Insider: Expected Value and Rare Drafting

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Expected Value is a term that is thrown around the MtG community frequently. “+EV” has become synonymous with “profitable” and grinding has an air of pride attached to it, and that’s whether you’re grinding events or trade tables. While the short-cut term is certainly good enough for communication between people who speak the same language, using its actual meaning and underlying calculations, we can find some interesting things.

Rare drafting is a Science, not an Art. The reason being, there are actually correct times to take a rare for its monetary value, and times when there isn’t. For years, at a standard LGS draft, $5 was the minimum monetary value a card would need to have for me to pick it over another playable. I figured, for a $15 draft, with 3 packs, if I get a $5 card, that pays for that pack, and recoups some investment.

I felt this way until Aaron Forsythe posted some very interesting information about M10 limited using extensive data from MTGO. You can find the data here:
A quick recap for those to lazy to click the hyperlink: He lists the top 25 cards in M10 limited, based on win% of the decks they were included in, during round 1 of sealed events.
In this particular set, 17 of the 25 listed are Rares and Mythics, 5 are Uncommons, and 2 are commons! Also of note, the entire range of the top 25 leads to win% >51%.

Here’s where things get interesting, with regards to Expected Value. In short, Expected Value is simply a weighted average, where we calculate the average outcome of a decision, based on the likelihood of each outcome. The problem with Magic (or reason for its success) is the probabilities of such outcomes are not known. But now we have an idea of the range of how good some of the best uncommons can really be in a limited environment.

Keep in mind, these probabilities aren’t precise for a couple reasons:
1) The data comes from M10 not M12.
2) Your individual win% is going to vary from the population of MTGO players as a whole.
However, it does give us an idea of how much better some cards can really be, and how much of an impact they can have on your chances of victory.

I’ll start with the LGS I play at, as an example. Suppose we open a pack that has a junk rare (Sundial of the Infinite, perhaps), a Foil Timely Reinforcements, and a Mind Control. If this was Pack-1-Pick-1, using my previous rule, the $5-6 foil would be the pick. But with this new information, let’s see how correct that is. At least in M10, and I think most would agree M12 wouldn’t be much different, Mind Control owners won 54.07% of their matches. How can we use this information to determine if the foil is worth the pick or not? It will depend on prize structure.
At my LGS, an 8-man pod gives 1 pack to a player who loses in the semi-finals, and $15 store credit to 2nd place, and $25 store credit to the winner. It is not uncommon for first and second to split at $20/$20. Supposing the rest of my draft is fairly average, just the presence of Mind Control alone, means I’ve got a 54+% chance of making it to the Semi’s alone, and a 27.61% chance of making the finals, and a 14.93% chance of winning it all.
The possible outcomes are:
1st Round elimination: No Prize, 45.9% likely
2nd Round elimination: 1 booster prize ($3 for simplicity), 24.8% likely
Loss in the finals: $15 prize, 14.4% likely.
Win the 8-man: $25 prize, 14.9% likely.

So, if we can accept these figures are a decent starting ground of comparison, we can calculate the value of this proposition.
$0*(0.459) + $3*(0.248) + $15*(0.144) + $25*(0.149)= $6.629 which, depending on who you are talking to, may or may not be more than a Foil Timely Reinforcements. In this case, I think it is correct to go with the Mind Control, but the prize structure will of course affect the decision. Other things to consider, is there’s an opportunity cost in picking Mind Control. I /didn’t/ get to add the Timely Reinforcements to my deck. So would the $5-6 foil, plus the value it adds to my deck make up the difference? It might, but I’m inclined to think that cards like Mind Control, gain so much edge, that it is tough to reject their strength. That being said, there are only a handful of Uncommons that this applies to. Namely, Mind Control, Delayed Blast Fireball, Oblivion Ring, Serra Angel, Acidic Slime (although I’d argue its not as strong in M12 as it was in M11 or M10), Overrun, and I would guess Sengir Vampire belongs on the list too. (Note: if you play Swiss-Pack-Per-Win, the numbers resoundingly favor the $6 card).

I fear that some of this article will be ignored, because the numbers aren’t going to fit your exact scenario. That’s likely true, we have to pick somewhere to start, and the data that’s available is about all we can do. FWIW, my limited match win% is in the 54-56% range as it is, so likely adding a Mind Control to my pool skews my win% up a bit farther (making it even more valuable than a Timely Reinforcements, although making the chances of winning without the Mind Control also reasonable). Finding your win% is easy, and work with it. Obviously, winning in later rounds should be tougher, but again, we have to work with averages here, because it’s all that’s available.

