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.