Berserking Better
How the prisoner's dilemma applies to chess
The prisoner’s dilemma is a classic problem in game theory. The setup goes like this: two suspects are being held by the police. The police have enough evidence to convict each of them on a minor charge, but not the main charge, so they separate them and offer each a bargain: talk and we’ll let you off easy.
If they both stay silent, they both get one year in prison. If they both talk, they both get two years in prison. If one stays silent and the other talks, the one who talks will get off free, and they one who stayed silent gets three years.
The outcomes are often shown as a 2-by-2 matrix:
What would you do, talk (defect) or stay silent (cooperate)? Say you’re Prisoner 1. If you assume that Prisoner 2 will defect, you’re better off defecting as well: you’ll get two years rather than 3. If you assume Prisoner 2 will cooperate, you’re still better off defecting: you’ll get off free rather than serving one year. It seems you’re better off defecting no matter what Prisoner 2 does.
If you look at it from Prisoner 2’s perspective, the exact same logic applies. You’re better off defecting no matter what Prisoner 1 does. So it seems if both prisoners behave logically, they will both defect and both end up serving two years in prison.
The weird part is, if neither of them defected, they would have ended up with one year each. It seems as though an inescapable chain of logic led them to an outcome that was worse for both.
At first, the prisoner’s dilemma seems to have little relevance for chess, because it hinges on the possibility of cooperation. Chess is a zero-sum game, meaning one player’s gain is the other player’s loss. If my position gets better, yours gets worse. If I win, you have to lose. In such a situation, it’s hard to see how cooperation is possible.
In a single game of standard chess, it’s generally not, but in tournaments involving multiple players things get more complicated. One area where the prisoner’s dilemma comes up is in lichess arena tournaments. In contrast to a traditional over-the-board tournament where each round starts at a fixed time, in an arena tournament, when you finish your game you’re immediately paired and start a new game. Rather than a fixed number of rounds, the tournament ends after a set time limit.
This is great for limiting downtime but it changes the tournament dynamics. In a traditional tournament it doesn’t really matter how long your games take. In an arena tournament, if two players are identical in terms of chess strength but one completes their games twice as quickly, the faster player will end up with twice as many points on average. Finishing your games quickly is a huge advantage. This point doesn’t seem to be fully appreciated by my opponents who WON’T RESIGN IN LOSING POSITIONS.
Where were we? Oh, right. Another twist is the possibility of “berserking”. If a player hits the Berserk button, they get half as much time for the game, but an extra point if they win. Imagine two games between players of equal ability. In game A, neither player berserks; in game B, both players berserk. The players in game B, having half as much time, will finish their game about twice as quickly on average. As a cherry on top, the winner of game B will get an extra point. This means that just by berserking, without changing their chess ability at all, the players in game B are amassing points more than twice as quickly as the players in game A.
But here comes the prisoner’s dilemma. What if one player berserks and the other doesn’t? The player who doesn’t berserk will still enjoy a faster game on account of their opponent having less time (though not as fast as if they berserked as well), but they’ll also have a much better chance of winning because they have twice as much time to think. You’d rather have two berserks than zero, but not berserking when your opponent does may be the best scenario of all.
There’s one more wrinkle to arena tournaments. If you win two games in a row you’re “on fire” and all your games are worth double points as long as you keep winning. This is a genius feature and makes me think whoever designed the rules was keenly aware of the risk of players just automatically berserking. By increasing the value of stacking up wins, they made berserking less attractive.
This is the feature that makes an arena game a clear-cut prisoner’s dilemma. If both players berserk, they’re both better off than if they didn’t; but if your opponent berserks and you don’t, that’s even better for you. Given the individual incentive to not berserk, neither player can be confident the opponent will berserk, which leads to many games where neither player berserks.
Given the self-destructive result of the prisoner’s dilemma, game theory researchers have looked for ways in which cooperation could arise. One of the ideas is to look at what happens if you play out the same scenario out multiple times with the same players. Maybe it’s worth cooperating if you know you’ll find yourself in the same situation with the same person again?
In arena tournaments you can play the same opponent multiple times, so it’s an example of the repeat scenario. Berserking the first time you face an opponent could be seen as a tacit invitation to cooperation, in effect saying, “Hey dude, it’s better for both of us if we both berserk, what do you say?” If the opponent berserks as well the deal is sealed and you can continue both berserking each time you face each other. Such an arrangement benefits both players, but it’s also risky because each side has an incentive to defect that only gets stronger as you get closer to the end of the tournament.
At the end of the day, it’s hard to say what the best strategy is. Unless you’re playing me, in which case you should definitely berserk. Don’t worry, I will too. Trust me.