Branching Out: Game Theory and Magic
Geordie Tait and Richard Grace really got into it last week, didn't they? I can't help but express some support for both of them. I think Richard was unduly harsh, but I also believe that the last two months of SCG articles have been sorely lacking in diversity. That's not to say that the articles are without merit. But it does mean that if you don't like those particular topics then you're out of luck. Mr. Grace seems like someone who feels they've got a handle on the theory of card advantage and doesn't care for Type I. Well, that means that many of the articles have been useless to him these past couple months. That's not to say they're useless for everyone.
What I see from Geordie is a defense of the content and quality. I agree. As a sometimes contributor (blatant attempt to get you to look at my past articles), I think that the StarCity articles are a cut above other sites in terms of both quality and quantity. However, I do believe the diversity is over time and tends to ebb and flow with the vagaries of the online world.
I want to present a new topic for discussion in an attempt to broaden the horizons of discussion. Game Theory in Magic. For those of you that don't know anything about game theory I'll link to a website and give a brief explanation. The simplest examples are usually the most interesting and instructive. For example, the classic prisoner's dilemma encapsulates many of the principles of game theory. We can also see how playing the game once and playing the game repeatedly encourage different courses of action. That's simple game theory, and it helps to explain a lot of decisions that are made in the world around us.
Sometimes, however, game theory leads us to a slightly less intuitive answer. The football game is the most common problem. Suppose that I'm the offensive coordinator of the Bucs (an admittedly painful job this season), and I'm playing against the Saints defensive coordinator. I have two choices on each play: Run or Pass. My opponent likewise has two choices: Defend Run or Defend Pass. If he chooses correctly I lose five yards. If I choose correctly then I gain five yards. What's the best strategy?
Well, in this case it's pretty obvious. I'll call plays completely randomly and he'll call defenses completely randomly. How you work through the math isn't particularly important but you can see that I have a 50% chance of gaining five yards and a 50% chance of losing five yards and thus my expected yardage on any given play is zero yards.
Let's change the example though. Let's assume that the Bucs just signed Barry Sanders out of retirement, and that our running game is going to be a lot better. Now, everything stays the same except, that if I call run when he calls defend pass, I will gain ten yards. How does this change my strategy? Most people would say that you should run more. But this is not in fact the case. It is logical (and proven true) that your optimal strategy will be the one in which your opponent is forced to guess at random. If we continued our old strategy of running and passing in equal portions then our opponent's best strategy would be to just defend run every play. In fact, this would yield the same zero yards per play as before. So... we need to run less to entice him to defend the pass so that Barry can actually get some yards. You can see all the math for this concept at this site.
So we can see that Game Theory gives us an explanation for real world occurrences as well as some counterintuitive results. But what does all of this have to do with Magic?
Unfortunately, I need to go back to Onslaught block to give a really concrete example (Mirrodin hasn't provided me with enough examples to draw any conclusions from). Onslaught Rochester was a much-maligned format. From a game theorists perspective it was fascinating (I have two friends who hate Magic but enjoy economics and game theory. When I explained to them the environment of Onslaught Rochester, they became very interested). Rochester has all the elements that you need for game theory, it just makes them much more complicated. Rochester also has one additional aspect that is intrinsic for analyzing the game. Perfect information. You'll note that in both the examples above it is assumed that you know nothing about your opponent's choice before he makes it but you are given full disclosure when he makes it. This aspect makes Rochester unique.
I had a theory during triple Onslaught that you could determine the winner at a table simply by looking at what colors they drafted and their relation to everyone else at the table. Obviously, this is an oversimplification. I'm sure at some point in triple Onslaught someone drafted U/G and got amazing cards and won. But overall I think this was a sound theory. If this is the case, then the individual card evaluations stop mattering and it becomes an issue of spacing and color distribution.
For the sake of our argument let's assume that all 8 people are going to avoid splashing (a relative rarity in triple Onslaught) and that each person is going to be forced into a dual colored deck with relatively equal numbers of each color. Let's further say that red can support 4.25 drafters (meaning that if four people draft it, they'll have slightly better then average cards, but if a fifth were to jump in the card quality would drop below par). Black can support 3.5 drafters. White can support 2.75. Green can support 3.25 and Blue can support 2.25 drafters. [I did the actual number crunching for Chicago in the second half of this article from last year. Tragically, GP: Boston in February was the last time we got enough info to do this sort of analysis. - Knut]
Let's further assume that for reasons outside of our analysis that all drafters need either Black or Red (Sparksmith... Cough! Cough!).
Now we can see why the"typical" Rochester draft would be
Seat 1: U/R (2.25/2 + 4.25/4 = 2.1875)
Seat 2: G/B (3.25/3 + 3.5/4 = 1.958)
Seat 3: R/W (4.25/4 + 2.75/3 = 1.979)
Seat 4: G/B (1.958)
Seat 5: U/R (2.1875)
Seat 6: B/W (3.5/4 + 2.75/3 = 1.792)
Seat 7: R/G (4.25/4 + 3.25/3 = 2.146)
Seat 8: B/W (1.792)
This is the optimal distribution, and you can see why U/R was so dominant. There's an extra wrinkle that needs to be thrown in though, and this is the tough one to quantify. Spacing is important. Let's say that Seat 2 and Seat 5 switched. Raise your hand if you think Seat 2 would be able to do anything. But Seat 1 gets much better, because he picks up the best Blue card in 15 of 24 draft packs (seven in the first and third round of packs and once in the second). We can also see that seat 7 benefits tremendously, because he now gets to pick the best Red card in 10 of 24 draft packs.
