Hare on Infinite Worlds: A Summary

In The Limits of Kindness (pp. 182-188, 2013), Caspar Hare considers some problems that arise if we are 'minimally considerate and benevolent' in a world that is infinite (i.e. contains infinitely many people). I will summarize these problems here.

Hare considers what would happen if, considering the possibility that the world is infinite, we were to try to have minimal good will towards all agents in the world. It seems plausible that this would require at least preferring a world that 'tightly pareto-dominates' another: in other words, it has the same population (i.e. everyone in one world has a counterpart in the other), and is at least as good for everyone in the world and strictly better for some people. But Hare argues in that if one prefers a tightly pareto-dominant world, then one is rationally required to prefer anonymously tightly pareto-dominated worlds. In Chapter 9 of the book, Hare defends the following:

Anonymous Benevolence

If decency requires of you that you be benevolent toward people of kind K (to prefer, other things being equal, K-pareto-dominant worlds), then decency and rationality together require of you that you be anonymously benevolent toward kind K-people (to prefer, other things being equal, anonymously K-pareto-dominant worlds). (p. 146, Hare, 2013)

The argument Hare gives for Anonymous Benevolence appeals to three principles: Minimal Benevolence, which says roughly that your being better off in a state of affairs is a reason to prefer it (p. 23), transitivity, which says 'if you prefer a to b, and you prefer b to c, then you prefer a to c' (p. 114, Hare, 2013), and Generalized Morphing. The first two principles are plausible, but it's worth outlining the third in more detail. Prior to his discussion of morphing, Hare - in Chapter 7 of the book - defends the following claim about essences:

Personal Essence is not Perfectly Fragile

All people could have been ever-so-slightly different along any natural dimension of qualitative sameness and difference. (p. 101, Hare, 2013)

In other words, an agent A remains either identical to or counter-part related to A* if there is only a very small qualitative difference between A and A* (see pp. 139-142, Hare, 2013 for a discussion of identity/counterpart theories). I take this claim to at least be plausible in most cases, but can discuss the principle in more detail elsewhere if this is useful.

Once we grant that personal essence is not perfectly fragile, we can introduce the concept of morphing. The idea here is basically that for any two non-counterpart agents A and B (I will focus here on counterparts rather than identities), there is some sequence of transitions wherein A is a counterpart of A1, A1 is a counterpart of A2, and so on, until we reach some An that is a counterpart of B. So the counterpart relation is not transitive.

Hare says that one world 'K-pareto-dominates' another iff all people of kind K exist in two worlds and no K-person is worse off in that world and at least one K-person is better off in that world. And one world anonymously K-pareto-dominates another if there exists a correspondence relation that pairs the K-people in two worlds, and the dominance relation holds between K-people and their pairs. Generalized Morphing says that if one world anonymously K-dominates another then there exists a morphing sequence such that each world in that sequence (non-anonymously) K-dominates its predecessor. This is because we can construct a sequence of worlds sufficiently similar that anonymity does not hold between them, given what we have said above about essences.

It is easy to see how Generalized Morphing will result in Anonymous Benevolence if we assume Minimal Benevolence and transitivity. Minimal benevolence guarantees that we will prefer each K-pareto-dominant world in the sequence, and so transitivity guarantees that we will prefer the anonymously K-pareto-dominant world to the anonymously K-pareto-dominated world. This kind of argument isn't entirely uncontroversial (e.g. it's similar to the sequential argument for the repugnant conclusion in population ethics) but in the interest of time let's assume that it works, and that Anonymous Benevolence is the result.

In Chapter 11 Hare considers what would happen if we tried to extend Anonymous Benevolence to cases where agents believe the world may be infinite. The problem is that, given our assumption about essences, infinite worlds anonymously K-pareto-dominate themselves. This is because if we have an infinite world where people have the names and utility of the integers, say, we can create a correspondence relation in finite worlds that will eventually pair them with someone of slightly greater utility (here I'm giving an overly simplistic account of the argument on p. 184). Here is the diagram that Hare uses to illustrate this application of morphing to infinite worlds:

p. 185, The Limits of Kindness, Hare (2013)

So we will end up either violating transitivity or saying that a world is better than itself. Notice that we can make finitely or infinitely many agents better off here. In the case above we make infinitely many agents better off, but - as Hare points out - we could make finitely many agents better off. Assume that positive integer agents all have the same amount of utility (+2, say), while negative integer agents have some lower amount of utility (+1, say). Then a correspondence relation that associates everyone with the integer above them will result in an infinite world that K-pareto-dominates itself but is better for only finitely many people.

