The new summary looks good =) Although I second Michael Dennis’ comment below, that the infinite regress of priors is avoided in standard game theory by specifying a common prior. Indeed the specification of this prior leads to a prior selection problem.
The formality of “priors / equilibria” doesn’t have any benefit in this case (there aren’t any theorems to be proven)
I’m not sure if you mean “there aren’t any theorems to be proven” or “any theorem that’s proven in this framework would be useless”. The former is false, e.g. there are things to prove about the construction of learning equilibria in various settings. I’m sympathetic with the latter criticism, though my own intuition is that working with the formalism will help uncover practically useful methods for promoting cooperation, and point to problems that might not be obvious otherwise. I’m trying to make progress in this direction in this paper, though I wouldn’t yet call this practical.
The one benefit I see is that it signals that “no, even if we formalize it, the problem doesn’t go away”, to those people who think that once formalized sufficiently all problems go away via the magic of Bayesian reasoning
Yes, this is a major benefit I have in mind!
The strategy of agreeing on a joint welfare function is already a heuristic and isn’t an optimal strategy; it feels very weird to suppose that initially a heuristic is used and then we suddenly switch to pure optimality
I’m not sure what you mean by “heuristic” or “optimality” here. I don’t know of any good notion of optimality which is independent of the other players, which is why there is an equilibrium selection problem. The welfare function selects among the many equilibria (i.e. it selects one which optimizes the welfare). I wouldn’t call this a heuristic. There has to be some way to select among equilibria, and the welfare function is chosen such that the resulting equilibrium is acceptable by each of the principals’ lights.
I’m not sure what you mean by “heuristic” or “optimality” here. I don’t know of any good notion of optimality which is independent of the other players, which is why there is an equilibrium selection problem.
I think once you settle on a “simple” welfare function, it is possible that there are _no_ Nash equilibria such that the agents are optimizing that welfare function (I don’t even really know what it means to optimize the welfare function, given that you have to also punish the opponent, which isn’t an action that is useful for the welfare function).
I’m not sure if you mean “there aren’t any theorems to be proven” or “any theorem that’s proven in this framework would be useless”.
Hmm, I meant one thing and wrote another. I meant to say “there aren’t any theorems proven in this post”.
I second Michael Dennis’ comment below, that the infinite regress of priors is avoided in standard game theory by specifying a common prior. Indeed the specification of this prior leads to a prior selection problem.
Just to make sure that I was understood, I was also pointing out that “you can have a well-specified Bayesian belief over your partner” even without agreeing on a common prior, as long as you agree on a common set of possibilities or something effectively similar. This means that talking about “Bayesian agents without a common prior” is well-defined.
When there is not a common prior, this lead to an arbitrarily deep nesting of beliefs, but they are all well-defined. I can refer to “what A believes that B believes about A” without running into Russell’s Paradox. When the priors mis-match then the entire hierarchy of these beliefs might be useful to reason about, but when there is a common prior, it allows much of the hierarchy to collapse.
The new summary looks good =) Although I second Michael Dennis’ comment below, that the infinite regress of priors is avoided in standard game theory by specifying a common prior. Indeed the specification of this prior leads to a prior selection problem.
I’m not sure if you mean “there aren’t any theorems to be proven” or “any theorem that’s proven in this framework would be useless”. The former is false, e.g. there are things to prove about the construction of learning equilibria in various settings. I’m sympathetic with the latter criticism, though my own intuition is that working with the formalism will help uncover practically useful methods for promoting cooperation, and point to problems that might not be obvious otherwise. I’m trying to make progress in this direction in this paper, though I wouldn’t yet call this practical.
Yes, this is a major benefit I have in mind!
I’m not sure what you mean by “heuristic” or “optimality” here. I don’t know of any good notion of optimality which is independent of the other players, which is why there is an equilibrium selection problem. The welfare function selects among the many equilibria (i.e. it selects one which optimizes the welfare). I wouldn’t call this a heuristic. There has to be some way to select among equilibria, and the welfare function is chosen such that the resulting equilibrium is acceptable by each of the principals’ lights.
I think once you settle on a “simple” welfare function, it is possible that there are _no_ Nash equilibria such that the agents are optimizing that welfare function (I don’t even really know what it means to optimize the welfare function, given that you have to also punish the opponent, which isn’t an action that is useful for the welfare function).
Hmm, I meant one thing and wrote another. I meant to say “there aren’t any theorems proven in this post”.
Just to make sure that I was understood, I was also pointing out that “you can have a well-specified Bayesian belief over your partner” even without agreeing on a common prior, as long as you agree on a common set of possibilities or something effectively similar. This means that talking about “Bayesian agents without a common prior” is well-defined.
When there is not a common prior, this lead to an arbitrarily deep nesting of beliefs, but they are all well-defined. I can refer to “what A believes that B believes about A” without running into Russell’s Paradox. When the priors mis-match then the entire hierarchy of these beliefs might be useful to reason about, but when there is a common prior, it allows much of the hierarchy to collapse.