## Tuesday, 16 November 2010

### Guest Article - Tournament Maths – How to Make a Profit on Magic Online

By Sebastian Parker

It's been a while since I contributed to the blog, but I have some spare time (since i've finished my degree and haven't got a job yet) to play magic. Unfortunately I haven’t been able to go to all the PTQ’s I had planned to because I stupidly won the first one I went to (OOPS...). So now I have turned my attention once again to Magic Online.

Last time I talked about block constructed vampires, making profit online from a small initial investment. There are several layers of probability maths in magic:

1) “in game” maths, which affects your plays, like the probability of drawing a removal spell influencing whether you block or not.
2)“deck construction” maths, which affects how many lands you put in your deck, how many birds of paradise is preferable etc.
3)“tournament” maths, which affects your deck choice, how likely you are to win a round, how likely you are to top8.

This time I want about tournament maths - how to go about making profit online.
First, I’d like to introduce the concept of expected value (EV). EV is a way of expressing the average outcome of uncertain events. The easiest way to explain is with an example game:

Alex and Norman are playing a bean game. Alex has to throw beans into a cup a 3 metres from where he’s standing. Alex will win every bean which lands in the cup. Norman thinks Alex will struggle to throw beans into the cup, so Norman and Alex each put 5 beans into the pot from which Alex will attempt to throw beans.
From Alex’s perspective, he’s paid 5 beans to play the game. He will win every bean which lands in the cup. His expected winnings are the number of attempts times 1 bean times the probability that he will successfully get a bean into the cup minus the cost of entry to the game, 5 beans.

EV= expected prizes – cost of entry

EV= 10*1 bean*P(success) – 5 beans

(the asterisk * indicates multiplication, P(success) indicates the probability that Alex will successfully throw a bean into the cup)

So if Alex is good at throwing beans into cups, i.e. if P(success) is more than 0.5, then Alex will have an EV of more than 0 * (i.e. positive) and as the game goes on expect to increase his collection of beans.
*(EV = 10*0.5 – 5 =0)

Now let’s relate this concept to magic – 2-man constructed queues. To enter costs 2 tix, the prize depends on the format but most at the moment are either Scars of Mirrodin packs (which you can sell to the trading bots ..Hogwarts.., .7BP or .Coruscant for 3.93 tix) or M11 packs (which are 3.3 I think).

In order to make profit in 2-mans, we can apply the same formula:

Scars pack queues:M11 queues:
EVscars = 3.93*P(win) scars – 2EVM11 = 3.3*P(win) M11 – 2

And rearrange to find the point at which we start to make profit (EV=0 is the point between profit and loss, so setting EV=0 gives us a minimum P(win) needed to make profit).

2/3.93 = P(win) scars = 50.9%
2/3.3 = P(win) M11 = 60.6%

So in order to make profit in 2 man queues, you’ll need to be winning more than these %’s.

So far so good, these are fairly simple calculations. Now let’s move it up a notch – 8 man constructed. 6tix entry, 3 rounds, single elimination, 5-3-2-2 scars packs. To keep things simple, we’ll assume and average win% rather than take into account the fact that winning players are tougher opponents.

We’ll start by listing all the outcomes and their probabilities and payouts:

OutcomeLikelihoodPrize
Lose 1st roundP(lose)0
Win 1st round, lose 2nd roundP(win)*P(lose)2*3.93
Win 1st round, win 2nd round, lose finalsP(win)*P(win)*P(lose)3*3.93
Win all three roundsP(win)*P(win)*P(win)5*3.93

Since there are only two outcomes to a match – win and lose, P(lose) = 1 - P(win)
EV = expected prizes – costs
EV8man = P(lose)*0 + P(win)*P(lose)*2*3.93 + P(win)*P(win)*P(lose)*3*3.93 + P(win)*P(win)*P(win)*5*3.93 – 6

This is easiest to do in excel, so I’ve written a spreadsheet to work it out for me. In order to make profit in an 8-man, you need a match win% of just 51%.

Let’s add another layer of complexity. When you’re drafting, the booster packs you open have random cards in them and the cards hold a value. The expected value of the cards inside a pack is the value of the cards you could open, multiplied by the likelihood of opening them. As such, I have another spreadsheet with all the card names of Scars of Mirrodin and how much you can sell them to cardbot for. Multiplying each value by the likelihood of opening that card yields a value of 1.08tix. So to buy a scars pack for 4tix and open it and sell the cards inside has an EV of -2.92tix. (We all know just opening packs straight up is like throwing money down the drain, and this is why – it costs nearly 3tix per booster!)

So when you’re drafting, it costs 3 packs and 2 tix (or 14 tix if you’re buying boosters). Your expected prizes are the value of the cards you open (3 Scars of Mirrodin booster packs worth = 3*1.08 = 3.24tix) plus the actual prizes you win.
Again, we list all the outcomes and their probabilities and payouts:

OutcomeLikelihoodPrize (4-3-2-2)Prize (8-4)
Lose 1st roundP(lose)00
Win 1st round, lose 2nd roundP(win)*P(lose)2*3.930
Win 1st round, win 2nd round, lose finalsP(win)*P(win)*P(lose)3*3.934*3.93
Win all three roundsP(win)*P(win)*P(win)4*3.938*3.93

Assuming you’re using packs you’ve won (which are worth 3.93tix to you) 2 tix + 3 packs = 2+3*3.93 = 13.79

EV4-3-2-2= P(lose)*0 + P(win)*P(lose)*2*3.93 + P(win)*P(win)*P(lose)*3*3.93 + P(win)*P(win)*P(win)*5*3.93 +3.24-13.79
EV8-4= P(lose)*0 + P(win)*P(lose)*0 + P(win)*P(win)*P(lose)*4*3.93 + P(win)*P(win)*P(win)*8*3.93 +3.24-13.79

Again, I’ll let the spreadsheet do the work. To make profit in a 4-3-2-2, you need to win more than 79% of your matches. To make profit in an 8-4, you need to win more than 64% of your matches. Remember, this is under the assumption that you’re selling all the cards you open (that you actually can – you’ll end up with a bunch of worthless rares like Dissipation Field along with all of the commons and lots of uncommons).