It seems irrational for our preferences to be intransitive or reflexive, and yet the alternative seems to be giving up plausible principles like Minimal Benevolence or Generalized Morphing. Hare considers several responses to this worry. Let's suppose we don't want to say that our preferences should be irrational (intransitive or reflexive), or that we should simply not care about individuals who are made worse off in infinite worlds (giving up Minimal Benevolence).

Hare considers the view that we should care about individuals because we are uncertain about whether the world is infinite or not, and if the world is finite then it is good to make people better off, even if it makes no difference conditional on the world being infinite. I want to explore this in more detail in a future post, but I think that Hare is right to point out that this seems like too fragile a basis on which to ground good actions or good intentions, especially if we think that these merely need to accord with Minimal Benevolence. (Hare considers one additional solution on p. 188 that I won't consider here, both for the reasons that Hare gives and because the solution itself strikes me as ad hoc).

This concludes my summary of the discussion of infinite worlds in Hare's The Limits of Kindness. I thought it would be useful to summarize this because some of the issues related to agent identity across world will probably be relevant to a post I want to put up over the next few days that is a follow up to my post on Agential Betterness.

Average Utilitarianism in Infinite Worlds

There are different forms of utilitarian and non-utilitarian theories in finite population ethics that it's worth trying to extend to the infinite case. On such view is average utilitarianism. Roughly speaking, average utilitarianism is the view that we ought to increase the average wellbeing of a population, rather than its total wellbeing. Is it possible to apply such a theory to populations that are infinitely large?

One natural way of doing this is simply to associate the value of an infinite population with its average utility. So the value of a world containing infintiley-many people at utility level 2 is just 2. One obvious problem for this view, and for average utilitarian views more generally, is that they say that it isn't better to have infinitely-many people at utility level 2 than to have finitely-many people at utility level 2. In addition, adding happy people whose lives are good but below utility level 2 - e.g. those who are at utility level 1 - will make a world worse according to this view.

There are a few ways that the average utilitarian might go about avoiding these consequences for her view. For example, we could argue that we can not only say that it's worse for non-existent agents that they don't exist if - were they to have existed - they would have been happy, but we can also say that non-existent agents have utility 0 at the worlds in which they don't exist. This view isn't always popular in population ethics, for reasons that I can go into in another post. But it would essentially allow us to turn different population problems in ethics into identical population cases. For example, suppose we were deciding between the following three worlds:

	$Jill$	$Jack$	$Sam$
$w_1$	1	2	-
$w_2$	2	-	1
$w_3$	-	1	2

This is the case that famously makes problems for many person-affecting views, because such views will say that $w_3 \succ w_2$, $w_2 \succ w_1$, and $w_1 \succ w_3$, violating transitivity. If we could treat all agents who exist in at least one world in our option set as a part of the relevant population and set their utility to level zero at the worlds where they don't exist, then we could instead treat these as single population cases each containing Jill, Jack, and Sam, where "$-$" means that an agent does not exist in the relevant world:

	$Jill$	$Jack$	$Sam$
$w_1$	1	2	0
$w_2$	2	0	1
$w_3$	0	1	2

In both this case and the case above, average utilitarians will derive the same ranking (equality) over worlds $w_1$, $w_2$, and $w_3$. But in the first case the average utility of the worlds will be 1.5, while in the second case it will be 1. This doesn't make a difference to the betterness rankings in the case above, but it will make a difference to cases in which we bring fewer agents into existence. For example, consider the following two worlds:

	$Jill$	$Jack$	$Sam$
$w_1$	9	-	-
$w_2$	-	6	6

If we don't consider the utility of non-existing agents then the average utility of the first world is 9, while the average utility of the second world is 6. But treat non-existing agents as agents at utility 0, then the average utility of the first world is 3 while the average of the second world is 4.

