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JGWeissman comments on A simple game that has no solution - Less Wrong
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Comments (123)
P1: .5C .5B
P2: Y
It's not a Nash equilibrium, but it could be a timeless one. Possibly more trustworthy than usual for oneshots, since P2 knows that P1 was not a Nash agent assuming the other player was a Nash agent (classical game theorist) if P2 gets to move at all.
I have no idea where those numbers came from. Why not "P1: .3C .7B" to make "P2: Y" rational? Otherwise, why does P2 play Y at all? Why not "P1: C, P2: Y", which maximizes the sum of the two utilities, and is the optimal precommitment under the Rawlian veil-of-ignorance prior? Heck, why not just play the unique Nash equilibrium "P1: A"? Most importantly, if there's no principled way to make these decisions, why assume your opponent will timelessly make them the same way?
If we multiply player 2's utility function by 100, that shouldn't change anything because it is an affine transformation to a utility function. But then "P1: B, P2: Y" would maximize the sum. Adding values from different utility functions is a meaningless operation.
You're right. I'm not actually advocating this option. Rather, I was comparing EY's seemingly arbitrary strategy with other seemingly arbitrary strategies. The only one I actually endorse is "P1: A". It's true that this specific criterion is not invariant under affine transformations of utility functions, but how do I know EY's proposed strategy wouldn't change if we multiply player 2's utility function by 100 as you propose?
(Along a similar vein, I don't see how I can justify my proposal of "P1: 3/10 C 7/10 B". Where did the 10 come from? "P1: 2/7 C 5/7 B" works equally well. I only chose it because it is convenient to write down in decimal.)
Eliezer's "arbitrary" strategy has the nice property that it gives both players more expected utility than the Nash equilibrium. Of course there are other strategies with this property, and indeed multiple strategies that are not themselves dominated in this way. It isn't clear how ideally rational players would select one of these strategies or which one they would choose, but they should choose one of them.