Recently, I started playing League of Legends, a free online game based off of Defense of the Ancients. Without getting into too much detail, players team up in groups of five to traverse to the other side of a map and destroy the opposing team's base. Although there are many obstacles along the way, your primary concern is avoiding battle against the other team in which you are overpowered. It doesn't take much of an imagination to see that you are pretty screwed are all by yourself in the corner of the map and the entire opposing team converges on your position.
All other things being equal, the general idea here is that you only want to engage in team battles where you can expect to break even if not be favored to win. That means you want to gather more of your teammates together before a battle than your opponents do. But your opponents know this too and will counteract accordingly. This I-know-that-you-know-that-I-know business can get out of hand fairly quickly, so I put together a video to describe the interaction and find the game's equilibria. I want you to watch it and consider how it relates to the 8-4 queues on Magic Online:
So we see that there are two reasonable strategies to take: you either bring your entire team to the fight or you just avoid the fight altogether. Both are reasonable strategies, and we see both being used in practice within the game.
But that is not the subject of this article. Right now, I want to investigate why people even bother with the 8-4 queues. Unfortunately, I do not have the answer—just some theories, and I want to present them to the community and hear your feedback on them so we can figure out what really going on here.
Before we can get to that point, though, we need to look into the game theory behind choosing to go into an 8-4 queue. In the video, the level of troops you could bring into the fight was something that you could choose, and the number you chose in influenced who would win the battle. Now imagine I turned substituted “level of units” with “level of play skill.” Obviously, as much as you might wish otherwise, you cannot magically change our ability from awful to amazing overnight. Indeed, whenever you enter a tournament, your play skill is not going to change over the next few hours. You are who you are. Practicing may make you better over the long term, but the time to practice has passed when it gets to game time.
So while you might lose out on your ability to set your play skill as you could in the game in the video, you can still opt out of playing the tournament altogether. No one is forcing you to play. So if you think you are going to get blasted out of the water and have no hope of making the finals (which is necessary to win any product out of the 8-4 queue), there is no need for you to even show up in the first place. As it turns out, people participate in tournaments anyway because they (gasp!) get pleasure out of the mere act of playing the game.
That said, Magic Online gives players a wide variety of options for which tournaments they play in. Even if you do not feel comfortable in the world of 8-4 draft queues, you can go down to the 4-3-2-2 or Swiss queues and theoretically face more reasonable competition. Since we have to figure Magic players enjoy playing, let's consider playing in one of these two queues as the “opt out” strategy. The question is why we see anyone at all willing to play in the 8-4 queues.
Consider this: To earn more packs on average out of an 8-4 queue than one of the other queues, you need to have an extremely high win percentage in drafts out of those queues. So if you are Luis Scott-Vargas, then 8-4 seems like a great home for you. If you are a terrible Magic player, then one of the lesser choices is more up your alley.
But what about the just-better-than-terrible group of players? Well, all of the players worse than them are in the lesser queues. That means if they participate in the 8-4s, they have no one to feed off of, and thus will not be able to reach the win percentage necessary to justify participating in an 8-4 draft.
Now let's move to the bad players. Again, we see that all of the players worse than them have moved to the lesser queues. If they stay in the 8-4s, they are going to be demolished by the below average players, average players, above average players, excellent players, professional players, and Luis Scott-Vargas. So you predictably move along to the lesser queues.
If we keep going through each of these iterated steps, we eventually discover that Luis Scott-Vargas is the only player sitting in an 8-4 queue. Even a professional player would not want to have anything to do with the 8-4s, because there won't be anyone in that queue worse than them. The best of the best—like Luis Scott-Vargas—will still be there and ready to demolish a lesser pro. So lesser pros can choose between having a sub-par time in the 8-4, or they can be the top dog in one of the other queues. Naturally, they should be steering clear of the 8-4.
This creates quite a problem for Luis Scott-Vargas, as he is now the only player left in his queue. Magic Online needs eight players to fire up a tournament, so it looks like he won't be able to play in the 8-4 either. Consequently, he has to move down to one of the other queues as well. Thus, we have reached an important finding—given the assumptions, 8-4 queues never fire in equilibrium.
“Wait!” you say. “8-4 queues do fire in real life! This article is stupid. I want my last ten minutes back.”
Well, I am obviously aware that people play in these 8-4 queues. The puzzle is explaining why. Note that I said “given the assumptions, 8-4 queues never fire in equilibrium.” That means we need to look at the assumptions and figure out which one of them is false.
A1) Players are rational.
Game theory only works if players are rational. Fortunately, game theoretical rationality is much easier to meet than the definition of rationality you likely have in your mind. To be rational, you must only meet two requirements (for the most part). First, you must be able to rank all of your outcomes. For example, if I were to ask you “do you prefer winning two packs or three?” and you were to respond with “I don't know,” then you are not rational. Basically everybody meets this requirement. Second, you preferences must be transitive. For example, you cannot tell me you prefer oranges to lemons, lemons to strawberries, and strawberries to lemons. Again, just about everyone meets this requirement. If you find someone who breaks it, you can usually just point out the flaw in their logic, and they will almost certainly correct the issue.