The conclusion is that it’s much easier to “go infinite” online playing 8-4’s than 4-3-2-2’s, which we all knew since we’re good players and we only play in 8-4’s anyway.
We could add another layer of complexity, which would be to account for winning players being tougher opponents – P(win round 1) being greater than P(win round 2) etc. But in order to come up with an estimate, we’d need a bunch of data on the ratings of players who enter the tournaments. (The ELO rating system used by the DCI uses a formula to calculate the probability of the outcome of a match given the difference in rating between the two players). At the moment, MTGO ratings are not published due to a previous culture online of ratings taunting – players being obnoxious to each other due to their low ratings etc. so calculating different win% for different rounds is impossible.

I hope you’ve enjoyed reading about the maths of magic tournaments, hopefully I was clear in my explanations. If you were skimming through for some actual online strategy, rather than just some abstract theory:

1)Force poison in draft, I mean it – take Plague Stinger over pretty much anything (even Sunblast Angel and Arc Trail for example). Do not pass infect creatures except to pick up Darksteel Axe, Untamed Might, Grasp of Darkness, Skinrender, Steel Hellkite, Wurmcoil Engine, (Spikeshot Elder if you might be Gr or Br) or money cards (Koth, Venser, Masticore, Elspeth). Cut the player to your left out of black and green as hard as you possibly can. Give them good metalcraft cards so they’ll stay out of poison. I have made 49tix in the 7 drafts i’ve played so far using this strategy and I intend to continue. (editor: must be lucky, I agree with the sentiment but I've found infect to be massively overdrafted and easy to beat if you want to).

2)Play UW control in Scars of Mirrodin block constructed – it’s dominating this year. Check out http://www.wizards.com/Magic/Digital/MagicOnline.aspx?x=mtg/digital/magiconline/whatshappening for daily events decklists. It also doesn’t have the same mirror match problem as last year since it’s about 100tix to put together not 12, so people are trying out cheaper (and worse) options.

3)Don’t play queues where the payout is M11 packs (pauper, classic, singleton) when you can play Scars (scars block, standard, extended, legacy) queues and have a much easier ride on the road to profit.

1. Who knew the best block deck was the one with 2 Planeswalkers :o

2. Good article, enjoyed it.

From a personal PoV, I'd also factor in time in your eV calculations...

Oh, and for 8-4 vs. 4-3-2-2 queues, consider that "conventional wisdom" among pros and serious players is that 8-4 queues are "better" and my gut tells me you get alot more noddies in 4322 - even just a few percent "easier" per match would really skew those stats.

When I drafted alot, I used to run 8-4 but that was mostly because you could split the finals, saving you about 30% of your tourney time. More time to play = more eV ;)

3. nice article seb, found it very intersting.

4. Thanks Russel, since this was supposed to be an introductory article I didn't want to overwhelm people with too much maths.
I'd be happy to write another post with the details how to take into account toughness of opponents and time.

5. Disagree on the poison comments. I've been going infinite online and my good results are fairly evenly spread between RW metalcraft, RB removal, RB poison, GB poison. Poison's certainly strong, but it can also be shallow if too many people go for it or the packs are weak. The day I take Plague Stinger over Arc Trail P1P1 though, has not yet dawned.

As for the stats, given that you started by saying taht you only need to win 50.9% of your matches in 2-mans to go infinite and then proceeded to start talking about 5-3-2-2 constructed queues, only to end up on 4-3-2-2 and 8-4 draft queues, if I'm reading it correctly. Comparing apples and oranges doesn't seem right!

6. Oh, and I just looked at the latest SoM block T8...I've never seen anything so depressing. Any format where there are FOUR 75 card clones in the top ten decks (and this is not mentioning that there are basically only two other decks in there, and those are all almost identical to each other as well) has got to be horrific to play. I've never seen anything quite that bad since, oh, well...ummm....Mirrodin block? :S

7. Most Block Constructed formats are godawful with only a single set in - especially now that the `large' sets are so small. This time last year it was Vampires and UW control and the year before was all about Naya. It'll open up when more sets come in but having such a small card pool doesn't allow for diversity without having a shallow power curve across the set.

8. Yeah, but it's not generally THAT bad, that you end up with so many 75 card clones.

9. @Richard,

Last year's Vampires was sickeningly omnipresent, as was naya the year before (noble heirarch, wooly thoctar, ajani vengeant was an unbeatable opening). This year is actually quite diverse in comparison if you look at multiple Daily's and the Premier Event - monoblue grand architect (costs~25tix) UW control (~90tix) Monored liquimetal coating (costs 4 koth ~80tix total, don't play this deck it sucks, i've tried it) and infect (~20tix) make up a decent metagame.

10. Fair enough. Perhaps it was just the one I looked at, where there were a grand total of three decks represented, if you count two that were very similar as being distinct.