Good question. Itâs easiest to imagine the one-dimensional spatial case like (...,L2, L1, me, R1, R2, âŠ) where {Li} are people to my left and {Ri} are those to my right. If I turn 180° this permutes the vector to (..., R1, me, L1, âŠ) Which is obviously an infinite number of permutations, but seems morally unobjectionable.
Thank you for the example. I have two initial comments and possibly more if you are interested. 1. In all of the literature on the problem, the sequences that we compare specify social states. When we compare x=(x_1,x_2,...) and y=(y_1,y_2,...) (or, as in your example, x=(....,x_0,x_1,x_2,...) and y=(...,y_0,y_1,y_2,...)), we are doing it with the interpretation x_t and y_t give the utility of the same individual/âgeneration in the two possible social states. For the two sequences in your example, it does not seem to be the case that x_t and y_t give the utility of the same individual in two possible states. Rather, it seems that we are re-indexing the individuals. 2. I agree that moral preferences should generally be invariant to re-indexing, at least in a spatial context (as opposed to an intertermporal context). Let us therefore modify your example so that we have specified utilities x_t,y_t, where t ranges over the integers and x_t and y_t represent the utilities of people located at positions on a doubly infinite line. Now I agree that an ethical preference relation should be invariant under some (and possibly all) infinite permutations IF the permutation is performed to both sequences. But it is hard to give an argument for why we should have invariance under general permutations of only one stream.
The example is still unsatisfactory for two reasons. (i) since we are talking about intergenerational equity, the t in x_t should be time, not points in space where individuals live at the same time: it is not clear that the two cases are equivalent. (They may in fact be very different.) (ii) in almost all of the literature (in particular, in all three references in the original post), we consider one-sided sequences, indexed by time starting today and to the infinite future. Are you aware of example in this context?
For the two sequences in your example, it does not seem to be the case that xt and yt give the utility of the same individual in two possible states. Rather, it seems that we are re-indexing the individuals.
This is true. I think an important unstated assumption is that you only need to know that someone has utility x, and you shouldnât care who that person is.
Now I agree that an ethical preference relation should be invariant under some (and possibly all) infinite permutations IF the permutation is performed to both sequences. But it is hard to give an argument for why we should have invariance under general permutations of only one stream.
Iâm not sure what the two sequences you are referring to are. Anonymity constraints simply say that if y is a permutation of x, then x~y.
in almost all of the literature (in particular, in all three references in the original post), we consider one-sided sequences, indexed by time starting today and to the infinite future. Are you aware of example in this context?
It is a true and insightful remark that whether we consider vectors to be infinite or doubly infinite makes a difference.
To my mind, the use of vectors is misleading. What it means to not care about temporal location is really just that you treat populations as sets (not vectors) and so anonymity assumptions arenât really required.
I guess you could phrase that another way and say that if you donât believe in infinite anonymity, then you believe that temporal location matters. This disagrees with general utilitarian beliefs. Nick Bostrom talks about this more in section 2.2 of his paper linked above.
A more mathy way thatâs helpful for me is to just remember that the relation should be continuous. Say s_n(x) is a permutation of _n_ components. By finite anonymity we have that x~s_n(x) for any finite n. If lim {n â infinity} s_n = y, yet y was morally different from x, the relation is discontinuous and this would be a very odd result.
I would not only say that âthat you only need to know that someone has utility x, and you shouldnât care who that person isâ is an unstated assumption. I would say that it is the very idea that anonymity intends to formalize. The question that I had and still have is whether you know of any arguments for why infinite anonymity is suitable to operationalize this idea.
Regarding the use of sequences: you canât just look at sets. If you do, all nontrivial examples with utilities that are either 0 or 1 become equivalent. You donât have to use sequences, but you need (in the notation of Vallentyne and Kagan (1997)), a set of âlocationsâ, a set of real numbers where utility takes values, and a map from the location set to the utility set.
