Insane prisoner's dilemma

Two perfectly rational human prisoners, let us call them Alice and Bob, have been caught and are about to be interrogated. In the meantime they are both being kept in the one cell.

The outcome will be as per normal prisoner's dilemma. If neither confesses, they both win. If one confesses and the other does not, the first one wins a lot and the second loses. If they both confess, they both lose. They will never meet again.

If this were normal prisoner's dilemma, the outcome game theory predicts is that they both confess. It doesn't matter how much they confer before hand, the game is set up so that this is the rational thing to do.

However, being human, we shall suppose they have a novel ability and a novel option while they are in the cell:

• They can tell what motivates the other player, which will be some linear combination of both of the players outcomes. Of course initially both prisoners only care about their own outcome.

• They can choose to go insane, modifying their own motivation to be a new linear combination of their own outcome and the other player's. From then on their actions will be guided by that new motivation (possibly to their detriment!). Note that their actions will only be irrational from the perspective of their initial motivation. From the perspective of their new motivation, their actions from then on will be perfectly rational.

If we only allow a limited set of linear combinations, working out the player's actions becomes fairly easy to solve.

What will happen is this: One of the players, say Alice, goes insane to the extent that, to her, the game becomes a coordination game. She will then not confess if she knows that Bob will not confess. Bob, seeing this, will also go insane to the same extent. The problem is now a coordination problem, and they have common knowledge that they both do not want to confess. So they don't confess, and both win.

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