Let's just take these two ways of getting out averages as our base, since we can always consider more methods later. How can we apply these views to infinite cases? One natural way to derive the average utility of agents in an infinite world is to look at the limit of the partial sum of the average utility of agents, if it exists. If agents come with no natural order then this requires absolute convergence. If they come with a natural order then we can appeal to that ordering when searching for the limit. We can express this way of calculating average utility in infinite worlds as follows, where $a(w)$ is the average utility of the agents at a world $w$ (in the interest of time I'm not being in any way careful here, so I'll probably need to reformulate these later): $$a\left( w_{i}\right) =lim \sum ^{\infty }_{n=1}\dfrac {u\left( p_{n}\right) }{n}$$

Or, if we are looking at the differences in average utility between identical agents or ordered agents at different worlds $w_i$ and $w_j$ (where $p^{w_i}_n$ is the nth agent in world $w_i$): $$a\left( w_{i}- w_{j}\right)=lim\sum ^{\infty }_{n=1}\dfrac {u\left( p_{n}^{w_i}\right) - u\left( p_n^{w_j}\right) }{n}$$

But what about cases where agents are non-identical and unordered? In such cases we could, instead of looking at the limit of the sequence, look at any possible limit of the sequence given all possible orderings. So if world $w_i$ contains infinitely many agents at utility 2 and infinitely many agents at utility 4, then the average utility at $w_i$ is the interval[2, 4]. One question for this view is how we should define functions like subtraction across intervals. For example, suppose that $a(w_i) = [2,4]$ and $a(w_j)=[3,5]$. What is $a(w_i)-a(w_j)$? If we are using standard interval arithmetic then the answer will be $[-3, 1]$, but it may not be plausible to use standard interval arithmetic in this case.

Average utilitarian theories can try to give a ranking of infinite worlds by saying that $w_i \succ w_j$ iff $a\left( w_{i}\right) > a\left( w_{j}\right)$, or that $w_i \succ w_j$ iff $a\left( w_{i}- w_{j}\right)$ is positive. The latter is more useful than the former in one respect since it can produce rankings between worlds where the average utility of agents in one or both worlds diverge, but the difference between the averages does not.

One problem with this view is that in some cases the limit of the partial sum of the utilities or the differences between the utilities will diverge. First, there are cases where the limit does not exist, as $u\left( w_{i}- w_{j}\right)$ diverges to positive or negative infinity. Second, there are cases where $u\left( w_{i}- w_{j}\right)$ oscillates between two values in the interval $[-\infty,+\infty]$.

In cases where it diverges to positive (negative) infinity we want to say that $w_i$ is better (worse) than $w_j$. This should be a condition that's easy enough to add to the theory.

Cases of oscillation are more difficult to deal with. We might be able to use alternative summation methods (e.g. Cesáro summation) to cause these sequences to converge in some cases, if the summation methods are consistent with the average utilitarian view. But it is unlikely that any summation method will rid us of all divergent cases of this sort.

Another method would be to use the midpoint between the limit inferior and the limit superior on the oscillating sequence in cases where these are not $-\infty$ and $\infty$ respectively, since in this case the midpoint will be indeterminate. If the limit inferior is $-\infty$ and the limit superior is not $\infty$ then the midpoint will be $-\infty$, if it is defined, and if the limit superior is $\infty$ and the limit superior is not $-\infty$, then the midpoint will be $\infty$, if it is defined. If the limit inferior and the limit superior are both finite, then the midpoint will exist and will be finite.

The problem with this midpoint view is that it seems to implicitly impose an order on the agents in the relevant worlds. Suppose that there are infinitely many x-agents and y-agents in worlds $w_1$ and infinitely many z-agents in $w_2$, and that their utilities are as follows:

	x-agents	y-agents	z-agents
$w_1$	2	4	-
$w_2$	-	-	3

Here $a(w_1)=[2,4]$ and $a(w_2)=3$. If we were to take the midpoint view then $a(w_1)=a(w_2)$ and so the two worlds are presumably just as good as one another. But this seems to assume some proportionality between x-agents, y-agents, and z-agents (namely a 1:1:2 proportionality) that does not exist in infintie worlds. Imagine, for example, what would happen if we discoverd that all of the z-agents in $w_2$ are in fact identical to a proper subset of the y-agents in $w_1$: i.e. $w_1$ is just $w_2$ plus infinitely many agents with utility 2 and finitely or infinitely many agents with utility 4. It seems like $w_2$ should not be deemed identical to $w_1$ in terms of average utility in this case, and yet this is what the midpoint view will say. Notice that this is also the result that we will get if we don't appeal to midpoints but define the utility of $w_1$ as the interval $[2,4]$ and then use standard interval arithmetic. In other words, if the average utilitarian wants to avoid problems like this one then he may not want to appeal to standard interval arithmetic.

