The backwards reasoning in this problem is the same as is used in the unexpected hanging paradox, and similar to a problem called Guess 2⁄3 of the Average. This is where a group of players each guess a number between 0 and 100, and the player whose guess is closest to 2⁄3 of the average of all guesses wins. With thought and some iteration, the rational player can conclude that it is irrational to guess a number greater than (2/3)100, (2/3)^2100, (2/3)^n*100, etc. This has a limit at 0 when n → ∞, so it is irrational to guess any number greater than zero.
“I think correct strategy gets profoundly complicated when one side believes the other side is not fully rational.”
Very true. When you’re not playing with “rational” opponents, it turns out that this strategy’s effectiveness diminishes after n=1 (regardless, the average will never be greater than 67), and you’ll probably lose if you guess 0 - how can you be rational in irrational times? If everybody is rational, but there is no mutual knowledge of this, the same effect occurs.
The kick is this: even if you play with irrationals, they’re going to learn—even in a 3rd grade classroom, eventually the equilibrium sets in at 0, after a few rounds of play. After the first round, they’ll adjust their guesses, and each round the 2⁄3 mean will get lower until it hits 0. At that point, even if people don’t rationally understand the process, they’re guessing 0.
That’s what equilibrium is all about—you might not start there, or notice the tendency towards it, but once it’s achieved it persists. Players don’t even need to understand the “why” of it—the reason for which they cannot do better.
That’s a little offshoot, not entirely sure how well it relates. But back to the TIPD...
“do you really truly think that the rational thing for both parties to do, is steadily defect against each other for the next 100 rounds?
Yes, but I’m not entirely sure it matters. If that’s where the equilibrium is, that’s the state the game is going to tend towards. Even a single (D, D) game might irrevocably lock the game into that pattern.
The backwards reasoning in this problem is the same as is used in the unexpected hanging paradox, and similar to a problem called Guess 2⁄3 of the Average. This is where a group of players each guess a number between 0 and 100, and the player whose guess is closest to 2⁄3 of the average of all guesses wins. With thought and some iteration, the rational player can conclude that it is irrational to guess a number greater than (2/3)100, (2/3)^2100, (2/3)^n*100, etc. This has a limit at 0 when n → ∞, so it is irrational to guess any number greater than zero.
“I think correct strategy gets profoundly complicated when one side believes the other side is not fully rational.”
Very true. When you’re not playing with “rational” opponents, it turns out that this strategy’s effectiveness diminishes after n=1 (regardless, the average will never be greater than 67), and you’ll probably lose if you guess 0 - how can you be rational in irrational times? If everybody is rational, but there is no mutual knowledge of this, the same effect occurs.
The kick is this: even if you play with irrationals, they’re going to learn—even in a 3rd grade classroom, eventually the equilibrium sets in at 0, after a few rounds of play. After the first round, they’ll adjust their guesses, and each round the 2⁄3 mean will get lower until it hits 0. At that point, even if people don’t rationally understand the process, they’re guessing 0.
That’s what equilibrium is all about—you might not start there, or notice the tendency towards it, but once it’s achieved it persists. Players don’t even need to understand the “why” of it—the reason for which they cannot do better.
That’s a little offshoot, not entirely sure how well it relates. But back to the TIPD...
“do you really truly think that the rational thing for both parties to do, is steadily defect against each other for the next 100 rounds?
Yes, but I’m not entirely sure it matters. If that’s where the equilibrium is, that’s the state the game is going to tend towards. Even a single (D, D) game might irrevocably lock the game into that pattern.