November 1, 2004

Iterated madness  

Will Baude has been talking about an iterated not-a-PD game over at Crescat, and I don't get it. He says that "even though a strategy is completely dominant in a single-play of the game, there can still be other possible equilibria, given the right set of prior expectations." No argument there. But those iterated strategies are still about maximizing utility, which is precisely the reason you can get a different outcome in an iterated PD (especially when it's infinitely repeated) than you would in a one shot game.

Here's Will's game:
  Astor trusts Astor betrays
Spade trusts 10,10 3,8
Spade betrays 8,3 0,0

The dominant strategy in both the one shot and the repeating game is to trust. Betraying just leads you to get less points, regardless of which player you are. In the iterated game, the strategy should be to get to (trust, trust) as soon as possible... so, if you're not sure if you'll get betrayed the first time out (which is silly, since your opponent would have no rational reason to betray) then shouldn't you respond with trust anyway, in the hopes that you can coordinate there? Even if you betray, there's no reason your opponent should betray on the next round, because even if you (irrationally!) keep betraying, she'll still get more points by trusting than she would be betraying.

The only way I can see to get to the equilibrium Will's talking about is if one of the players is irrational or if for some reason you have incomplete information (ie you can't calculate what's rational behavior on your opponent's part). Absent one of these circumstances, Will's strategy seems oddly like a case of beating your opponent into submission when she was on your side to begin with -- and paying 2 utils per turn for the thrill!


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