Every time I have heard utilitarianism discussed, it is generally assumed that utility should be averaged. While this sounds reasonable, the reasoning isn't particularly strong.
Harsanyi's theorem is often used as a justification of utilitarianism and basically involves averaging of individual utilities.
Other aggregation procedures don't seem applicable. Geometric averaging of individual utilities would be an odd thing to do, for instance. I can't quite give you the reason why it would be odd, but I have an instinct that it would do weird things like massively favour the best off or the worst off, or something like that.
There's a lot of work on judgement aggregation that involves using procedures different from arithmetic averaging, but I don't know whether one could apply it to aggregating utilities...
If we start, as utilitarians do, from the idea that everyone is of equal moral importance, we do need a method of aggregating individual utilities that treats each person's utility outcome for a given action as having equal importance. The simplest mathematical function that achieves this is taking the arithmetical mean. While other procedures are available (for instance, taking the mean of the cubes of utility), the arithmetical mean as simplest has a default presumption in its favour.
Well, averaging isn't exactly implied by Harsanyi's theorem, as suggested in the accepted answer, but only a linear combination of utilities. As Risse points out:
Harsanyi’s theorem [...] implies the existence of certain coefficients for a given profile of utility functions, but for a different profile, we obtain different coefficients. However, a complete formulation of utilitarianism requires the multi-profile format. For it is utilitarian doctrine that each person count equally. An explicit formulation of this claim would stipulate that the aggregation be indifferent between two distributions that only differ in terms of the distribution of the overall utility across persons. But such a condition must compare and thus refer to several profiles at once and cannot be captured in the single-profile format. Therefore, as discussed here, Harsanyi’s theorem cannot make a complete case for utilitarian summation. So we should think of Harsanyi’s theo- rem as providing an argument for the summation method as such, while not implying anything about the weights given to the individuals. An argument for equality must then come from elsewhere.
(Emphasis in original.)
Nor was that paper the first to notice the issue; Amartya Sen is a [far] more illustrious critic; as quoted by Mongin:
the lack of a multi-profile equivalent was part of Sen's objections to Harsanyi's approach. Let us briefly review the reasons for which he claimed that the Impartial Observer and the Aggregation Theorems are not proper axiomatisations of utilitarianism.
What is needed is an axiomatisation that (1) permits independent formulation of individual utilities, and (2) which has the invariance property of having the set of ai [ = the weights given to the individuals in the utility sum] determined independently of the utility functions to be aggregated (and in particular having ai = 1) (1986, p 1124).
[...]
Sen's point [(2)] here appears to be that the standard Aggregation Theorem takes the (n + 1)-utilities of the individuals and the social observer as given, and therefore makes the weights a dependent on the chosen profile. That is, Harsanyi follows the traditional Bergson-Samuelson approach to the "social welfare function." We agree with Sen that this is inappropriate for an axiomatisation of utilitarianism. One major reason that this is so is just alluded to at the end of Sen's quotation and can be made explicit as follows. In contrast with affine or weighted sum rules, which have been rarely defended in the tradition of ethics and political philosophy, classical utilitarianism always involves uniform weights, be they equated to 1 - as in Bentham's sum rule - or to 1/n - as in the (also time-honoured) average utility rule, Harsanyi's own favourite. To go from the affine conclusion of the Aggregation Theorem to classical utilitarianism one must satisfy a symmetry requirement which cannot, strictly speaking, be stated in the language of single profiles.
Now, it is actually possible to derive a multi-profile version of Harsanyi's theorem and Mogin proceeds to derive it, essentially relying on the welfarism axioms. Essentially, in this context, a mixture-preserving (MP) function replaces the NVM utility. However while this is technically an interesting result (and the author also discusses how it skirts Arrow's impossibility theorem), the issue is that it's not really a philosophical or factual leap because:
MP representations of the individuals' VNM preferences are cardinal in the sense of being unique up to positive affine transforms but not necessarily cardinal in the more relevant sense of measuring the intensity of preferences. [... And] evidence is by and large unfavourable to the claim that MP functions adequately measure preference differences, as independently revealed to the experimenter.