Regarding permutations of one or two sequences. One form of anonymity says that x ~ y if there is a permutation, say pi, (in some specified class) that takes x to y. Another (sometimes called relative anonymity) says that if x is at least as good as y, then pi(x) is at least as good as pi(y). These two notions of anonymity are not generally the same. There are certainly settings where the fullblown version of the relative anonymity becomes a basic rationality requirement. This would be the case with people lined up on an infinite line (at the same point in time). But it is not hard to see its inappropropriateness in the intertemporal context: you would have to rank the following two sequences (periodic with period 1000) to be equivalent or non-comparable
This connects to whether denying infinite anonymity implies that âtemporal location mattersâ. If x and y above are two possible futures for the same infinite-horizon society, then I think that any utilitarian should be able to rank x above y without having to be critisized for caring about temporal location. Do you agree? For those who do not, equity in the intertemporal setting is the same thing as equity in the spatial (fixed time) setting. What those people say is essentially that intergenerational equity is a trivial concept: that there is nothing special about time.
If you do not think that the sequences x and y above should be equivalent in the intergenerational context then I would be very interested to see another example of sequences (or whatever you replace them with) that are infinite permutations of each other, but not finite permutations of each other, and where you do think that equivalence should.
P.S
Regarding continuity arguments, I assume that the usefulness of such arguments depends on whether you can justify your notion of continuity by ethical principles rather than that they appear in the mathematical literature. Take x(n)=(0,0,....,1,0,0,...) with a 1 in the n:the coordinate. For every n we want x(n) to be equivalent to (1,0,0,....). In many topologies x(n) goes to (0,0,0,....), which would then give that (0,0,...) is just as good as (1,0,0,....).
The question that I had and still have is whether you know of any arguments for why infinite anonymity is suitable to operationalize this idea.
Maybe I am missing something, but it seems obvious to me. Here is my thought process; perhaps you can tell me what I am overlooking.
For simplicity, say that A is the assumption that we shouldnât care who people are, and IA is the infinite anonymity assumption. We wish to show A IA.
Suppose A. Observe that any permutation of people canât change the outcome, because itâs not changing any information which is relevant to the decision (as per assumption A). Thus we have IA.
Suppose IA. Observe that itâs impossible to care about who people are, because by assumption they are all considered equal. Thus we have A.
Hence A IA.
These seems so obviously similar in my mind that my âproofâ isnât very insightful⊠But maybe you can point out to me where I am going wrong.
One form of anonymity says that x ~ y if there is a permutation, say pi, (in some specified class) that takes x to y. Another (sometimes called relative anonymity) says that if x is at least as good as y, then pi(x) is at least as good as pi(y). These two notions of anonymity are not generally the same.
I hadnât heard about this â thanks! Do you have a source? Google scholar didnât find much.
In your above example is the pi in pi(X) the same as the pi in pi(y)? I guess it must be because otherwise these two types of anonymity wouldnât be different, but that seems weird to me.
If x and y above are two possible futures for the same infinite-horizon society, then I think that any utilitarian should be able to rank x above y without having to be critisized for caring about temporal location. Do you agree?
I certainly understand the intuition, but Iâm not sure I fully agree with it. The reason I think that x better than y is because it seems to me that x is a Pareto improvement. But itâs really not â there is no generation in x who is better off than another generation in y (under a suitable relabeling of the generations).
I would be very interested to see another example of sequences (or whatever you replace them with) that are infinite permutations of each other, but not finite permutations of each other, and where you do think that equivalence should.
(0,1,0,1,0,1,...) and (1,0,1,0,1,0,...) come to mind.
The problem in your argument is the sentence â...any permutation of people canât change the outcome...â. For example: what does âany permutationâ mean? Should the stream be applied to both sequences? In a finite context, these questions would not matter. In the infinite-horizon context, you can make mistakes if you are not careful. People who write on the subject do make mistakes all the time. To illustrate, let us say that I think that a suitable notion of anonymity is FA: for any two people p1 and p2, p1âČs utility is worth just as much as p2âČs. Then I can âproveâ that A FA by your method. The A â FA direction is the same. For FA â A, observe that if for any two people p1 and p2, p1âČs utility is worth just as much as p2âČs, then it is not possible to care about who people are.
