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?
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?
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.
Eliezer, would your ideas from this post apply here?
There could be many acceptable negotiating equilibria between what you think is the ‘fair’ point on the Pareto boundary, and the Nash equilibrium. So long as each step down in what you think is ‘fairness’ reduces the total payoff to the other agent, even if it reduces your own payoff even more.
If I’m not too confused, the Nash equilibrium is [P1: A], and the Pareto boundary extends from [P1: B, P2: Y] to [P1: C, P2: Y]. So the gains from trade give P1 1-3 extra points, and P2 0-2 extra points. As others have pointed out, a case could be made for [P1: C, P2: Y], as it maximizes the total gains from trade, but maybe, taking your idea of different concepts of fairness from the linked post, P2 should hold P1 hostage by playing some kind of mixed X,Y strategy unless P1 offers a “more fair” split.
Is that behavior by B the kind of thing that the reasoning in the linked post endorses?
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 think this should get better and better for P1 the closer P1 gets to (2/3)C (1/3)B (without actually reaching it).
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.
Eliezer, would your ideas from this post apply here?
If I’m not too confused, the Nash equilibrium is [P1: A], and the Pareto boundary extends from [P1: B, P2: Y] to [P1: C, P2: Y]. So the gains from trade give P1 1-3 extra points, and P2 0-2 extra points. As others have pointed out, a case could be made for [P1: C, P2: Y], as it maximizes the total gains from trade, but maybe, taking your idea of different concepts of fairness from the linked post, P2 should hold P1 hostage by playing some kind of mixed X,Y strategy unless P1 offers a “more fair” split.
Is that behavior by B the kind of thing that the reasoning in the linked post endorses?