Why do we sum utilities, instead of aggregating in some other way?

Question

Utilitarianism, as I understand it, works roughly as follows:

• Every person has a utility function, which describes how strongly they prefer any given scenario, sometimes informally described as the "happiness" of a person.
• To evaluate the relative ethics of two possible sets of consequences, you compute the utility function of each (affected) person in each situation, sum the utilities, and compare the total utility in each situation.
• Whichever situation has the greater total utility is ethically preferable.

There are a number of variations on this, but most of them use some kind of utility-function-like-thing and sum it over the participants in much the same way as above. For example, you might use expected utility instead of just utility. To avoid wordiness, I will be using the term "happy" as an informal adjective for the utility function, in whatever form a particular system defines it (which may be very different from actual happiness).

Why do we sum the utility? If we do something else with the utility instead of summing it, we can get materially different ethical systems:

• We can take the minimum. In this case, the ethical system tries to ensure the least-happy person is as happy as possible. This may be more egalitarian than standard Utilitarianism, but could also fail to account for overall societal well-being by favoring universal mediocrity.
• We can take the maximum. In this case, the system tries to ensure the happiest person is as happy as possible. I can't really defend this as a "reasonable" form of ethics, but I have occasionally seen people make arguments like this.
• We can take the median. This makes the system much less egalitarian than standard Utilitarianism because it ignores the extremes almost entirely. But it might benefit the middle class.
• We can take the mean. This is basically just the sum with a correction for population size. It may have the advantage of disfavoring reproduction when living standards are inadequate to support children. But this can also be done, more crudely, by allowing the utility function to be negative.
• We could use any number of more complicated statistical aggregations, with varying effects on the system, or somehow combine multiple aggregations into a single value.

What is the philosophical argument for taking the sum, as opposed to some other form of statistical aggregation?

(For that matter, how do we even know that summing the utilities of different people is a legitimate operation? Maybe we can determine whether Alice or Bob is happier through some kind of magical brain scan, and perhaps we can even put this on some kind of relative scale, but "the combined utility of Alice and Bob" doesn't seem to have a physical referent under most definitions of utility. Before we can begin to discuss trying to maximize this number, we have to know that it means something.)

Answer 1

There is no objective measure of 'good', or even 'pleasure' and therefore no proper objective utility measure that is not really constructed to serve some other end. Summing measures is meant to be an oversimplified stereotype, but as a matter of course, you can shape utility functions any way you want, and happiness can be measured any way you want. So you can always adapt the utility function itself so that summation is the appropriate operation.

For instance, the equivalent of your min-max convention can be accomplished with summation, if you add a highly weighted factor for equality to the definition of happiness, assuming liberal guilt is real and inequality reduces the returns on one's good fortune instead of increasing it. Or you can use a very regressive curve that allows for huge increases in happiness near the low end with returns diminishing quickly, because this is probably realistic for many causes of happiness: a tiny bit more food when you are starving is immensely satisfying.

This kind of messing with the machinery kind of makes Utilitarianism and classical economics into formless metaphorical principles instead of prescriptive systems. But there is no objective way to say they are wrong, or that they are even departures from the 'real' Utilitarianism.