(There is a third requirement—that your preferences over reduced lotteries must be transitive as well—but this gets complicated and does not much apply to the discussion at hand.)
It looks like we are fine here, so I don't think this is the assumption that is causing the problem.
A2) Players naturally understand the equilibrium and play accordingly.
Here's a poll for those of you who are reading this article: did you intuitively understand the logic I discussed in the video and applied to choosing the appropriate queue? There is a range to this. Unless you have taken a game theory class before or studied the market for lemons in an economics class, you probably could not formally prove this concept before. If you have, you fall into group (a). However, it would not surprise me at all if you had figured out 90% of the logic on your own, before you had even read the first two paragraphs of this article. That would put you into group (b). Another group had never thought of this but just bought into it in the last five minutes. This is group (c). A final group—group (d)—will flame me. (Whatever.)
The question is how prevalent the last two groups are. If the majority of people playing in 8-4s fall into groups (c) and (d), then Luis Scott-Vargas has a bunch of party buddies to play with. Check that. If the majority of people playing in 8-4s fall into group (c), then LSV had a bunch of party buddies to play with. After this article is published, all of them will be gone. But if everyone falls into group (d), then LSV can sleep soundly at night, knowing that he will have plenty of victims for the rest of time, as those players are entirely unaware that this interaction is going on.
However, if everyone belongs to (a) or (b), then the 8-4 queues should be empty, and LSV is sad, and this is not the incorrect assumption.
A3) 8-4s produce more efficient than 4-3-2-2 drafts, and this does not bias players toward the 8-4s.
This used to be a decent explanation back before the implementation of the Swiss drafts that award one pack to each player for each match they win. 8-4 queues give the players a total of twelve packs. 4-3-2-2 queues payoff eleven. As such, the 8-4s seem like a deal while 4-3-2-2s look like a rip off. This might have driven some people to believe that they should go to 8-4s. While the crowding out equilibrium I have been discussing still applies to this situation—meaning the 8-4 queues should be void of players—I can understand how people in group (b) might not realize this. Heck, I am in group (a), and I didn't even figure it out until I sat down and started writing this article. (I have not included the proof for this, as it is a bit more convoluted than the one I went over earlier.)
The reason I say this used to be a decent explanation is because Swiss drafts now exist, and those give twelve packs in total, just like the 8-4s do. The fact that people still participate in 4-3-2-2 queues is baffling in itself, but that is a second puzzle best left to another article.
A4) Players have do not have an inflated sense of their abilities.
If everyone thinks they are as good as Luis Scott-Vargas, then everyone will play in 8-4s. I think this is a likely explanation. But note that even if you are just slightly worse than the best player in the world, you should still play in the Swiss queues following the same logic as before. And if there were eight people tied for the best Magic players in the world, they should all want to descend on the Swiss queues as well.
(To see this note, that if we cloned LSV seven times, then each would expect to win 50% of his matches in an 8-4 all LSV queue. But if all of the LSVs took on separate Swiss queues, their win percentages will rise above 50%, and they will earn more packs on average.)
A5) Players don't see 8-4 as a status symbol.
I get a kick every time someone parades around a message board saying (paraphrased) “I'm so special—I play on the 8-4 queues!” Congratulations, cupcake—if you value being able to say that you play in the elite-ish 8-4 queues, then you have screwed up the model. But it is unclear whether he actually places value on being able to say this or if he is just saying it because it is true. And if players place value on being telling other they are 8-4 players, nothing stops them from choosing the Swiss queue and then lying later on.
A6) Players don't care how they win.
I ought to present this as a comparison. When I play League of Legends, here is my order of preferred outcomes:
(win a good game) > (lose a good game) > (win a bad game) > (lose a bad game)
I like to win, but stomping over awful competition isn't much fun. As such, I would rather lose a close game then destroy everyone in sight. I think the rest of the ordering is very straightforward.
Supposing that better players are concerned with the quality of their games and worse players only care about cobbling together whatever wins they can, then I can see how an equilibrium would form with the good players in the 8-4 camp and the worse players in the Swiss queue. This also caters to the good players who really want to get quality practice in before a PTQ or other important event, as the 8-4 queue will filter out bad players who are unlikely to be in attendance at such an event.
While this explanation is plausible, I would like to note that there is no tangible reward to winning a game in League of Legends. Win or lose, I don't really go home with anything more or less had the opposite outcome occurred. Once they start putting together seasonal tournaments, however, things are going to change. At that point, I will not care how I win, just so long as I win. Same goes with Magic Online, at least for me personally. Your mileage may vary.
A7) My math is right.
I'm just throwing this in there as to cover my rear end in case I blew it somewhere along the way. But I really don't think I did, so I doubt this is the answer.
The Big Question
So what is going on here? What false assumption is making people play in the 8-4 queues? Is it a combination of assumptions? Is there another assumption that I missed? If you play in 8-4s, why do you do it?
Game theory cannot explain everything. In fact, most of the time, these models are better at explaining what doesn't cause things to happen, narrowing down the possible list of causal mechanisms. I hope we are moving a step in that direction today.
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