your EV is EV(X+.05) - EV(X).  To have a net positive impact on your overall EV for the draft, then, the rare has to be worth at least as much value you're giving up by passing the good card.  Fortunately, we don't need different versions of the EV change table for different changes in win probability; over any small range the EV function is almost a straight line, so the change in your EV for a '10% bomb' is pretty close to double the EV change for a '5% great card'.  The following tables show the change in EV for all three drafts, for that hypothetical .05 win probability card.

X   DeltaSwiss(X) Delta4322(X) Delta84(X)
0   0.02175 0.0146 0.0002
.05   0.06225 0.0422 0.003
.1   0.09825 0.0686 0.0125
.15   0.12975 0.0945 0.0321
.2   0.15675 0.1208 0.0651
.25   0.17925 0.148 0.114
.3   0.19725 0.1765 0.1801
.35   0.21075 0.2063 0.2631
.4   0.21975 0.2367 0.3609
.45   0.22425 0.2668 0.469
.5   0.22425 0.2947 0.5808
.55   0.21975 0.3183 0.6872
.6   0.21075 0.3349 0.7777
.65   0.19725 0.3416 0.8405
.7   0.17925 0.3356 0.8642
.75   0.15675 0.314 0.8382
.8   0.12975 0.2752 0.7546
.85   0.09825 0.2178 0.6093
.9   0.06225 0.1424 0.4033
.95   0.02175 0.0505 0.1442

Keep in mind that the values in all these tables are expressed in terms of packs, since that's what the draft queues pay out in; to convert to ticket value you'll need to multiply by whatever the value in tickets of a single pack of the set is.  For rough estimates in what follows, I'm going to use 3.5 tickets as the value of a pack.

For most players, the most important parts of these tables are from about X=.4 (that is, a 40% game win percentage) to about X=.7 (a 70% game win percentage) for the Swiss and 4322 queues, and only up to about X=.6 for the 8-4 queue; if you're winning 70% of your games in 8-4 drafts, then you've already gone infinite off of prize packs alone and you have much better things to do with your time than read this article!  Even over these ranges, though, the different queues show striking differences.  The Swiss EVs are fairly consistent over that range, right around .2 packs no matter what your 'normal' win percentage is.  This means that if a raredraft changes your win probability by 5%, it only needs to be worth about three quarters of a ticket to be worth it; if the pick changes your win probability by 10%, it has to be worth about 1.5 tickets, regardless of what your 'normal' game win percentage is.  For instance (signals aside), it's likely to be mathematically correct to take Birds of Paradise ($1.65 as I write this) over e.g. Overrun in an M10 draft, but probably wrong to take Ball Lightning ($1) over it.  On the other hand, the 8-4 queue shows a much more dramatic range: if you think you're about 40% against the field, then that '5% rare' needs to be worth about a third of a pack — in other words, a little bit more than 1 ticket for the 5% rare, or about 2 tickets for the 10% range.  But if you think you're about 60% against the 8-4 field, then the 5% rare needs to be worth about .75 packs, or nearly 3 tickets, before it's the right pick.  A raredraft with a 10% change in win probability has to be worth more than 5 tickets to be worthwhile: if you think you're 60% in an 8-4 queue, you're better off picking Fireball over something like a Dragonskull Summit!  4-3-2-2, of course, splits the difference, ranging from about .7 to 1.2 tickets for the 5% win probability change, or 1.5 to 2.5 tickets for a 10% change.

Of course, all of these are estimates; none of this is meant to be a hard-and-fast guideline.  That being said, if you're interested in maximizing the value of your drafts — or even just interested in a little bit of the mathematics behind the game — I hope this has been interesting, and maybe even useful.  Thanks for reading, and may all your drafts be full of bomb rares!


As a math guy, I like this by ArchGenius at Mon, 10/26/2009 - 11:24
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As a math guy, I like this kind of article.

If nothing else, it shows how difficult it is to make a profit through rare-drafting.