This âproofâ was not meant to illustrate anything besides the fact that if we are not careful, we will be wasting our time.
I did not get a clear answer to my question regarding the two (intergenerational) streams with period 1000: x=(1,1,...,1,0,1,1,,...) and y=(0,0,...,0,1,0,0,,...). Here x does not Pareto-dominate y.
Regarding (0,1,0,...) and (0,1,0,...): I am familiar with this example from some of the literature. Recall in the first post that I wrote that the argumentation in much of the literature is not so good? This is the literature that I meant. I was hoping for more.
I will try to make the question more specific and then answer it. Suppose you are given two sequences x=(x_1,x_2,âŠ) and y=(y_1,y_2,âŠ) and that you are told that x_t is not necessarily the utility of generation t, but that it could be the utility of some other generation. Should your judgements then be invariant under infinite permutations? Well, it depends. Suppose I know that x_t and y_t is the utility of the same generation â but not necessarily of generation t. Then I would still say that x is better than y if x_t>y_t for every t. Infinite anonymity in its strongest form (the one you called intergenerational equity) does not allow you to make such judgements. (See my response to your second question below.) In this case I would agree to the strongest form of relative anonymity however. If I do not know that x_t and y_t give the utility of the same generation, then I would agree to infinite anonymity. So the answer is that sure, as you change the structure of the problem, different invariance conditions will become appropriate.
Thank you for the clarification and references â it took me a few days to read and understand those papers.
I donât think there are any strong ways in which we disagree. Prima facie, prioritizing the lives of older (or younger) people seems wrong, so statements like âI know that xt and yt is the utility of the same generationâ donât seem like they should influence your value judgments. However, lots of bizarre things occur if we act that way, so in reflective equilibrium we may wish to prioritize the lives of older people.
Wait a minute. Why should knowing that x_t and y_t are the utility of the same generation (in two different social states) not influence value judgements? There is certainly not anything unethical about that, and this is true also in a finite context. Let us say that society consists of three agents. Say that you are not necessarily a utilitarian and that you are given a choice between x=(1,3,4) and y=(0,2,3). You could say that x is better than y since all three members of society prefers state x to state y. But this assumes that you know that x_t and y_t give the utility of the same agent in the two states. If you did not know this, then things would be quite different. Do you see what I mean?
But this assumes that you know that xt and yt give the utility of the same agent in the two states. If you did not know this, then things would be quite different
No, you would know that there is a permutation of x which Pareto dominates y. This is enough for you to say that x>y.
I understand and accept your point though that people are not in practice selfless, and so if people wonder not âwill someone be better offâ but âwill I specifically be better offâ then (obviously) you canât have anonymity.
Things would not be all that different with three agents. Sorry. But let me ask you: when you apply Suppesâ grading principle to infer that e.g. x=(1,3,4) is better than y=(2,0,3) since there is a permutation of xâ of x with xâ>y, would you not say that you are relying on the idea that everyone is better off to conclude that xâ is better than y? I agree of course that criteria that depend on which state a specific person prefers are bad, and they cannot give us anonymity.
I agree that the immediate justification for the principle is âeveryone is better offâ, but as you correctly point out that implies knowing âidentifyingâ information.
It is hard for me to justify this on consequentialist grounds though. Do you know of any justifications? Probably most consequentialist would just say that it increases total and average utility and leave it at that.
I am not sure what you mean by consequentialist grounds. Feel free to expand if you can.
I am actually writing something on the topic that we have been discussing. If you are interested I can send it to you when it is submittable. (This may take several months.)
By the way, one version of what you might be saying is: âboth infinite anonymity and the overtaking criterion seem like reasonable conditions. But it turns out that they conflict, and the overtaking criterion seems more reasonable, so we should drop infinite anonymity.â I would agree with that sentiment.
