Interesting idea!

In light of the relativity of simultaneity, that whether A happens before or after B can depend on your reference frame, you might have to just choose a reference frame or somehow aggregate over multiple reference frames (and there may be no principled way to do so). If you just choose your own reference frame, your theory becomes agent-relative, and may lead to disagreement about what's right between people with the same moral and empirical beliefs, except for the fact they're choosing different reference frames.

Maybe the point was mostly illustrative, but I'd lean against using any kind of average (including mean, median, etc.) without special care for negative cases. If the average is negative, you can improve it by adding negative lives, as long as they're better than the average.

The question of how to aggregate over time may even have important consequences for population ethics paradoxes. You might be interested in reading Vanessa Kosoy's theory here in which she sums an individual's utility over time with an increasing penalty over life-span. Although I'm not clear on the justification for these choices, the consequences may be appealing to many: Vanessa, herself, emphasizes the consequences on evaluating astronomical waste and factory farming.

Hey Bob, good post. I've had the same thought (i.e. the unit of moral analysis is timelines, or probability distributions of timelines) with different formalism

The trolley problem gives you a choice between two timelines ($t,\overline{t}\in T$). Each timeline can be represented as the set containing all statements that are true within that timeline. This representation can neatly state whether something is true within a given timeline or not: “You pull the lever” $\in t$, and “You pull the lever” $\notin \overline{t}$. Timelines contain statements that are combined as well as statements that are atomized. For example, since “You pull the lever”, “The five live”, and “The one dies” are all elements of $t$, you can string these into a larger statement that is also in $t$: “You pull the lever, and the five live, and the one dies”. Therefore, each timeline contains a very large statement that uniquely identifies it within any finite subset of $T$. However, timelines won’t be our unit of analysis because the statements they contain have no subjective empirical uncertainty.

This uncertainty can be incorporated by using a probability distribution of timelines, which we’ll call a forecast ($f,\overline{f}\in F$). Though there is no uncertainty in the trolley problem, we could still represent it as a choice between two forecasts: $f$ guarantees $t$ (the pull-the-lever timeline) and $\overline{f}$ guarantees $\overline{t}$ (the no-action timeline). Since each timeline contains a statement that uniquely identifies it, each forecast can, like timelines, be represented as a set of statements. Each statement within a forecast is an empirical prediction. For example, $f$ would contain “The five live with a credence of 1”. So, the trolley problem reveals that you either morally prefer $f$ (denoted as $f\succ \overline{f}$), prefer $\overline{f}$ (denoted as $f\prec \overline{f}$), or you believe that both forecasts are morally equivalent (denoted as $f\sim \overline{f}$).

(In light/practice of advice I've read to just go ahead and comment without always trying to write something super substantive/eloquent, I'll say that) I'm definitely interested in this idea and in evaluating it further, especially since I'm not sure I really thought about this in an explicit way before (since I generally just think "average per each person/entity's aggregate [over time] vs. sum aggregate of all entities," without focusing that much on a distinction between an entity's aggregate over time and that same entity's average over time). Such an approach might have particular relevance under models that take a less unitary/consistent view of human consciousness. I'll have to leave this open and come back to it with a fresh/rested mind, but for now I think it's worth an upvote for at least making me recognize that I may not have considered a question like this before.