This is one furthter problem for average utilitarianism in infinite worlds that I want to briefly mention here. The first is that finite additions of utility will be washed out on the view given above, because the average of 2, 2, 2, 2, ... and the average of 4, 2, 2, 2, ... will both be 2. This seems bad, because we presumably do make the world better off if we add a finite amount of utility to finitely-many individuals. One way of getting around this, however, may be to treat the difference in value of two worlds $v$ as the pair $\left( a(w_i - w_j), b(w_i - w_j) \right)$, where $a(w_i - w_j)$ is as defined above, and $b(w_i - w_j)= \sum(u\left( p_{n}^{w_i}\right) - u\left( p_n^{w_j}\right))$, i.e. the difference between the utilities of the agents of the two worlds. So $a(w_i) \neq a(w_j)$ iff $b(w_i - w_j)$ is $\pm \infty$ and $a(w_i) = a(w_j)$ iff $b(w_i - w_j)$ is finite. We can then say that $w_i \succcurlyeq w_j$ iff $a(w_i - w_j)>0$ or if $a(w_i - w_j)=0$ and $b(w_i - w_j)\geq 0$, i.e. we don't look at the absolute values except in cases where the averages are the same.

Anyway, of this is super scrappy and I'll need to fix it at some point in the future. But I just wanted to sketch what an average utilitarian view might look like in infinite worlds, and to point out that trying to formulate such a view shows that it's not clear how to extend this theory to infinite worlds.

Is Agential Betterness Sufficient?

Here I will discuss a principle very similar to Lauwers and Vallentyne's Weak Catching-Up, which I will call Agential Betterness. I want to be able to show that if Agential Betterness is true, then there is no further principle P such that (i) P delivers a betterness verdict in any double-infinity pairs (as defined below), and (ii) P is consistent with Agential Betterness. In other words, any principle for ordering worlds must either be silent on all double infinity pairs, or it must be inconsistent with Agential Betterness.

Although Lauwers and Vallentyne formulate Weak Catching-Up in terms of locations, I will only be concerned with cases in which the relevant locations are agents (we can return to generalizations of these results to locations other than agents in a later post):

Agential Betterness (AB): If (1) U and V contain the same agents, and (2) for all possible enumerations of agents, the lower limit, as T approaches infinity, of the sum of the values of U-V of agents 1 to T is greater than zero $($i.e., $lim_{T\rightarrow\infty} inf \sum_{t=1,..,T}(U_t - V_t) > 0)$, then U is strictly better than V. And if $lim_{T\rightarrow\infty} inf \sum_{t=1,..,T}(U_t - V_t) \geq 0$ then U $\succcurlyeq$ V).

Agential Betterness is restricted to cases in which all of the agents are the same between the worlds we are comparing. I will argue that this makes denials of this principle highly implausible, even if we would wish to reject a similar principle that applied to distinct populations.

Pairs (or sets) of worlds can have fully distinct populations, fully overlapping populations, or populations that partly overlap. I will call pairs of the first kind disjoint, pairs of the second kind fully intersecting (rather than 'identical', since these worlds may have distinct structures, even if they contain identical agents), and pairs of the third kind partially intersecting.

If U and V contain the same agents, Agential Betterness will fail to deliver a verdict iff U and V form a double-infinity pair. Lauwers and Vallentyne define double infinity pairs for fully intersecting worlds. But we can try to generalize the concept of double-infinity pairs to worlds that don't fully intersect. Lauwers and Vallentyne define a world U-V as follows:

"For two worlds, U and V, with the same locations, U+V (respectively, U-V) is a world with the same locations, with the value at each location equal to value at U at that location plus (respectively, minus) the value at V at that location" (p. 15)

Building on this, we can define U+V and U-V for all worlds, and not merely fully intersecting worlds, as follows: For two worlds, U and V, U+V (respectively, U-V) is the set of all possible worlds with the value at each shared location equal to value at U at that location plus (respectively, minus) the value at V at that shared location, and the value at each non-shared location equal to the value at U at that location plus (respectively, minus) the value at V at some non-shared location.

Given this, we can define double-infinity pairs for any pairs of worlds (and not just those that fully intersect) as follows:

Double-Infinity Pairs (def): For any worlds U and V, if (1) there is some world in {U-V} such that the sum of the non-negative values of that world is infinite, and (2) there is some world in {U-V} such that the sum of the negative values of that world is infinitely negative, then U and V are a double-infinity pair.