Like you said this analysis is good as a hard-and-fast guideline. As a model of a typical draft situation I don't think it's very accurate. That's because in any single-elimination or swiss type event, you're probability of winning each round changes. The last round of a 8-4 draft is most often much more difficult than the first round. If you're playing swiss, the last round with a 2-0 record is going to be much more difficult than a last round with a 0-2 record. Of course, modeling that type of situation is an order of magnitude more difficult than modeling constant win percentages. And, in the end, it's probably a waste of time.

I don't think many drafters look all that closely at the expected value of winning prizes. If they did, they would probably see that they have to do a lot better at drafting than sealed deck or constructed in order to win the same level of prizes.

Rare-Drafting is kind of like the lottery. Players don't look at the probability of opening a Baneslayer Angel or Lotus Cobra, but they draft because it gives them the feeling that they could get a "free" draft by simply being lucky and opening one of those cards. Even the thrill of the possibility of opening one of these cards makes drafting worth it despite the difficulty in making a profit. And even if I don't open that Baneslayer Angel, there is always that possibility that I can win my way to another chance at it.

I suspect most drafters are by Shaterri at Mon, 10/26/2009 - 12:35
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I suspect most drafters are actually aware, at least in the back of their heads, that drafting isn't as profitable as sealed deck; but even a four-round sealed represents a substantially higher investment in time than a draft does, and I suspect that many people's threshold between 'feasible to spend an evening on' and 'too much time', especially for weekday evenings, falls somewhere in between the two. (I'd put mine, for instance, at about 3 hours or so, which lets me draft occasionally but would hardly be enough for a sealed event.)

Constructed I'm less convinced of; not only is the initial outlay substantial, but my impression has been that the competition level in constructed queues is still sufficiently high (precisely because of that barrier-to-entry) that it can be inordinately hard to do reasonably well in the heavily-played-formats. (My record in constructed events outside of the handful of Kaleidoscope events I've played is something like 2-8, and I'm a solid 18-1900 level player RL.)

And the thrill that you're talking about is a fascinating phenomenon in its own right. Really, it's another facet of the well-established economic principle that people have a highly non-linear utility function for money: a ten-ticket rare has an appeal that's much more than 10 times that of a one-ticket rare. It's a major part of the reason why lotteries are so popular (along with, of course, a general misunderstanding of the math behind them).

I don't like the stat by BDirg (not verified) at Mon, 10/26/2009 - 11:43
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I don't like the stat analysis of rare drafting. The major problem is that X is arbitrary, i.e. completely made up in all cases. Say it's p2p1 and you are looking at a super playable card vs a rare draft...well you don't really know how much that playable card will help your deck since you still have half the deck to draft. You have no idea how much a single card will help your chances of winning and if you never draw it (has happened to all of us) it helped your chances by 0%.

I have a very simple rule when decided between playable vs for the tix: if the card is worth a pack (3-4 tix) take it, otherwise go the playable. If it's a bunch of chaff vs a 2 tix card I take the 2 tix every time, but I don't think that was the point of your article.

It SHOULD be common sense that since 4-3-2-2 pays out 11 packs and Swiss pays out 12 with the same entry fee that in the long run your expected average payout will ALWAYS be higher with Swiss (or 8-4), yet 4322 is still the most popular format. The common argument is that Swiss is too long and 8-4 is too hard so people go with 4322. It makes no mathematical sense, but people still do it. You would think being able to lose the first round and still win 2 packs would be attractive to people, but I guess it just isn't.

Swiss has the problem that by ArchGenius at Mon, 10/26/2009 - 12:00
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Swiss has the problem that even if you win all your games, you still can't do better than you started.

With 4-3-2-2 you at least have a chance of doing better than breaking-even. That chance is enough to make it more appealing to some people than swiss. Even if it is worse in the long run.

I believe the popularity of 4-3-2-2 is more about Psychology than Statistics..