Forget overtaking. Infinite anonymity (in its strongest form â the one you called intergenerational equity) is incompatible with the following requirement: if everyone is better off in state x=(x_1,x_2,..) than in state y=(y_1,y_2,..), then x is better than y. See e.g. the paper by Fleurbaey and Michel (2003).
Good question. Itâs easiest to imagine the one-dimensional spatial case like (...,L2, L1, me, R1, R2, âŠ) where {Li} are people to my left and {Ri} are those to my right. If I turn 180° this permutes the vector to (..., R1, me, L1, âŠ) Which is obviously an infinite number of permutations, but seems morally unobjectionable.
Thank you for the example. I have two initial comments and possibly more if you are interested. 1. In all of the literature on the problem, the sequences that we compare specify social states. When we compare x=(x_1,x_2,...) and y=(y_1,y_2,...) (or, as in your example, x=(....,x_0,x_1,x_2,...) and y=(...,y_0,y_1,y_2,...)), we are doing it with the interpretation x_t and y_t give the utility of the same individual/âgeneration in the two possible social states. For the two sequences in your example, it does not seem to be the case that x_t and y_t give the utility of the same individual in two possible states. Rather, it seems that we are re-indexing the individuals. 2. I agree that moral preferences should generally be invariant to re-indexing, at least in a spatial context (as opposed to an intertermporal context). Let us therefore modify your example so that we have specified utilities x_t,y_t, where t ranges over the integers and x_t and y_t represent the utilities of people located at positions on a doubly infinite line. Now I agree that an ethical preference relation should be invariant under some (and possibly all) infinite permutations IF the permutation is performed to both sequences. But it is hard to give an argument for why we should have invariance under general permutations of only one stream.
The example is still unsatisfactory for two reasons. (i) since we are talking about intergenerational equity, the t in x_t should be time, not points in space where individuals live at the same time: it is not clear that the two cases are equivalent. (They may in fact be very different.) (ii) in almost all of the literature (in particular, in all three references in the original post), we consider one-sided sequences, indexed by time starting today and to the infinite future. Are you aware of example in this context?
Thank you for the thoughtful comment.
This is true. I think an important unstated assumption is that you only need to know that someone has utility x, and you shouldnât care who that person is.
Iâm not sure what the two sequences you are referring to are. Anonymity constraints simply say that if y is a permutation of x, then x~y.
It is a true and insightful remark that whether we consider vectors to be infinite or doubly infinite makes a difference.
To my mind, the use of vectors is misleading. What it means to not care about temporal location is really just that you treat populations as sets (not vectors) and so anonymity assumptions arenât really required.
I guess you could phrase that another way and say that if you donât believe in infinite anonymity, then you believe that temporal location matters. This disagrees with general utilitarian beliefs. Nick Bostrom talks about this more in section 2.2 of his paper linked above.
A more mathy way thatâs helpful for me is to just remember that the relation should be continuous. Say s_n(x) is a permutation of _n_ components. By finite anonymity we have that x~s_n(x) for any finite n. If lim {n â infinity} s_n = y, yet y was morally different from x, the relation is discontinuous and this would be a very odd result.
I would not only say that âthat you only need to know that someone has utility x, and you shouldnât care who that person isâ is an unstated assumption. I would say that it is the very idea that anonymity intends to formalize. The question that I had and still have is whether you know of any arguments for why infinite anonymity is suitable to operationalize this idea.
Regarding the use of sequences: you canât just look at sets. If you do, all nontrivial examples with utilities that are either 0 or 1 become equivalent. You donât have to use sequences, but you need (in the notation of Vallentyne and Kagan (1997)), a set of âlocationsâ, a set of real numbers where utility takes values, and a map from the location set to the utility set.
