Wednesday, November 23, 2011

Game Theory: Part VII: Cyclic Games: The Iterated Prisoner's Dilemma II: (Electric Bogaloo: )The End of the Game

We have seen that sometimes, the choice to cooperate in the hopes of future cooperation outweighs the rewards of defection. But what if there is no reason to cooperate in the future? What if future cooperation is impossible?
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Let us revisit the Iterated Prisoner's Dilemma. We have seen that, eventually, the benefits of cooperation with retaliation against defectors outweigh the benefits of constant defection. But let us look at what happens when rational individuals are told that the game will be ending on the current turn.

Let us look at the reasons any such individual would have for cooperating throughout the game. Past cooperation does not apply, because it has already happened. Future cooperation does not apply, because there is no future. And there is no way to influence what one's partner will do on the current turn. Therefore, the rational way to play this turn is to treat the game as though it were the simple dilemma, and defect.

So, on the final turn (assuming both players know it is the final turn), both players will defect.

Now, let's see what happens if the players are told that the game will be ending after the next turn.

Both players know the proof above that the plays for next turn will be two defections. There is nothing that either player can do to prevent it. So, the rational thing to do is to ignore that turn, and treat this turn as though it were the last.

And we can carry this back to infinity. If both players know how long the game is going to last, or even that they will know when the game is going to end, from the start, neither player has any reason to cooperate with the other! Sometimes, ignorance truly is bliss.

Tune in... sometime... for a somewhat-unfair solution to this.

2 comments:

  1. So, then, the player would only ever cooperate if he thinks the game will have infinite turns?

    Secular humanism comes to mind - the practice of cooperating, even with the knowledge that life's length (the number of turns, in other words) is finite.

    ReplyDelete
  2. Not quite. Even if both players know that the game is finite, they will cooperate as long as neither can know when the final turn will be.

    ReplyDelete

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