The thing is that in the long by Shaterri at Mon, 10/26/2009 - 12:24
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The thing is that in the long run, your expected average payout won't always be higher with Swiss than 4-3-2-2s; the transition between the two happens at a win percentage (about 60%) that's well within most players' reach. What makes 4-3-2-2 less appealing there isn't that it doesn't beat out Swiss — it does — but that once you hit that range you're probably better off playing 8-4s. I suspect the main factor attracting people to 4-3-2-2s over 8-4s is the lower variance, but going into much detail on that topic in this article would've only made it even more complicated, and I figured people's eyes were likely to be glazing over as it was. :-)

As far as the statistical analysis of raredrafting goes, it's not meant to be a hard-and-fast formula; but I do think that it offers, or at least confirms, some useful guidelines with respect to mid-range rares. Just because X isn't wholly known doesn't mean it's 'completely made up'; MTGO even gives a pretty good means for estimating your X vs. the various queues by checking your game history.

I agree with you. Over the by ArchGenius at Mon, 10/26/2009 - 12:46
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I agree with you. Over the long run, Swiss will beat out 4-3-2-2. However most players don't look at the long run. That's my point. No matter how much you tell them it's not mathematically sound, people will still do it.

I've had similar experiences with teaching students why buying a lottery ticket is a bad idea.

As for 8-4, it can burn people. Back before Swiss was an option, I played a lot 4-3-2-2 drafts and got to the point where I was just about breaking even over the long run (over 100 drafts). Then I decided to try 8-4 because the payout is better. I tried 10 8-4 drafts in a row and ended up winning a total of 4 packs.

Now I realize that if I kept at it, over the long run my numbers would improve, but losing 9 out of 10 drafts was enough to convince me to stay away from 8-4 drafts. Even though I think I've gotten much better at drafting since then, I still don't see myself going back to 8-4 drafts. I imagine there are others out there who have had similar experiences with the 8-4 drafts.

thoughts by speks at Mon, 10/26/2009 - 13:59
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Very informative article even if it hurts my head from reading it haha, i'll have to come back and read the rest when my headache is gone.

I agree with one of the other commentator that X can vary alot depending on the pack. Like in the 1st pack 1st pick, even if you open a 10% bomb ie: overrun, you still can't be sure that you'll be able to play with it because of the steep triple green requirement. Sure you can force green, but it won't be a 10% bomb if people are fighting for your color. So if theres even a decent rare in the pack, say a Ball Lightning ($1) just to stick with your examples, I think the higher EV play would still be to take the rare and ship the bomb and make a mental note that sending your neighbor a strong signal into green and allow yourself more openess to reading the signals that are coming to you. In pack 2 however, this becomes more of a coinflip, depending on the strength of the bomb vs price of the rare. In pack 3, the balance shifts heavily to the bomb, if you open a bomb in your color in pack 3, it is very likely to help you win additional games in the draft, the rare has to be at least be able to pay for a pack or 2 to be able to even worth consider taking here.

So i guess over the course of 3 packs, X shifts from 1 end of the spectrum to the other, so it kinda balances out for your analysis purposes. But still, I think it is worth mentioning that X is not a constant.

Oh, absolutely. Each pick has by Shaterri at Tue, 10/27/2009 - 01:22
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Oh, absolutely. Each pick has to be evaluated on its own merits, and the DeltaX of any given pick (that is, the change in win percentage) isn't much more than a rough guesstimate — my '5%' and '10%' figures are meant to be back-of-the-envelope starting points more than anything else, good enough to crank through the formulas and come out with a reasonable range for final numbers.