Regarding permutations of one or two sequences. One form of anonymity says that x ~ y if there is a permutation, say pi, (in some specified class) that takes x to y. Another (sometimes called relative anonymity) says that if x is at least as good as y, then pi(x) is at least as good as pi(y). These two notions of anonymity are not generally the same. There are certainly settings where the fullblown version of the relative anonymity becomes a basic rationality requirement. This would be the case with people lined up on an infinite line (at the same point in time). But it is not hard to see its inappropropriateness in the intertemporal context: you would have to rank the following two sequences (periodic with period 1000) to be equivalent or non-comparable
x=(1,1,....,1,0,1,1,...,1,0,1,1,...,1,......) y=(0,0,....,0,1,0,0,...,0,1,0,0,...,0,......)
This connects to whether denying infinite anonymity implies that âtemporal location mattersâ. If x and y above are two possible futures for the same infinite-horizon society, then I think that any utilitarian should be able to rank x above y without having to be critisized for caring about temporal location. Do you agree? For those who do not, equity in the intertemporal setting is the same thing as equity in the spatial (fixed time) setting. What those people say is essentially that intergenerational equity is a trivial concept: that there is nothing special about time.
If you do not think that the sequences x and y above should be equivalent in the intergenerational context then I would be very interested to see another example of sequences (or whatever you replace them with) that are infinite permutations of each other, but not finite permutations of each other, and where you do think that equivalence should.
P.S
Regarding continuity arguments, I assume that the usefulness of such arguments depends on whether you can justify your notion of continuity by ethical principles rather than that they appear in the mathematical literature. Take x(n)=(0,0,....,1,0,0,...) with a 1 in the n:the coordinate. For every n we want x(n) to be equivalent to (1,0,0,....). In many topologies x(n) goes to (0,0,0,....), which would then give that (0,0,...) is just as good as (1,0,0,....).
Maybe I am missing something, but it seems obvious to me. Here is my thought process; perhaps you can tell me what I am overlooking.
For simplicity, say that A is the assumption that we shouldnât care who people are, and IA is the infinite anonymity assumption. We wish to show A IA.
Suppose A. Observe that any permutation of people canât change the outcome, because itâs not changing any information which is relevant to the decision (as per assumption A). Thus we have IA.
Suppose IA. Observe that itâs impossible to care about who people are, because by assumption they are all considered equal. Thus we have A.
Hence A IA.
These seems so obviously similar in my mind that my âproofâ isnât very insightful⊠But maybe you can point out to me where I am going wrong.
I hadnât heard about this â thanks! Do you have a source? Google scholar didnât find much.
In your above example is the pi in pi(X) the same as the pi in pi(y)? I guess it must be because otherwise these two types of anonymity wouldnât be different, but that seems weird to me.
I certainly understand the intuition, but Iâm not sure I fully agree with it. The reason I think that x better than y is because it seems to me that x is a Pareto improvement. But itâs really not â there is no generation in x who is better off than another generation in y (under a suitable relabeling of the generations).
(0,1,0,1,0,1,...) and (1,0,1,0,1,0,...) come to mind.
The problem in your argument is the sentence â...any permutation of people canât change the outcome...â. For example: what does âany permutationâ mean? Should the stream be applied to both sequences? In a finite context, these questions would not matter. In the infinite-horizon context, you can make mistakes if you are not careful. People who write on the subject do make mistakes all the time. To illustrate, let us say that I think that a suitable notion of anonymity is FA: for any two people p1 and p2, p1âČs utility is worth just as much as p2âČs. Then I can âproveâ that A FA by your method. The A â FA direction is the same. For FA â A, observe that if for any two people p1 and p2, p1âČs utility is worth just as much as p2âČs, then it is not possible to care about who people are.
This âproofâ was not meant to illustrate anything besides the fact that if we are not careful, we will be wasting our time.
I did not get a clear answer to my question regarding the two (intergenerational) streams with period 1000: x=(1,1,...,1,0,1,1,,...) and y=(0,0,...,0,1,0,0,,...). Here x does not Pareto-dominate y.
Regarding (0,1,0,...) and (0,1,0,...): I am familiar with this example from some of the literature. Recall in the first post that I wrote that the argumentation in much of the literature is not so good? This is the literature that I meant. I was hoping for more.
