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.