Immediately before the statement of Theorem I in section III.
Yes, that’s what it means. I don’t see how that makes it unmeaningful.
In my mind, there’s a meangingful difference between construction and description- yes, you can describe any waveform as an infinite series of sines and cosines, but if you actually want to build one, you probably want to use a finite series. And this result doesn’t exclude any exotic methods of constructing utility functions; you could multiply together the utilities of each individual in the pool and you’d end up with an aggregate utility function that could be expressed as a linear combination of the individual utilities (and the ones vector), with the weights changing every time you add another individual to the pool or add another outcome to be considered.
More relevant to the discussion, though, is the idea of the aggregator should not introduce novel preferences. This is an unobjectionable conclusion, I would say, but it doesn’t get us very far: if there are preferences in the pool that we want to exclude, like a utility monster’s, setting their weight to 0 is what excludes their preferences, not abandoning linear combinations, and if the system designer has preferences about “fairness” or so on, then so long as one of the agents in the pool has those preferences, the system designer can incorporate those preferences just by increasing their weight in the combination.
But in both cases, the aggregator would probably be created through another function, and then so long as it does not introduce novel preferences it can be described as a linear combination. Instead of arguing about weights, we may find it more fruitful to argue about meta-weights, even though there is a many-to-one mapping (for any particular instance) from meta-weights to weights.
Let K represent the row vector with all 1s (a constant function). Let “pseudogamble” refer to column vectors whose elements add to 1 (Kx = 1).
I’d recommend the use of “e” for the ones vector, and if the elements add to 1, it’s not clear to me why it’s a “pseudogamble” rather than a “gamble,” if one uses the terminology that column vectors where only a single element is 1 are “outcomes.”
I find preferences much clearer to think about as “tradeoffs”- that is, column vectors that add to 0, which are easily created by subtracting two gambles, but now the scaling is arbitrary and the sign of the product of a utility row vector and a tradeoff column vector unambiguously determines the preference for the preference.
For instance, if x, y, and z are outcomes
Alphabetical collision!
Given a linearly independent set of row vectors, it is possible to find a column vector whose product with each row vector is independently specifiable. In particular, you can find column vectors x and y such that Kx=Ky=1, Ax>Ay for all initial utility functions A, and Sx<Sy, where S is the aggregate utility function.
you could multiply together the utilities of each individual in the pool and you’d end up with an aggregate utility function that could be expressed as a linear combination of the individual utilities (and the ones vector), with the weights changing every time you add another individual to the pool or add another outcome to be considered.
Unlikely, unless there are at least as many agents as outcomes.
if the system designer has preferences about “fairness” or so on, then so long as one of the agents in the pool has those preferences, the system designer can incorporate those preferences just by increasing their weight in the combination.
Yes. In fact, I think something like that will be necessary. For example, suppose there is a population of two agents, each of which has a “hedon function” which specifies their agent-centric preferences. One of the agents is an egoist, so his utility function is his hedon function. The other agent is an altruist, so his utility function is the average of his and the egoist’s hedon functions. If you add up the two utility functions, you find that the egoist’s hedon function gets three times the weight of the altruist’s hedon function, which seems unfair. So we would want to give extra weight to the altruist’s utility function (you could argue that in this example you should use only the altruist’s utility function).
if the elements add to 1, it’s not clear to me why it’s a “pseudogamble” rather than a “gamble,” if one uses the terminology that column vectors where only a single element is 1 are “outcomes.”
Unlikely, unless there are at least as many agents as outcomes.
It’s unlikely that the weights of existing agents would change under either of those cases, or that the multiplication could be expressed as a weighted sum, or that the multiplication would have axiom 2-ness?
If you add up the two utility functions, you find that the egoist’s hedon function gets three times the weight of the altruist’s hedon function, which seems unfair.
Indeed. The problem is more general- I would classify the parts as “internal” and “external,” rather than agent-centric and other, because that makes it clearer that agents don’t have to positively weight each other’s utilities. If you have a ‘maltruist’ whose utility is his internal utility minus the egoist’s utility (divided by two to normalize), we might want to balance their weight and the egoist’s weight so that the agents’ internal utilities are equally represented in the aggregator.
Such meta-weight arguments, though, exist in an entirely different realm from this result, and so this result has little bearing on those arguments (which is what people are interested in when they resist the claim that social welfare functions are linear combinations of individual utility).
It’s unlikely that the weights of existing agents would change under either of those cases, or that the multiplication could be expressed as a weighted sum, or that the multiplication would have axiom 2-ness?
