Probability, Standard and Bannings

Feature Article from Steve Guillerm
Steve Guillerm
6/11/2012 10:22:00 AM
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A few weeks ago I spoke of embracing variance in Magic. It turns out that humans are inherently bad at probability, and so it requires a conscious effort to overcome what we feel to be true. For example, how often have you experienced a situation in which something either happened, or didn't, and your brain just short-cutted to assuming it's a 50-50 chance? Here's one that might throw you for a loop if you haven't seen it before.

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The Monty Hall problem is loosely based on the old game show Let’s Make a Deal, named after the show’s host. The set-up is such: there are 3 doors to choose from. 1 has a fabulous prize behind it, such as a new car. Two have a booby prize, such as a goat. You choose a door at random (say, door 1), and then the host opens another door to reveal a goat, and then gives you the option of either sticking with your choice, or switching to the third door, the remaining unrevealed door. The question is, should you stick to your choice, or switch, and what is the probability of winning with either choice?

Intuitively, most people will say that because there are two doors, there's an equal chance. In reality, you should choose to switch, as the probability of winning with your initial choice was 1/ 3, and therefore switching to the other door gives you a win probability of 2/3. If you disagree, check it out on Wikipedia, but let's get back to Magic.

One of the harsh truths of a Magic tournament is that, barring ties, 25% of the field will be eliminated from top-8 contention after two rounds. The ol' 0-2 drop. Of course, that's not us, right? We're better than those guys, right? Well, it turns out that even if we're good enough to win 75% of our matches against a random opponent, probability's out to to get us, and we're going to start the tournament 0-2 once every sixteen times. 75% against the field is actually pretty extraordinary, but what it means is that even if you're improving, getting better and better, you're still going to suffer setbacks. Try your best not to let them get you down.

This is very simplified, and obviously a real tournament isn't going to be a homogeneous mass of players. Some will be scrubs like Steve Guillerm and Frank Lepore, some will be champs like Jackie Lee or Craig Wescoe. Then again, you wouldn't be able to determine your exact probability even if the room's players were homogeneous. Mostly, it's just to show you that even good, solid players can go 0-2 pretty often.

Let's look at the other side of the coin, winning a tournament. Let's say that there's a player named Jon who is far and away the best player in the room. His win percentage is 90% (with 10% going to the luck factor) during the Swiss rounds, and against the stiffer competition of the top 8, he still manages to win 80% of the time. What is his probability of winning the tournament? We'll assume that X-0, double-draw makes it, or X-1, draw the last round makes it. Here are the calculations for a 7-round tournament:

First, we can calculate the probability that he goes 5-0. That's simply (0.9)^5 = .59049

We then add the probability that he goes 5-1 in the first 6 rounds. There are 5 different ways for this to happen, as the loss would not happen in the sixth round.

That is 5*(0.9)^4*(0.1) = 0.32805

The total probability that a 90% match-winning player makes the top 8 is 91.85%

Once there, the probability of winning is (0.8)^3=0.512=51.2%

In total, the probability of winning the whole thing is 91.85% times 51.2%, or 47.03%

So, even if you're far and away the best player in the room, there's still a lot of room to miss. I don't mean for this to get you down, but rather to understand that as much as Magic is a game of skill, as long as luck and variance exist, a tournament win is hardly guaranteed, even to the best players.

Perhaps one of the most frustrating aspects of competitive Magic is that it's really hard to track progress. How good am I? Am I better or worse now than I was 4 months ago? You used to be able to track your rating on WotC's website, but they've done away with rating in favor of Planeswalker Points, and besides, rating wasn't particularly useful anyway. For example, if I went to a PTQ and made it to the top 4, I might see my rating go up by 75 points, only to watch it trickle down as going 3-1 at my local FNM lost me points.

Tracking your average finish at an event might not necessarily do you any good, either. If you go to a large tournament once a month, your sample size is going to be tiny, and it'd be hard to track improvements compared to confounding factors like choice of deck, or attention to the format. Between January and April, I attended 3 Standard GPs, and Day Two-ed all three, finishing in the money for two of them. In that same time, I went 0-2 drop in all three of the Modern PTQs I attended. I'm not a Standard master, nor am I a terrible PTQ grinder. It's partially the amount of attention I paid to the formats, and partially just a strange blip on the radar. I don't expect I'll crash and burn at my next Modern event any more than I can guarantee that I'll succeed at my next Standard event.

Instead of worrying about individual tournament results, you should work on improving individual skills. Do you find yourself missing triggers? Play some games with friends outside of sanctioned play, so that they can help you improve. Whether the method is gently reminding you, or smacking you with a rolled-up newspaper, most friends will relish the chance to tell you your faults, but you'll show them, because you're actually learning.

Do you find yourself describing match-ups in terms of percentages? As in, “My deck beats Delver 60% of the time!” Stop That right now. It's not actually helping you one bit. Instead of thinking in terms of how often you win, determine what makes you win or lose in a given match. Remember that two different people can play the exact same deck, and play very differently. Besides, if your deck wins 60% of the time, you lose 40% of the time! Why are you assuming that 40% of the time you'll just lose?