One additional factor you missed, incidentally, is that when you choose to pick a rare over the better card for your deck it often increases the DeltaX of the next rare-draft decision, as the falloff from the metaphorical '23rd card' to a 25th or even 26th card can be much steeper than the gap between the 22nd and 23rd. And contrariwise, if you think you can be assured of having plenty of depth in your draft — if you're obviously somehow the only black drafter at the table, for instance — then you can rare-draft more agressively than you might otherwise simply because your opportunity cost is drastically reduced. I've even had the joy of rare-drafting something midrange over a good card that tabled.

i like 4322 cause my 75% 1st by Anonymous (not verified) at Mon, 10/26/2009 - 15:29
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i like 4322 cause my 75% 1st round win and 66% of the next draft is better than the 0% for the next draft in 8-4

swiss is time consuming but i enjoy for the first week of release or if i have the time to play three games, i would prefer leagues.....grrrrrr

i use to rare draft the crap out of sets, all the way down to a 1450 rating! now those cards are worth nothing. likewise i hve had many a draft where i am taking one rare, 4 uncs and all the rest commons and poof, 1720+ it really depends on the set of variables in front on me. i was passed a p1p2 ablaze, took it, tried to go red, thought that was a good signal, turns out i was wrong bad move for me.

lots of math and stats, woozy head.

Yeah I have found that by Paul Leicht at Mon, 10/26/2009 - 15:38
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Yeah I have found that drafting rares is very inconsistent across sets. Some sets reward raredrafting in the sense that the Rares can be bombs worth splashing for (ARB for example). Others the exemplary commons and uncommons outweigh the slight value advantage of the Rare by a good margin. It is always hard to pass a card like Fireball, Lightning Bolt, etc. Woozy head indeed. :)

I don't play to "make money" by Urzishra (not verified) at Mon, 10/26/2009 - 15:48
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I don't play to "make money" or care about the estimated value of said prize winnings of drafts. I draft because its fun and the quickest and easiest way to play sanctioned events on MTGO.

i play in the 4-3-2-2 because I can consistently win the first match enough of the times for me that it is basically paying 1 pack and tix for me to play MTGO, which I don't mind too much.

I don't think anyone actually by Shaterri at Tue, 10/27/2009 - 01:04
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I don't think anyone actually plays to make money; given the outlay involved and the variance, it's likely impossible for even the best drafters to clear minimum wage — and that's assuming that they can successfully multitable! That being said, it's IMHO still worth the while to know when picking 'for profit' is a good idea, if only because it minimizes the extent to which drafts eat into your savings. If anything, I probably rare-draft less than I abstractly 'should' because most of my value comes in understanding how to draft a set and in playing the games.

Nice article! by Felorin at Mon, 10/26/2009 - 16:35
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I like analyzing probabilities too. I think you did a good job choosing which simplifying assumptions to make to get reasonable ballpark estimates here. That said, I'll mention another of the factors people could use to adjust this analysis to suit their own case, which is that a given rare has different value to different people.

If your normal use of rares is to sell them to help fund future drafting, their value is the price you can sell them at (which varies depending on whether you sell to bots, or invest time to find another player who'll buy them for a bit more). If you're keeping them for collection and/or constructed, and you absolutely would have bought that rare if you hadn't cracked it in draft, then your value is the price the card sells at, which is a bit higher. This value falls off dramatically once you already have 4 in your collection (or 1, if you never play anything but Elder Dragon Highlander & other singleton formats!)

For someone like me, who keeps the rares to use and enjoy but in many/most cases wouldn't buy that rare if I didn't open one, the value's harder to estimate, and is based on some nebulous estimate of how much happiness and fun I derive from owning the card. The ratio of this value to the value of a dollar changes over time also, as my income goes up (or occasionally down - dang recession!)

One could also factor in the psychological (happiness and fun) value of winning more games/matches/prizes vs. the value of the rares. Or in my case (as some others have concluded), the value of maximizing the rate at which one increases ones skills at drafting and winning. I feel that if I sacrifice some percentage of my games to raredraft, I'm not going to learn quite as well the nuances of how I would have played that same draft had I focused 100% on winning and ignoring every card's cash value. I want to improve my skills as much as possible as fast as possible, so I usually won't consider taking a rare over a card that would improve my deck more unless it's worth at least 10 tix.

