While it may be intuitive to reduce population ethics to a single lottery, this is incorrect; instead, it can only be reduced to n repeated lotteries, where n is the number of people...
This post will argue that within the framework of hedonic utilitarianism, total utilitarianism should be preferenced over average utilitarianism. Preference utilitarianism will be left to future work. We will imagine collections of single experience people (SEPs) who only have a single experience that gains or loses them a certain amount of utility
Both average and total utilitarianism begin with an axiom that seem obviously true. For total utilitarianism this axiom is: "It is good for a SEP with positive utility to occur if it doesn't affect anything else". This seems to be one of the most basic assumptions that one could choose to start with - it's practically equivalent to "It is good when good things occur". However, if it is true, then average utilitarianism is false, as a positive, but low utility SEP may bring the average utility down. It also leads to the sadistic conclusion, that if a large number of SEPs involve negative utility, we should add a SEP who suffers less over adding no-one. Total utilitarianism does lead to the repugnant conclusion, but contrary to perceptions, near zero, but still positive utility is not a state of terrible suffering like most people imagine. Instead it is a state where life is good and worth living on the whole.
On the other hand, average utilitarianism starts from its own "obviously true" axiom, that we should maximise the average expected utility for each person independent of the total utility. We note that average utilitarianism depends on a statement about aggregations (expected utility), while total utilitarianism depends on a statement about an individual occurrence that doesn't interact with any other SEPs. Given the complexities with aggregating utility, we should be more inclined towards trusting the statement about individual occurrences, then the one about a complex aggregate. This is far from conclusive, but I still believe that this is a useful exercise.
So why is average utilitarianism flawed? The strongest argument for average utilitarianism is the aforementioned "obviously true" assumption that we should maximise expected utility. Accepting this assumption would reduce the situation as follows:
Original situation -> expected utility
Given that we already exist, it is natural for us really want the average expected utility to be high and for us to want to preference it over increasing the population seeing as not existing is not inherently negative. However, while not existing is not negative in the absolute sense, it is still negative in the relative sense due to opportunity cost. It is plausibly good for more happy people to exist, so reducing the situation as we did above discards important information without justification. Another way of stating the situation is as follows: While it may be intuitive to reduce population ethics to a single lottery, this is incorrect; instead, it can only be reduced to n repeated lotteries, where n is the number of people. This situations can be represented as followed:
Original situation -> (expected utility, number of SEPs)
Since this is a tuple, it doesn't provide an automatic ranking for situations, but instead needs to be subject to another transformation before this can occur. It is now clear that the first model assumed away the possible importance of the number of SEPs without justification and therefore assumed its conclusion. Since the strongest argument for average utilitarianism is invalid, the question is what other reasons are there for believing in average utilitarianism? As we have already noted, the repugnant conclusion is much less repugnant than it is generally perceived. This leaves us with very little in the way of logical reasons to believe in average utilitarianism. On the other hand, as already discussed, there are very good reasons for believing in total utilitarianism, or at least something much closer to total utilitarianism than average utilitarianism.
I made this argument using SEPs for simplicity, but there's no reason why the same result shouldn't also apply to complete people. I also believe that this line of argumentation has implications for the anthropic principle.
The following was originally towards the start of the article. I think this is still an interesting approach, but I'm not convinced that it has any benefits over noting that it seems absurd that creating SEPs with positive utility could be bad or criticising the sadistic conclusion. I also think that my attempted formalisation needs a bit more work to make it correct. I also think that imagining combining universes provides a very interesting method for thinking about this problem.
Let's begin by considering a relatively simple argument for total utilitarianism. If we have a group of SEPs who experience different amounts of utility, we can quantify how good or bad each group is in utilitarian terms by imagining how much negative or positive utility a single SEP would need to have in order to balance out the existence of the group and result in the existence of the new group being neither good nor bad. If we accept that groups can cancel out like this, then this pushes us towards an aggregate model of utilitarianism because, for example, doubling the number of SEPs in a group where all the members have positive utility, seems very helpful when we want this group to cancel out a SEP with negative utility. Once we accept that sheer weight of numbers can allow a group with small, but positive utility, to cancel out SEPs with arbitrarily large amounts of negative utility and that the SEP required to cancel out a group is a valid measure of how good a group is in utilitarian terms, we have pretty much proven the repugnant condition. If if the actual aggregate function doesn't end up being a total function, proving the repugnant condition would provide us with a total-like function and would also disarm the most severe objection to total utilitarianism. (There is also a very convincing argument where you argue it is better to have a million people with 99 utility, then 100 utility and then you keep repeating until you end up with a ridiculously large number of people with small utility, I'll add a link if I can find it).
Let's try to state our assumptions more clearly and see if they are justified. Firstly, for a SEP with any amount of negative utility is it the case that there will be some amount of SEPs with small positive utility who would lead to a neutral universe if no other SEPs existed? Firstly, I'll note that this happens in both average and aggregate utilitarianism. Secondly, this seems pretty much equivalent to the torture vs. dust specks problem. We can convert the large negative utility into dust specks, then let each SEP with a small positive utility cancel out a dust speck.
Secondly, is the ability to cancel out a person with negative utility a good metric to measure how good a situation is in utilitarian terms? Let's suppose we were able to show something a bit more general, that if universe1 is better than universe2 and universe3 is better than universe4 then universe1&3 is better than universe2&4 where universea^b has all the SEPs in universea and universeb. Furthermore, if universe1 is just as good as universe2 then universe1&c is just as good as universe2&c. If these axioms are true, then it will be perfectly valid to cancel out groups equivalent to empty universe, before comparing the remaining SEPs in order to determine whether one universe is better than another. Why would we believe this? Well, it seems rather strange to think that whether a SEP should occur or not depends on what is happening elsewhere in the universe given that one SEP experiencing utility does not affect how another SEP experiences utility. It seems bizarrely inconsistent that we might want one half of the universe to be what would be a worse universe (an empty universe) if it existed on it's own.