Agreed that it’s insufficient, but I think it shows that there’s no way to specify strategies that work regardless of other players’ strategies, and I agree that this generalizes to better solution concepts, which I agree “make the equilibrium selection problem worse”.
I’d also point out an oft-noted critical failure of Nash Equilibria, which is that they assume infinite computation, and (therefore) no logical uncertainty. A game can pay out the seventeenth digit of the BB(200) to player 1 and the eighteenth digit to player 2, and we must assume these are known, and can be used to find the NE. I haven’t thought through the following through completely, but it seems obvious that this issue can be used to show why NE is not generally a useful/valid solution concept for embedded agents, because they would need models of themselves and other agents their own size to predict goals / strategies.
I’m saying that non-uniqueness of the solution is part of the conceptual problem with Nash equilibria.
Decision theory doesn’t exactly provide a “unique solution”—it’s a theory of rational constraints on subjective belief, so, you can believe and do whatever you want within the confines of those rationality constraints. And of course classical decision theory also has problems of its own (such as logical omniscience). But there is a sense in which it is better than game theory about this, since game theory gives rationality constraints which depend on the other player in ways that are difficult to make real.
I’m not saying there’s some strategy which works regardless of the other player’s strategy. In single-player decision theory, you can still say “there’s no optimal strategy due to uncertainty about the environment”—but, you get to say “but there’s an optimal strategy given our uncertainty about the environment”, and this ends up being a fairly satisfying analysis. The nash-equilibrium picture of game theory lacks a similarly satisfying analysis. But this does not seem essential to game theory.
Pretty sure we’re agreeing here. I was originally just supporting cousin_it’s claim, not claiming that Nash Equilibria are a useful-enough solution concept. I was simply noting that—while they are weaker than a useful-enough concept would be—they can show the issue with non-uniqueness clearly.
Agreed that it’s insufficient, but I think it shows that there’s no way to specify strategies that work regardless of other players’ strategies, and I agree that this generalizes to better solution concepts, which I agree “make the equilibrium selection problem worse”.
I’d also point out an oft-noted critical failure of Nash Equilibria, which is that they assume infinite computation, and (therefore) no logical uncertainty. A game can pay out the seventeenth digit of the BB(200) to player 1 and the eighteenth digit to player 2, and we must assume these are known, and can be used to find the NE. I haven’t thought through the following through completely, but it seems obvious that this issue can be used to show why NE is not generally a useful/valid solution concept for embedded agents, because they would need models of themselves and other agents their own size to predict goals / strategies.
I’m saying that non-uniqueness of the solution is part of the conceptual problem with Nash equilibria.
Decision theory doesn’t exactly provide a “unique solution”—it’s a theory of rational constraints on subjective belief, so, you can believe and do whatever you want within the confines of those rationality constraints. And of course classical decision theory also has problems of its own (such as logical omniscience). But there is a sense in which it is better than game theory about this, since game theory gives rationality constraints which depend on the other player in ways that are difficult to make real.
I’m not saying there’s some strategy which works regardless of the other player’s strategy. In single-player decision theory, you can still say “there’s no optimal strategy due to uncertainty about the environment”—but, you get to say “but there’s an optimal strategy given our uncertainty about the environment”, and this ends up being a fairly satisfying analysis. The nash-equilibrium picture of game theory lacks a similarly satisfying analysis. But this does not seem essential to game theory.
Pretty sure we’re agreeing here. I was originally just supporting cousin_it’s claim, not claiming that Nash Equilibria are a useful-enough solution concept. I was simply noting that—while they are weaker than a useful-enough concept would be—they can show the issue with non-uniqueness clearly.