Now let us assume the following principle, which I take to be plausible:

Agent Permutation (AP): For any property p or relation r that we use to enumerate the agents of an infinite world U, and for any arbitrary possible enumeration e of U, there exists a world U' such that (i) U and U' are fully intersecting, and (ii) p/rwill enumerate U' by e

Agent Permutation essentially says that there is no property or relation that we can use to enumerate or order agents in an infinite world that will always pick out the same enumeration (or indeed block any possible enumeration) in a world containing exactly the same agents. Agent permutation holds if worlds can retain the same agents in a world while adjusting whether they have property P (or to what degree they have property P, if P is gradable) or bear relation R to the relevant relatum or relata.

There are some properties or relations that will violate agent permutation. For example, if we can appeal to haecceitistic properties then we could simply enumerate the agents of a world by associating each agent of the infinite world with an integer. But if the betterness relation must appeal to a qualitative property of agents like their spatio-temporal location, then it seems plausible that we will be able to find a world that alters the relevant qualititative properties of agents (note that if adjusting the property changes the identity of agents, then Agential Betterness will no longer apply). I think it will be useful to discuss this principle, and some proposed properties and relations, in more detail in a future post.

If enumerating U by enumeration e results in U', then we will say that U' is a permutation of U (such that U is also a permutation of itself). Notice that since U' does not differ from U in any respect other than the enumeration of its agents, agent identity is preserved between U and U'. In other words, we suppose that merely changing the order in which evaluate agents does not alter the identity of those agents. (Notice also that we are only considering countably infinite worlds at the moment.)

Finally, I will assume that the betterness ordering is transitive, defined as follows:

Transitivity (T): For any worlds U, V, W: if U $\succcurlyeq$ V and V $\succcurlyeq$ W, then U $\succcurlyeq$ W

The claims I want to consider in the next post are as follows:

Claim 1: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in fully intersecting double-infinity pairs, and (ii) P is consistent with Agential Betterness.

Claim 2: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in any double-infinity pairs, and (ii) P is consistent with Agential Betterness.

Claim 3: If Agent Permutation and Transitivity are true, then there is no principle P such that (i) P delivers a betterness verdict in disjoint or partly intersecting infinite worlds (including worlds that are not double infinities), and (ii) P is consistent with Agential Betterness.

We might need further assumptions, or to adjust the ones given above, but I want to think about each of these claims in turn and make adjustments along the way.

Showing that claim 1 is true would be enough to show that if AB is necessary then it's also sufficient, because AB delivers a verdict about the betterness of world pairs in all and only those world pairs that are fully intersecting non-double infinity pairs (i.e. it leaves no 'gaps' for additional principles in fully intersecting non-double infinity pairs, and so if no other principle delivers verdicts in fully intersecting double infinity pairs, then AB is sufficient).

I will return to these three claims in a future post.

Theorem 6 and Restricted Transfers

Either moral evaluands within worlds (of which people and spacetime regions are natural candidates) come with a morally relevant order, or they don't. If they don't then we need some rule for comparing infinite worlds that doesn't rely on there existing any such ordering. In the last post, I considered the Full Weak Catching-Up rule. This rule seems plausible, but it posits incomparability between certain infinite worlds, which Lauwers and Vallentyne call 'double infinity' pairs: pairs such that "(1) the sum of the non-negative values of U-V is infinite, and (2) the sum of the negative values of U-V is infinitely negative" (p. 32).

Before returning to the issue of incomparability in later posts, it's worth asking if we have good reasons to think that Full Weak Catching-Up is necessary and sufficient for (weak) world betterness. In their paper, Lauwers and Vallentyne prove the following theorem:

Theorem 6: There is only one ranking rule, for worlds with the same locations, that satisfies Transitivity, Sum, Loose Pareto, Zero Independence, and Restricted Transfers. It is the Full Weak Catching-Up rule.

Lauwers and Vallentyne show (Theorem 4) that Transitivity, Sum, Loose Pareto and Zero Independence [1] generate the same ranking rule as the conditional formulation of the Full Weak Catching-Up rule, and (Theorem 2) that this ranking rule generates incomparability if and only if a pair of worlds is a double infinity pair. Given this, to prove that the biconditional Full Weak Catching-Up rule is the only rule that satisfies these four principles plus Restricted Transfers (Theorem 6) they show that (i) the Full Weak Catching-Up rule satisfies Restricted Transfers, and (ii) Restricted Transfers plus the other conditions mentioned above entail that double infinity pairs are incomparable.[2]

Restricted Transfers (RT): If locations have no natural structure, then, for any three worlds, U, U*, and V, having the same locations, if (1) U is better than V, and (2) U* is obtainable from U by some (possibly infinite) number of restricted transfers, then U* is better than V.