Where does that leave us? Well, my $5 gut-check rule, seems to be a fine ‘rule of thumb’ for the types of prize structures I’m used to, but it isn’t an Art, it’s a Science. It either IS correct to take the Foil Timely, or it isn’t. If you plan on keeping drafting as inexpensive as possible, making the correct choice more often than not, is going to be crucial. The most amazing statistic in Aaron’s post is: “Only 70 of the 229 cards in M10 have win % over 50%.” We don’t know how many of those are Rares, but if the ratio is similar to the top 25 cards, we could expect there to be only 20 non-Rares in that group of 70, meaning that most cards hurt your chances of winning (statistically speaking).

Next week I’m going to interview a local Dealer who moves most of his product on EBay, and does most of his buying through networking at the LGS. He’s able to live completely off this work, and we’re going to pick his brain for how he was able to build such a successful business plan. If there is any questions you want answered in that interview, with respect to networking, buying cards/collections, using EBay/Paypal or anything else, let me know in the comments and I’ll bring them up in the interview.

Thanks for reading, and happy drafting!
Chad Havas (@torerotutor on twitter)

9 thoughts on “Insider: Expected Value and Rare Drafting

  1. You forgot that you can still win without the mind control and need to calculate an ev for picking timely reinforcements. Then you ev will be value of timely foil + ev of winning. I too lazy to do all the math but I know your chances of winning alone are 9.67% if you don’t pick mind control which means you which means your ev for picking timely is $5(value of timely)+ ev(not picking mind control) which is at least $5+(9.67%*25)= ~$7. So even if prize was only given to the winner, pick the timely. I’m guessing the final number isaround 8.50 which is smth like a 30% greater ev than picking mind control.

    1. actually, i addressed that directly. Because timely wasn't in M10 we dont have data on that, but i said about it :
      " Other things to consider, is there’s an opportunity cost in picking Mind Control. I /didn’t/ get to add the Timely Reinforcements to my deck. So would the $5-6 foil, plus the value it adds to my deck make up the difference? It might, but I’m inclined to think that cards like Mind Control, gain so much edge, that it is tough to reject their strength."

      1. True, but if you can just calculate the EV of picking mind control vs the EV of not picking mind control. YOu dont necessarily need the winning percentage of the card you take instead. So EV of mind control is $6.629. You can calculate your percentages of winning without picking mind control in the same way.
        assuming sigle elim:
        No prize: 54.1% ikely
        Winning 8 man: .459*.459*.459=9.67%
        Getting 2nd: .459*.459*.541=11.40%
        2nd round elim: 0.459*0.541=24.83%
        So EV of not picking mind control=9.67*25+11.40*15+24.83*3=2.42+1.71+.745= 4.875.
        Therefore by picking mind control, you increase your EV by 6.629-4.875=1.754. Therefore as long as you pick any card worth more than $1.75 over mind control, your EV is improved. You can do this calculation with any card to figure out what cards are worth picking over it.
        THis becomes more complicated with swiss 8 ans, but this is a rough approach to single elim.

        1. Michael, i applaud the knowledge and effort shown here. This is slightly different than your typical P vs 1-P probabilities as you would have given two mutually exclusive events. While taking mind control or not, are mutually exclusive, they are choices, not chance events. The chance event is whether or not you win, GIVEN you have mind control in your pool, vs the chance you win GIVEN you don't have mind control in your pool. If you're familiar with probabilities, which it seems like you are (but for the rest reading, in case htier not), we'd write that as P(Match Win|Mind Control in Deck)=0.541 that DOES NOT mean that P(Match Win|Mind Control Not in Deck)=0.459 as you indicated above. In order to determine such probabilities, you would need to use Bayes theorem, which says that P(A|B)=P(A and B)/P(B) Unfortunately we don't have access to P(A and B) but we could deduce P(B) using set analysis, and try and work backwards. Seems like a lot of work to accomplish what I feel my estimation confirmed accurately, but i might look into this.

        2. Just to clarify my point further…. You've claimed that the win% without mind control is 0.459 as a 1-P probability of the mind control win% of 0.541. In reality, the complement to P(WIN|Mind control) is P(Lose|Mindcontrol) so all it means is your chance of losing with mind control in your deck is 45.9%, not that winning without it is 45.9%.

  2. Questions for the local dealer:
    – Besides eBay, do you have any other online presence?
    – What resources do you use to calculate a good buying price for collections?
    – Any tips on how to write your eBay auction in order to maximize profit?
    – Is this really the only thing you do for a living? If not, how do you combine with other job(s)?
    – How much time (and money) do you need to invest on a monthly basis to be able to live from it?
    – Any particular tips for someone trying the same, but with a 5-to-9 job?

  3. Really looking forward to the article!

    – When and how did you realize that this could be your only employment?
    – Do you use any kind of advertising?
    – Where do you search for cards to buy?
    – What legal aspects would you like to share for other starters (I assume this would not be applicable for me, being operational in Europe, but it might be interesting for US readers…)

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