Because everyone knows about these facts (implicitly, if not explicitly) before the draft, we get to see true game theory in practice during the first few packs. Triple Onslaught was much maligned for being boring. I disagree. I think Triple Onslaught just had all of its maneuvering and importance concentrated into the first two to four packs. I looked at Pro Tour Chicago's third Draft (I should really make Knut look up these urls... of course maybe that's the only reason he publishes me...) and check to see what color each deck was (unfortunately this was the only tournament I could find this information for). The results were surprising. Let's look at the top 4 tables (when I did this analysis I went even further and assigned a primary, secondary and tertiary color. The colors below are listed in that order. The participants are also listed in seat order):
Budde, Kai Blue Red
Harvey, Eugene Green White Black
Lebedowicz, Osyp Blue Red
Fries, Stefan Green Black
Bruneau, Derek Red Black
Crosby, Joseph White Black
Witt, Alexander Green Red
Reinhardt, Fabio Green Black
At this table, we can see that Black and Green are overdrafted and that White is underdrafted. Also of note is that Kai is getting six seats of blue cards because Osyp picked Blue/Red so close. As you would expect, Kai went 4-0 with Eugene going 3-1. Eugene got excellent position for his White and adequate spacing for Green, where as Joseph, the other White drafter, was fighting tooth and nail for Black.
Shachaf, Yuval Black Green
Martin, Quentin Red Green
Turian, Michael Blue Red
Hegstad, Brian Black White
Herzog, Nicolai Green Red Blue
Snepvangers, Abraham White Red
Krempals, Craig Green Black
Finkel, Jon White Red
This table featured our very own Mike Turian being the only real Blue drafter at this table. Mike went 3-1 at the table, along with Jon and Nicolai. This is a good example of how play skills and card selection and luck can end up mattering. Jon had a White/Red deck in a marginal location. Red was overdrafted somewhat, and he was getting great white only in the second round. Yet he finished better then the 1-3 Brian Hegstad, who had two non-Black drafters on either side of him, and good White cards in rounds one and three.
In fact, if you look at the decks, it's obvious that Finkel's deck was not that great. He had to maindeck Airdrop Condor (with three goblins), Pearlspear Courier (with six soldiers), and Crown of Awe, and he had a single Sparksmith and a single Pinpoint Avalanche with no real bombs. Compare that to Brian's deck that had Cabal Archon, two Daunting Defenders, three Nantuko Husks, a Harsh Mercy, and Improvised Armor.
Other tables had similar outcomes. I believe that at the top of the game, the play skill is so close (John Larkin, Dirk Baberowski and Raphael Levy were luminaries at table 11), that Rochester drafting comes down to positioning, synergy, and card evaluation. Of these three, I think the evidence shows that positioning is of vital importance, and while it isn't the be all end all of deck building (far from it, in fact), it is extremely important in determining your chances. In fact, it is this idea of positioning that results in strategies like cutting in drafts. Any time you are trying to isolate a color, you are implicitly sacrificing some early power for position, in the hopes that this will translate into even greater power later. [Sigh, just think of how interesting the information from splash-happy Amsterdam could have been, if we were only given the decklists. - Knut, beating the dead horse some more]
So what are the game theory aspects of your decision-making? They're based largely on the two principles I've already talked about. Spacing and color distribution. If you see a chance in Rochester to force the table to draft in a pattern that would be advantageous to you, then you should do it. For example, suppose you're fourth and the first three picks were: Red, Black, Green. On the face of it drafting a Red card here could be fantastic. You could ensure spacing that is advantageous (anything greater than two is pretty good for Red). But you need to think about some things for a second.
What choices are you leaving your neighbors? More then likely you're signing the death certificate for one of them and possibly yourself. That Green player is now stuck with a pretty bad set of choices. He can draft nicely and take Blue or White, but this is decidedly sub-optimal in triple Onslaught. He could take Black in competition with the person on his right, or Red in competition with you. Additionally, he knows most of what you know, so he'll realize that pushing you out of Red is probably more plausible, since trying to push the Black drafter out of Black would put that player in the same quandary as the Green Drafter is in now. Even here it's probably worth the gamble to draft Red, because the worst that could happen is that you're poorly spaced in a very good color. You might also encourage people after you to shy away from Red.
A lot of people are probably reading this and being highly critical of my"Actual cards don't matter" attitude. However, I believe that it's very important in Magic to understand the decisions we make when all else is equal. We make a lot of decisions in drafts that are marginal. Making these marginal decisions correctly is what separates Kai Budde from Jeff Hall. However, I think it's important to think about those situations in which you've got multiple choices that are all roughly equal. What are you saying to the table? What choices are you leaving the people beside you? Those aspects of the game transcend card valuation.
Mirrodin has totally screwed up this theory. With half of the set devoted to cards that can be played in many, if not all colors, and the ease that comes with splashing colors, it has become very difficult to determine color valuations and proper signals. Does my first pick Bonesplitter mean I'm going White? Does that Shatter I drafted in round one even mean that I'm particularly attached to Red? If any one wants to send me detailed Mirrodin Rochester results, I would love to analyze them the same way I did with the Triple Onslaught draft. The underlying principles should be the same just harder to spot.