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.