I forgot the reference for relative anonymity: See the paper by Asheim, dâAspremont and Banerjee (J. Math. Econ., 2010) and its references.
Fair enough. Let me phrase it this way: suppose you were blinded to the location of people in time. Do you agree that infinite anonymity would hold?
I will try to make the question more specific and then answer it. Suppose you are given two sequences x=(x_1,x_2,âŠ) and y=(y_1,y_2,âŠ) and that you are told that x_t is not necessarily the utility of generation t, but that it could be the utility of some other generation. Should your judgements then be invariant under infinite permutations? Well, it depends. Suppose I know that x_t and y_t is the utility of the same generation â but not necessarily of generation t. Then I would still say that x is better than y if x_t>y_t for every t. Infinite anonymity in its strongest form (the one you called intergenerational equity) does not allow you to make such judgements. (See my response to your second question below.) In this case I would agree to the strongest form of relative anonymity however. If I do not know that x_t and y_t give the utility of the same generation, then I would agree to infinite anonymity. So the answer is that sure, as you change the structure of the problem, different invariance conditions will become appropriate.
Thank you for the clarification and references â it took me a few days to read and understand those papers.
I donât think there are any strong ways in which we disagree. Prima facie, prioritizing the lives of older (or younger) people seems wrong, so statements like âI know that xt and yt is the utility of the same generationâ donât seem like they should influence your value judgments. However, lots of bizarre things occur if we act that way, so in reflective equilibrium we may wish to prioritize the lives of older people.
Wait a minute. Why should knowing that x_t and y_t are the utility of the same generation (in two different social states) not influence value judgements? There is certainly not anything unethical about that, and this is true also in a finite context. Let us say that society consists of three agents. Say that you are not necessarily a utilitarian and that you are given a choice between x=(1,3,4) and y=(0,2,3). You could say that x is better than y since all three members of society prefers state x to state y. But this assumes that you know that x_t and y_t give the utility of the same agent in the two states. If you did not know this, then things would be quite different. Do you see what I mean?
No, you would know that there is a permutation of x which Pareto dominates y. This is enough for you to say that x>y.
I understand and accept your point though that people are not in practice selfless, and so if people wonder not âwill someone be better offâ but âwill I specifically be better offâ then (obviously) you canât have anonymity.
Things would not be all that different with three agents. Sorry. But let me ask you: when you apply Suppesâ grading principle to infer that e.g. x=(1,3,4) is better than y=(2,0,3) since there is a permutation of xâ of x with xâ>y, would you not say that you are relying on the idea that everyone is better off to conclude that xâ is better than y? I agree of course that criteria that depend on which state a specific person prefers are bad, and they cannot give us anonymity.
Thanks Lawrence, this is a good point.
I agree that the immediate justification for the principle is âeveryone is better offâ, but as you correctly point out that implies knowing âidentifyingâ information.
It is hard for me to justify this on consequentialist grounds though. Do you know of any justifications? Probably most consequentialist would just say that it increases total and average utility and leave it at that.
I am not sure what you mean by consequentialist grounds. Feel free to expand if you can.
I am actually writing something on the topic that we have been discussing. If you are interested I can send it to you when it is submittable. (This may take several months.)
Good question; now that I try to explain it I think my definition of âconsequentialistâ was poorly defined.
I have changed my mind and agree with you â the argument for finite anonymity is weaker than I thought. Good to know!
I would be interested to hear your insights on these difficult problems, if you feel like sharing.
By the way, one version of what you might be saying is: âboth infinite anonymity and the overtaking criterion seem like reasonable conditions. But it turns out that they conflict, and the overtaking criterion seems more reasonable, so we should drop infinite anonymity.â I would agree with that sentiment.
Forget overtaking. Infinite anonymity (in its strongest form â the one you called intergenerational equity) is incompatible with the following requirement: if everyone is better off in state x=(x_1,x_2,..) than in state y=(y_1,y_2,..), then x is better than y. See e.g. the paper by Fleurbaey and Michel (2003).