Unlikely that the multiplication could be expressed as a weighted sum (and hence by extension, also unlikely it would obey axiom 2).
I agree in general, because we would need the left inverse of the combined linearly independent individual utilities and e, and that won’t exist. We do have freedom to affinely transform the individual utilities before taking their element-wise product, though, and that gives us an extra degree of freedom per agent. I suspect we can do it so long as the number of agents is at least half the number of outcomes.
Oh, I see what you mean. It should be possible to find some affinely transformed product that is also a linear combination if the number of agents is at least half the number of outcomes, but some arbitrary affinely transformed product is only likely to also be a linear combination if the number of agents is at least the number of outcomes.
Immediately before the statement of Theorem I in section III.
In my mind, there’s a meangingful difference between construction and description- yes, you can describe any waveform as an infinite series of sines and cosines, but if you actually want to build one, you probably want to use a finite series. And this result doesn’t exclude any exotic methods of constructing utility functions; you could multiply together the utilities of each individual in the pool and you’d end up with an aggregate utility function that could be expressed as a linear combination of the individual utilities (and the ones vector), with the weights changing every time you add another individual to the pool or add another outcome to be considered.
More relevant to the discussion, though, is the idea of the aggregator should not introduce novel preferences. This is an unobjectionable conclusion, I would say, but it doesn’t get us very far: if there are preferences in the pool that we want to exclude, like a utility monster’s, setting their weight to 0 is what excludes their preferences, not abandoning linear combinations, and if the system designer has preferences about “fairness” or so on, then so long as one of the agents in the pool has those preferences, the system designer can incorporate those preferences just by increasing their weight in the combination.
But in both cases, the aggregator would probably be created through another function, and then so long as it does not introduce novel preferences it can be described as a linear combination. Instead of arguing about weights, we may find it more fruitful to argue about meta-weights, even though there is a many-to-one mapping (for any particular instance) from meta-weights to weights.
I’d recommend the use of “e” for the ones vector, and if the elements add to 1, it’s not clear to me why it’s a “pseudogamble” rather than a “gamble,” if one uses the terminology that column vectors where only a single element is 1 are “outcomes.”
I find preferences much clearer to think about as “tradeoffs”- that is, column vectors that add to 0, which are easily created by subtracting two gambles, but now the scaling is arbitrary and the sign of the product of a utility row vector and a tradeoff column vector unambiguously determines the preference for the preference.
Alphabetical collision!
Agreed.
Unlikely, unless there are at least as many agents as outcomes.
Yes. In fact, I think something like that will be necessary. For example, suppose there is a population of two agents, each of which has a “hedon function” which specifies their agent-centric preferences. One of the agents is an egoist, so his utility function is his hedon function. The other agent is an altruist, so his utility function is the average of his and the egoist’s hedon functions. If you add up the two utility functions, you find that the egoist’s hedon function gets three times the weight of the altruist’s hedon function, which seems unfair. So we would want to give extra weight to the altruist’s utility function (you could argue that in this example you should use only the altruist’s utility function).
It may contain negative elements.
It’s unlikely that the weights of existing agents would change under either of those cases, or that the multiplication could be expressed as a weighted sum, or that the multiplication would have axiom 2-ness?
Indeed. The problem is more general- I would classify the parts as “internal” and “external,” rather than agent-centric and other, because that makes it clearer that agents don’t have to positively weight each other’s utilities. If you have a ‘maltruist’ whose utility is his internal utility minus the egoist’s utility (divided by two to normalize), we might want to balance their weight and the egoist’s weight so that the agents’ internal utilities are equally represented in the aggregator.
Such meta-weight arguments, though, exist in an entirely different realm from this result, and so this result has little bearing on those arguments (which is what people are interested in when they resist the claim that social welfare functions are linear combinations of individual utility).
Ah! Of course.
Unlikely that the multiplication could be expressed as a weighted sum (and hence by extension, also unlikely it would obey axiom 2).
I agree in general, because we would need the left inverse of the combined linearly independent individual utilities and e, and that won’t exist. We do have freedom to affinely transform the individual utilities before taking their element-wise product, though, and that gives us an extra degree of freedom per agent. I suspect we can do it so long as the number of agents is at least half the number of outcomes.
Oh, I see what you mean. It should be possible to find some affinely transformed product that is also a linear combination if the number of agents is at least half the number of outcomes, but some arbitrary affinely transformed product is only likely to also be a linear combination if the number of agents is at least the number of outcomes.