That said, or course as the amount a card increases your win percentage drops from 5% down to 1% or below, the threshhold for raredrafting gets a lot lower. In a pack full of junk, I'll often take a rare. If a pack offers me the upgrade of my 23rd playable from a 2/2 bear to a 2/3 dude, again I could happily take a rare there. If it isn't the 3rd pack, I'll likely be offered a better upgrade later anyway. Of course if it's a rare that goes for 10 or 15 cents, I might take the modest playable over it anyway, especially if I already own a playset. Though having a few extras to trade has a little value for me too.

I think the fact that you *could* come out ahead in a 4-3-2-2 or an 8-4 being so often cited by people who generally do not *actually* come out ahead in many of their drafts is somewhat comical. And another example of non-linear psychological factors. To me, getting 3 prize packs is 75% as good as getting 4, and getting 2 prize packs is 50% as good. For many people though, that "I beat the system" rush of getting 4+ boosters is a Big Deal. Even if they started out with $14+ in real cash, and converted it to a "currency" that can't buy food or movie tickets but only another draft & thus is less valuable from a realistic economic analysis.

While I think it's plausible I could "go infinite" if I abandoned my career and dedicated myself to drafting until my bank account ran dry, the loss of income would hardly be compensated by the value of my booster pack winnings. I prefer to analyze drafting in terms if it offering varying sizes of discount on my next draft. To a person saying "I can do this free, or I have to pay in money sometimes", the difference between 4 packs and 3 is huge. For me, 1 pack is a decent savings, 2 packs is better than a 50% off coupon, and 3 packs says "Your next draft is 2 bucks, unless it's a nix tix week for that format, then congrats!" 4 packs merely becomes "2 dollar draft plus 1 pack towards the draft afterwards". (Since I don't sell packs to get the tix to get a freebie).

So while as mentioned above, the 4-3-2-2 has less variance, I like the swiss drafts for even lower variance still. Though I'm probably over the 60% win threshhold where the non-swiss formats both give better EV. If you factor in that for some (many) players, the experience of playing a sanctioned match for potential prizes has entertainment value, 8-4 and 4-3-2-2 both offer 1 to 3 of those experiences, whereas swiss gives you 3 matches every time. If you come up with some estimate of the cash value of the fun you have playing more rounds, you can figure out how much that skews you towards swiss drafts. (Of course if your time is scarce and you're mostly or entirely focused on prizes, the potential to get more drafts per unit time would skew you more towards the non-swiss drafts. This is also true if you enjoy the card picking and/or deck building far more than you enjoy the gameplay of the matches.) There's also the issue of consistency of budgeting time. Having a very busy life, it's more practical for me to budget in a fairly fixed amount of time spent on a leisure activity than a random amount.

I also approach the choice as a min-maxer. A draft where I at least win something is more psychologically satisfying to me than one where I don't win anything. If I were to play in 8 mans and win 70% of my games, I'd get more free or highly discounted drafts than I'd get playing in swiss. But about half the time I'd be going back to the rest of my life thinking "I didn't win anything this time". If I want to "maximize the minimum result", in swiss drafts I almost ALWAYS win at least booster, there's been only maybe 1 or 2 times when I didn't. Then I go about the rest of my life thinking "I won some stuff", and it's usually at least 2 boosters, which lifts my spirits a bit. I also rarely hit the case where I have to pay a full $14 to play a draft, which I think is a bit of a high price to pay for that quality and quantity of entertainment, most of my drafts cost $10 or $6 or less. (I like to stockpile boosters from the release 16 man swiss too, but that's another whole subject. Great prize payouts on those, not too hard to get 10 boosters and then feed them into my drafting habits.)

I might pay less in the long term if my drafting skills are over that threshhold. But it would be more uneven, and for me that un-evenness might make the game less enjoyable overall. I still intend to try 8-4 queues eventually and see if the empirical experience matches my theoretical thoughts or not - when/if I get more serious about Magic the 8-4 queues are the place to be, if I get THAT serious about it. Time will tell. I will say that given the average skill differences one may be likely to find in different queue types, even from a simple pack-win EV standpoint, it's not as simple as comparing straight across the chart. If it were the case that I can win 65% of the time in Swiss, but only 55% of the time in 8-4 queues because of higher caliber of opponents, my EV is still higher in Swiss for the moment.

don't forget us by got stuff to do (not verified) at Mon, 10/26/2009 - 18:38
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don't forget people that have stuff to do.. they think it's wise to join a 4-3-2-2 so they can draft some nice cards and have a shot at two tix before having to look busy again :)

This is a big group of people.