When you're looking at an opening hand, are you just checking for lands and spells, or are you actually plotting out your first few turns? Far too often, I've seen people keep a 1-lander, especially on the draw, and assume that they'll “get there,” drawing a land in the next two turns. I think a lot of this goes back to that root assumption of yes/no questions being 50-50. “Will the card I draw be a land?” Well, if your deck actually only has 20 lands, the probability is not 50%, it's 19/53, or around 36%. The probability of not drawing a land twice in a row? Over 40%. You're assuming you'll “get there,” even though you've only got under a 60% chance to get your second land drop on time. Ouch. Yeah, most Standard decks play closer to 24 lands, but I've got Legacy on my mind, and we play about 20 lands, and still fall prey to the same fallacy.

Random bits of extra probability: as you may or may not know, Magic cards are printed as sheets, and then cut apart. For any given set, the rare sheet contains 1 copy of each Mythic, and 2 copies of each Rare. Large sets (Avacyn Restored, M13), the rare sheet contains 121 cards, 15 mythics, and 53 rares, each printed twice. Therefore, the probability of opening a Mythic is 15/121, or about 1 in 8. In the small sets (New Phyrexia), the rare sheet is a smaller 80 cards: 10 Mythics, and 35 rares, each printed twice. The probability is exactly 1 in 8. Because of the smaller rare sheet, your odds of opening an Elesh Norn are better than opening a Bonfire of the Damned from a random pack.

So why do small set rares shoot up in value compared to regular ones? Well, because despite the better odds, we as players generally open fewer of them. When we draft a new block, we spend 3-4 months cracking 3 packs of the first set, and when the second set comes around, we're only opening 1 pack of it, while still opening 2 packs of the first set. By the time we get to the third set, we've finally gotten down to opening only one pack of the first set, but we open a single pack of the third set for only a few months before the new Core Set comes out.

I have no official confirmation of this, but I imagine the reason why Rise of the Eldrazi and Avacyn Restored are stand-alone large sets is partially to offset this problem, but in turn Worldwake and Dark Ascension become the de facto “third sets” in terms of packs opened. The upshot? If there are cards you think you might want in the future, consider picking them up now if they're in Dark Ascension. The prices have largely settled, and it's unlikely people will be drafting too much more DKA-INN, especially once M13 comes out.

The State of Standard

Just a few days ago, I was asked whether Standard was due for a banning, and I said that I didn't see a banning coming. However, after this weekend's results, I'm beginning to change my mind. In Indianapolis, we saw 4 traditional U/W Delver decks in the top 8, along with 2 “U/W Midrange,” which is essentially an evolution or variant. The mana-base is practically the same, and aside from a main-deck Day of Judgment, the spells are very similar. Cutting the smallest guys for a few big ones and Planeswalkers doesn't really make it a new deck.

 Snapcaster Mage
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Magic MTG Card
Magic MTG Card Snapcaster Mage Magic MTG Card
Magic MTG Card

In fact, this weekend, I played against Matt Costa at the SCG Open in Worcester in a feature match. Game one, his deck was classic U/W Delver, but in Game two, he'd transformed his deck into something a lot closer to a control deck, similar to these U/W Midrange decks. I was definitely thrown for a loop, and it showed off just how flexible the archetype is. Though the event was won by Solar Flare, we saw 6 U/W Delver decks in that top 8 as well, for a total of 12 U/W Aggro-Control decks out of 16, 75%.

The cries for the strike of the banhammer have been far and wide, but it's tough to say exactly what should get hit. Mana Leak is certainly not an option, as it's an important safety valve, and Cavern of Souls works to prevent it from being too dominant. Vapor Snag would be quickly replaced by Unsummon, and I cannot ever imagine Unsummon making the banned list. When it comes down to it, we're looking at either Ponder, Delver of Secrets, or Snapcaster Mage. Despite the undeniable power of a 3/2 flyer for 1 mana, we've seen decks moving away from Delver and still succeeding.

Of the two remaining candidates, both have “defenses” that WotC may or may not heed.

Snapcaster is a marquee rare, and an invitational card to boot. Players might get upset if the card were banned. Ponder's going to rotate at the end of the summer, and perhaps it'd be safe to just wait it out. The precedent set by the Jace/Stoneforge banning is that neither of these really matter. At the height of it, Jace, the Mind Sculptor was a $100 card, and no amount of concern over hurt feelings or wallets could save him. He was scheduled to rotate out with close friend Stoneforge Mystic only 4 months later, but 4 more months of Caw-Blade was too much to stomach.

If I had to make a prediction, my first guess would be “Ponder is banned”, followed by “no changes,” and “Ponder and Snapcaster Mage are banned” a distant third. I don't believe they'll cut Snapcaster Mage from Standard and leave Ponder running rampant. The ban-list announcement happens June 20th, so we've got one more weekend for someone to shake up Standard. Keep an eye on what happens this weekend at the TCGPlayer MaxPoint Diamond 5k in San Diego!

Steve Guillerm
@SteveExplosion on Twitter and MTGO

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