A 'restricted transfer' is, according to Lauwers and Vallentyne, "(1) a transfer of a positive amount of value from a location with positive value to a location with negative value such that (2) after the transfer, the donor location still has non-negative value and the recipient location still has non-positive value." (p. 33). For example, suppose the people <$p_1$, $p_2$, $p_3$, $p_4$,...> in $w_1$ have utility: <$3, -2, 3, -2,...$>. We could perform restricted transfers to turn $w_1$ into <$2, -1, 2, -1,...$> or into <$1, 0, 1, 0,...$>. If we performed infinitely many restricted transfers then we could also turn world $w_1$ into <$2, 0, 2, 0,...$> (we would do this by transferring 1 from $p_1$ and $p_3$ to $p_2$, and 1 from $p_5$ and $p_7$ to $p_4$, and so on).[3] But we could not turn $w_1$ into <$1, 1, 1, 1,...$> as this would mean that at least some of the recipient locations have gone from non-positive to positive value. And we could not turn it into <$-1, 0, - 1, 0,...$> as this would mean that at least some of the donor locations have from from non-negative to negative value. We can only ever use restricted transfers to bring the values of the locations of a world closer to zero.

Lauwers and Vallentyne opt against giving a defense of the Restricted Transfers principle in their paper. However, there are a couple of issues I think it might be useful to discuss relating to Theorem 6 and Restricted Transfers. The first is whether Restricted Transfers is indeed plausible. And the second is to see if we can show that Full Weak Catching-Up (or some similar principle for comparing worlds whose locations lack a natural ordering) is entailed by different (possibly weaker) principles.

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[1]   These conditions, as given by Lauwers and Vallentyne, are as follows:
Transitivity:If a world U is at least as good as V, and V is at least as good as W, then U is at least as good as W.
Sum: If, for each of two worlds, the sum of the values at their locations exists and is finite, then the first world is at least as good as the second world if and only if its sum is at least as great.
Loose Pareto If two worlds U and V have the same locations and each location has at least as much goodness in U than it does in V, then $ is at least as good as V.
Zero Independence: If U, V, and W are worlds with the same locations, then U is at least as good as V if and only if the world U+W is at least as good as V+W.

[2]    See pp. 46-7 of the Lauwers and Vallentyne paper for this proof. The proof of Theorem 2 appeals to the axiom of choice to guarantee the existence of free ultrafilters. I will hopefully return to questions surrounding free (non-principal) ultrafilters in a future post on hyperreals in infinite ethics.

[3]    As Lauwers and Vallentyne point out, unrestricted transfers will not always preserve value rankings. For example, we can - via unrestricted transfers - turn <0, 1, 1, 1,...> into <1, 1, 1, 1,...>. The latter is better than the former since it Pareto dominates it, but the former cannot be better than itself. However, these cases do not arise if we only appeal to restricted transfers.

The Full Weak Catching-Up Rule

Agential Pareto says that if two worlds contain identical people and one world Pareto dominates the other, then that world is strictly better than the other. It seems plausible that satisfying Agential Pareto is sufficient for world betterness (i.e. if $w_1$ Pareto dominates $w_2$ then $w_1$ is strictly better than $w_2$). But it seems less plausible that satisfying Agential Pareto is necessary for world betterness. After all, surely I can make a world better by making one person finitely worse off and infinitely many people better off. But Agential Pareto would not say that this results in a better world.

In this paper Lauwers and Vallentyne discuss a stronger rule than Agential Pareto, which they call Full Weak Catching-Up (for our purposes, "locations" are people):

Full Weak Catching-Up (FWW): If $U$ and $V$ have the same locations, and locations have no natural structure, then $U$ is at least as good as $V$ if and only if, for all possible enumerations of locations, the lower limit, as $T$ approaches infinity, of the sum of the values of $U - V$ at locations $1$ to $T$ is at least as great as zero $($i.e., $lim_{T\rightarrow\infty} inf \sum_{t=1,..,T}(U_t - V_t) \geq 0)$.