I just want to reiterate this by Umii at Mon, 10/26/2009 - 18:53
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I just want to reiterate this point. I know Swiss drafts are better EV than 4322, but my time costs money. Swiss drafts seem to attract slower players, and often run 3 hours, while 4322 drafts are usually over in less than 1.5 hours. If my time is worth $10/hr (conservatively), 4322s are the obvious choice. The only time I really like Swiss is when a set just comes out, and I need to get a feel for the format.

time by speks at Mon, 10/26/2009 - 19:13
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Just to clarify, swiss draft does not attract slower players, the players are exactly the same.

The longer duration is because nobody gets eliminated in a Swiss, so there are 4 matches every round that needs to play out to its conclusion, chances are pretty good that at least one of those match is going to take the full hour.

In 4-3-2-2, and 8-4 drafts, every round, the number players and matches halves, so chances are that the round isn't going to take the whole hour for 2 matches to finish. And the finals is just you vs your opponent, theres no wait time.

I know what you're saying, by Umii at Tue, 10/27/2009 - 01:54
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I know what you're saying, and thought about qualifying that it's just my perception in my original post. But it always seems like every round of a swiss goes to time due to slow play, whereas the first round of a 4322 is usually pretty quick. And in a swiss, I might walk away to get a snack/do chores, further lengthening it.

Oh, I understand completely by Shaterri at Tue, 10/27/2009 - 01:09
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Oh, I understand completely — this is a factor I alluded to in the article a couple of times, but the truth is that it's so hard to estimate that I didn't even want to try here. I think what you're talking about is a smaller-scale version of the same phenomenon that I was talking about in one of the other replies with respect to why people play more drafts than sealed-deck events online; it's just much easier to scrape together 2 hours and maybe a little longer for a 4-3-2-2 draft than it is to manage a guaranteed three hours and often a bit more for a Swiss.

tix = packs by got stuff to do (not verified) at Mon, 10/26/2009 - 18:39
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Tix = packs in the above post

I only raredraft when I know by The D.K. at Mon, 10/26/2009 - 21:07
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I only raredraft when I know for a fact that the particular rare I'm drafting is worth a pack or more. For example, I was Swissing the M10 drafts and got a Liliana Vess in pack 2, although I was red/green. I figured, "Hey, it's 5 tickets in the aftermarket, that's enough to pay for this pack and then some if I don't win." (Which is a big deal for me, since I'm so new to drafting, I gotta make a return somehow.)

Awesome article. Seriously by Tremor (not verified) at Mon, 10/26/2009 - 22:43
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Awesome article. Seriously awesome. I'd often weighed the opportunity cost of rare-drafting in my mind, but never took the time to work out the math and figure out exactly where the cutoff should be.

I agree with ArchGenius about by Amar at Tue, 10/27/2009 - 01:45
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I agree with ArchGenius about the psychology point.

Perhaps you lose more than 5% moving from 4-3-2-2 to 8-4. 10% seems like the minimum difference, knowing who they attract. And when practicing for a GP or other serious event, top players want 8-4s BECAUSE of the competition.

Approaching it from another angle, I think I'm probably about a 2/3s favorite going into any game in a Swiss draft. (Varies by format, mindset, etc, but generally speaking that's my estimate.) A result of just over 2 packs per draft payout average seems to confirm that. Before swiss existed, I was a bit lower, very close to 2 packs per draft in 4-3-2-2. So losing 5% there seems fair. But I don't like my odds at all in 8-4. I suspect if I had the money to sink into it, I would find over 10 drafts that I'm at a very low payout like ArchGenius discovered.

