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.)

Things would not be all that different with three agents. Sorry. But let me ask you: when you apply Suppes' grading principle to infer that e.g. x=(1,3,4) is better than y=(2,0,3) since there is a permutation of x' of x with x'>y, would you not say that you are relying on the idea that everyone is better off to conclude that x' 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.

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?

I will try to make the question more specific and then answer it. Suppose you are given two sequences x=(x_1,x_2,…) and y=(y_1,y_2,…) 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 – 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.

Forget overtaking. Infinite anonymity (in its strongest form – 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).

I forgot the reference for relative anonymity: See the paper by Asheim, d'Aspremont and Banerjee (J. Math. Econ., 2010) and its references.

The problem in your argument is the sentence "...any permutation of people can't change the outcome...". For example: what does "any permutation" 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, p1's utility is worth just as much as p2's. Then I can "prove" that A FA by your method. The A -> FA direction is the same. For FA -> A, observe that if for any two people p1 and p2, p1's utility is worth just as much as p2's, then it is not possible to care about who people are.

This "proof" 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 would not only say that "that you only need to know that someone has utility x, and you shouldn't care who that person is" 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 can't just look at sets. If you do, all nontrivial examples with utilities that are either 0 or 1 become equivalent. You don't have to use sequences, but you need (in the notation of Vallentyne and Kagan (1997)), a set of "locations", 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 "temporal location matters". 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,....).

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/generation 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?

I am curious about your definitions: intergenerational equity and finite intergenerational equity. I am aware of that some literature suggests that finite permutations are not enough to ensure equity among an infinite number of generations. The quality of the argumentation in this literature is often not so good. Do you have a reference that gives a convincing argument for why your notion of intergenerational equity is appropriate and/or desirable? I hope this does not sound like I am questioning whether your definition is consistent with the literature: I am only asking out of interest.