This rule - unlike Agential Pareto - says that if two worlds $w_1$ and $w_2$ have identical populations, then $w_1 \succcurlyeq w_2$ only if the sum of the differences of utilities across agents in two worlds converges to some $n \geq 0$ (it is somewhat similar to the weak people criterion later discussed by Arntzenius on p. 55 of this paper).

Since Full Weak Catching-Up is both a necessary and sufficient condition for (weak) world betterness, if we endorse this rule then not only do worlds need to satisfy the rule in order to be weakly better than one another, but we also cannot place further restrictions on what it takes for one world to be (weakly) better than another. This is at least somewhat concerning, since there will be many cases in which Full Weak Catching-Up delivers no verdict. For example, consider the following two infinite worlds that contain exactly the same people ($p_1$, $p_2$, $p_3$,...):

	$p_1$	$p_2$	$p_3$	$p_4$	$p_5$	$\dotsb$
$w_1$	3	0	3	0	3	$\dotsb$
$w_2$	1	1	1	1	1	$\dotsb$

Since not all possible enumerations of these populations satisfy Full Weak Catching-Up (e.g. the enumeration 0, 0, 0, 0,... of $w_1$ and the enumeration 3, 3, 3, 3,... of $w_1$ compared with $w_2$), Full Weak Catching-Up says that $w_1$ is not weakly better than $w_2$ and $w_2$ is not weakly better than $w_1$. So if Full Weak Catching-Up is true then there are evaluative gaps in our theory: worlds like $w_1$ and $w_2$ seem to be incomparable in terms of moral betterness. How far this incomparability extends and whether it is wrong to deem worlds like $w_1$ and $w_2$ incomparable are topics for future posts.

Three Conflicting Principles in Infinite Worlds

When we try to compare infinitely-large populations of people with non-zero utility, principles that coincide in the finite case are sometimes not jointly satisfiable. For example, consider the following three highly plausible, restricted principles (I can expand on why each of them is plausible when I consider which we should reject)::

Agential Pareto: If the populations of $w_1$ and $w_2$ contain exactly the same people and (i) for each person $P_i$ in $w_1$ and $w_2$, $U_{w_1}(P_i) \succcurlyeq U_{w_2}(P_i)$, and (ii) there exists a person $P_j$ in $w_1$ and $w_2$ such that $U_{w_1}(P_j) \succ U_{w_2}(P_j)$, then $w_1$ is strictly better than $w_2$ ($w_1 \succ w_2$).

Agential Isomorphism: If the populations of worlds $w_1$ and $w_2$ are completely disjoint and there is a one-to-one mapping from $w_1$ to $w_2$ such that each person at $w_1$ has a corresponding person in $w_2$ that has at least as much value in $w_2$ as they do in $w_1$, then $w_1$ is not strictly better than $w_2$ ($w_1 \nsucc w_2$).

Transitivity: If $w_1 \succcurlyeq w_2$ and $w_2 \succcurlyeq w_3$ then $w_3 \nsucc w_1$

But if worlds can contain infinitely many people then we can construct a problem case for these three principles. Suppose that $x$'s and $y$'s are different groups of agents, and we are to that there are infinitely many of these agents (i.e. the worlds contain infinitely many $x_1$ agents, infinitely many $x_2$ agents and so on). Now consider the following three worlds:

	$x_0$	$x_1$	$x_2$	$y_0$	$y_1$
$w_1$	1	1	2	0	0
$w_2$	0	0	0	1	2
$w_3$	1	2	2	0	0

We can see that, by Agential Pareto, $w_3 \succ w_1$. But Agential Pareto is silent on the relations between $w_1$ and $w_2$ since they contain distinct populations.

But since the populations of $w_1$ and $w_2$ are distinct, and so are the populations of $w_2$ and $w_3$, Agential Isomorphism applies. And it is easy to see that $w_1$ and $w_2$ are isomorphic copies of one another, and that $w_2$ and $w_3$ are also isomorphic copies of one another. So $w_1$ is at least as good as $w_2$ by Agential Isomorphism. And $w_2$ is at least as good as $w_3$ by Agential Isomorphism. But now we have it that $w_1 \succcurlyeq w_2$, $w_2 \succcurlyeq w_3$ and $w_3 \succ w_1$, which violates Transitivity.

In the finite population cases, these three principles are jointly satisfiable. But in infinite population cases, any theory that attempts to aggregate over people - which come in no natural order - will violate one of these three principles in the kind of case given above. I'll try to expand on this problem in the next post.