Ok, I think what’s going on is that we have different ideas in mind about how two people make joint decisions. What I have in mind is something like Nash Bargaining solution or Kalai-Smorodinsky Bargaining Solution (both described in this post), for which the the VNM-equivalent weights do change depending on the set of feasible outcomes. I have to read your comment more carefully and think over your suggestions, but I’m going to guess that there are situations where they do not work or do not make sense, otherwise the NBS and KSBS would not be “the two most popular ways of doing this”.
Note that, as expected, in all cases we only consider options on the Pareto frontier, and those bargaining solutions could be expressed as the choice made by a single agent with a normal utility function. You’re right that the weights which identify the chosen solution will vary based on the options used and bargaining power of the individuals, and it’s worth reiterating that this theorem does not give you any guidance on how to pick the weights (besides saying they should be nonnegative). Think of it more as the argument that “If we needed to build an agent to select our joint choice for us and we can articulate our desires and settle on a mutually agreeable solution, then we can find weights for our utility functions such that the agent only needs to know a weighted sum of utility functions,” not the argument “If we needed to choose jointly and we can articulate our desires, then we can settle on a mutually agreeable solution.”
The NBS and KSBS are able to give some guidance on how to find a mutually agreeable solution because they have a disagreement point that they can use to get rid of the translational freedom, and thus they can get a theoretically neat result that does not depend on the relative scaling of the utility functions. Without that disagreement point (or something similar), there isn’t a theoretically neat way to do it.
In the example above, we could figure out each of our utilities for not paying our part for the ticket (and thus getting no chance to win), and decide what weights to put on based on that. But as the Pareto frontier shifts- as more tickets or more animals become available- our bargaining positions could easily shift. Suppose my utility for not buying in is .635, and your utility for not buying in is 1; I gain barely anything by buying a ticket (b is .025, c is .005), and you gain a lot (b is .24, and c is .86), and so I can use my indifference to making a deal to get my way (.025*.24>.005*.86).
But then the Ibis becomes available, as well as a ticket that offers a decent chance to get it, and I desperately want to get an Ibis. My indifference evaporates, and with it my strong bargaining position.
In situations where social utility will be aggregated, one way or another, then we don’t really have a d to get rid of our translational freedom. In cases where the disagreement point is something like “everybody dies” it’s not clear we want our metaethics (i.e. how we choose the weights) to be dependent on how willing someone is to let everybody die to not get their way (the old utility monster complaint).
Think of it more as the argument that “If we needed to build an agent to select our joint choice for us and we can articulate our desires and settle on a mutually agreeable solution, then we can find weights for our utility functions such that the agent only needs to know a weighted sum of utility functions,”
I still disagree with this. I’ll restate/expand the argument that I made at the top of the previous thread. Suppose we want to use NBS or KSBS to make the joint choice. We could:
Compute the Pareto frontier, apply NBS/KSBS to find the mutually agreeable solution, use the slope of the tangent at that point to derive a set of weights, use those weights to form a linear aggregation of our utility functions, program the linear aggregation into a VNM AI, have the VNM AI recompute that solution we already found and apply it, or
Input our utility functions into an AI separately, program it to compute the Pareto frontier and apply NBS/KSBS to find the mutually agreeable solution and directly apply that solution.
It seems to me that in 1 you’re manually doing all of the work to make the actual decision outside of the VNM framework, and then tacking on a VNM AI at the end to do more redundant work. Why would you do that instead of 2?
You disagree with the statement that we can, or you disagree with the implication that we should?
Why would you do that instead of 2?
In practice, I don’t think you would need to. The point of the theorem is that you always can if you want to, and I’m not sure why this result is interesting to Nisan.
(Note also that this approach works for other metaethical approaches besides NBS/KSBS, and that you don’t always have access to NBS/KSBS.)
Yeah, I thought you meant to imply “should”. If we’re just talking about “can”, then I agree (with some caveats that aren’t very important at this point).
Ok, I think what’s going on is that we have different ideas in mind about how two people make joint decisions. What I have in mind is something like Nash Bargaining solution or Kalai-Smorodinsky Bargaining Solution (both described in this post), for which the the VNM-equivalent weights do change depending on the set of feasible outcomes. I have to read your comment more carefully and think over your suggestions, but I’m going to guess that there are situations where they do not work or do not make sense, otherwise the NBS and KSBS would not be “the two most popular ways of doing this”.
Ah, I think I see where you’re coming from now.
Note that, as expected, in all cases we only consider options on the Pareto frontier, and those bargaining solutions could be expressed as the choice made by a single agent with a normal utility function. You’re right that the weights which identify the chosen solution will vary based on the options used and bargaining power of the individuals, and it’s worth reiterating that this theorem does not give you any guidance on how to pick the weights (besides saying they should be nonnegative). Think of it more as the argument that “If we needed to build an agent to select our joint choice for us and we can articulate our desires and settle on a mutually agreeable solution, then we can find weights for our utility functions such that the agent only needs to know a weighted sum of utility functions,” not the argument “If we needed to choose jointly and we can articulate our desires, then we can settle on a mutually agreeable solution.”
The NBS and KSBS are able to give some guidance on how to find a mutually agreeable solution because they have a disagreement point that they can use to get rid of the translational freedom, and thus they can get a theoretically neat result that does not depend on the relative scaling of the utility functions. Without that disagreement point (or something similar), there isn’t a theoretically neat way to do it.
In the example above, we could figure out each of our utilities for not paying our part for the ticket (and thus getting no chance to win), and decide what weights to put on based on that. But as the Pareto frontier shifts- as more tickets or more animals become available- our bargaining positions could easily shift. Suppose my utility for not buying in is .635, and your utility for not buying in is 1; I gain barely anything by buying a ticket (b is .025, c is .005), and you gain a lot (b is .24, and c is .86), and so I can use my indifference to making a deal to get my way (.025*.24>.005*.86).
But then the Ibis becomes available, as well as a ticket that offers a decent chance to get it, and I desperately want to get an Ibis. My indifference evaporates, and with it my strong bargaining position.
In situations where social utility will be aggregated, one way or another, then we don’t really have a d to get rid of our translational freedom. In cases where the disagreement point is something like “everybody dies” it’s not clear we want our metaethics (i.e. how we choose the weights) to be dependent on how willing someone is to let everybody die to not get their way (the old utility monster complaint).
I still disagree with this. I’ll restate/expand the argument that I made at the top of the previous thread. Suppose we want to use NBS or KSBS to make the joint choice. We could:
Compute the Pareto frontier, apply NBS/KSBS to find the mutually agreeable solution, use the slope of the tangent at that point to derive a set of weights, use those weights to form a linear aggregation of our utility functions, program the linear aggregation into a VNM AI, have the VNM AI recompute that solution we already found and apply it, or
Input our utility functions into an AI separately, program it to compute the Pareto frontier and apply NBS/KSBS to find the mutually agreeable solution and directly apply that solution.
It seems to me that in 1 you’re manually doing all of the work to make the actual decision outside of the VNM framework, and then tacking on a VNM AI at the end to do more redundant work. Why would you do that instead of 2?
You disagree with the statement that we can, or you disagree with the implication that we should?
In practice, I don’t think you would need to. The point of the theorem is that you always can if you want to, and I’m not sure why this result is interesting to Nisan.
(Note also that this approach works for other metaethical approaches besides NBS/KSBS, and that you don’t always have access to NBS/KSBS.)
Yeah, I thought you meant to imply “should”. If we’re just talking about “can”, then I agree (with some caveats that aren’t very important at this point).