So my line is drawn at 65% Swiss, 60% 4-3-2-2 (which I wouldn't do since I can add and enjoy playing), and 40% 8-4. Which plants my butt firmly in the Swiss queues, thank you.

But to the point of the actual column, a 5% drop is only an 85c payout difference? That's considerably less than I'd assumed, and no single pick is going to hurt a draft too much. Obviously no one passes chase Rares in Swiss, but this means a 1tix Rare is worth it over a middling pick. That's good to know.

single raredraft can cost by Undeadgod (not verified) at Tue, 10/27/2009 - 09:59
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Honestly, in my opinion, a single raredrafted card can cost you a lot. Say you grab that Vess pack 3, while drafting red white instead of a lightning bolt, because you can sell her for 5 tickets. The bolt is not coming back, the chances of being passed another is slim. Also still worse, is the probability that that pass will come back to haunt you. This is less visible on V3, because without the table view, you do not get to see the names of who you pass each pack to. I miss this greatly as a drafter, since not only is it fair to be able to know this (similar to paper drafts), but also how a young drafter will learn the consequences of passing. The V3 system of "blind passes" only encourages (or is a benefit to) raredrafters, due to the fact that no one knows before the games who sucked up all the rares, and dramatically changed the pack environment, so no one gets called out on it. In paper, raredrafts still happen, and people get on with their lives, but if the person doing it is sitting to your right, his/her passes will tell you, and if you have to face them, your play-style will change accordingly even before the game.

Just my 2 cents, but I feel this feature should definitely return on v3. The visible consequence of your known (not assumed) passes coming back to haunt you is a needed learning experience for people who want to become better drafters. Or at the very least it was for me.

keep passing by Undeadgod (not verified) at Tue, 10/27/2009 - 10:01
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By the way, please keep passing me bolts in M10 drafts...

I think the simplifications by Hibiki (not verified) at Tue, 10/27/2009 - 14:30
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I think the simplifications the model does make it useless.

The competition in Swiss is very different from the competition in 8-4. So having an expected match win percentage that works on all three drafts format is just going to lead to poor results. Someone that wins 65% of their swiss matches will probably lose a ton of money in 8-4, because chances are they'd be under 50% in 8-4s. The difference in quality is that high.

So it is quite possible for 4-3-2-2s to be the optimal solution: All you need is to have an opposition in 4-3-2-2s that is very similar to the swiss in quality, and much easier than the 8-4s.

dont even get me ramped up on by Anonymous (not verified) at Wed, 10/28/2009 - 09:03
Anonymous's picture

dont even get me ramped up on foil rare drafting!oy if i crack one, look out, and if you pass it to me !!!!

crazy train

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It's interesting to look at your 2nd chart compared to my own drafting.

I've done 9 ZZZ drafts so far.

2 Swiss:
(83% match-win % & 5 pack winnings total)

7 4-3-2-2:
(63% match-win & 16 pack winnings total)

I speculate that I could consistently put up a 70% match-win in Swiss (2.352 pks / event), 60% in 4-3-2-2 (1.988 pks / event), and maybe 50% in 8-4 (1.5 pks / event.) [Maybe my speculation is off and the competition doesn't vary that much between each queue-type..]

I played swiss initially then switched to 4-3-2-2. The reason was time commitment. My experience in my two swiss was that rounds were consistently going near the full hour every time, causing these to take substantially longer than a single-elim queue. The secondary reason was that winning more than you came in with is available in 4-3-2-2 and is not in the swiss.

It's readily apparent that while all 3 have the same entry cost, swiss & 8-4 pay out 12 packs where 4-3-2-2 pays only 11 (Yuck!) For now though I'll probably stick with 4-3-2-2 with the goal of improving drafting & gameplay enough to do well in the 8-4s.

wasnt there speak of going by Anonymous (not verified) at Wed, 10/28/2009 - 12:22
Anonymous's picture

wasnt there speak of going 5322???