 Harsanyi’s second argument, the “aggregation theorem”, is about a social planner who, facing risky prospects, maximizes expected social welfare and wants to respect individual preferences about prospects. Harsanyi (1955) shows that these two conditions imply that social welfare must be a weighted sum of individual utilities, and concludes that this is another argument in favor of utilitarianism. Recent evaluation of this argument and its consequences may be found in Broome (1991), Weymark (1991). In particular, Broome uses the structure of this argument to conclude that social good must be computed as the sum of individual goods, although this does not preclude incorporating a good deal of inequality aversion in the measurement of individual good. Diamond (1967) has raised a famous objection against the idea that expected utility is a good criterion for the social planner. This criterion implies that if the social planner is indifferent between the distributions of utilities, for two individuals, (1,0) and (0,1), then he must also be indifferent between these two distributions and an equal probability of getting either distribution. This is paradoxical, this lottery being ex ante better since it gives equal prospects to individuals. Broome (1991) raises another puzzle. An even better lottery would yield either (0,0) or (1,1) with equal probability. It is better because ex ante it gives individuals the same prospects as the previous lottery, and it is more egalitarian ex post. The problem is that it seems quite hard to construct a social criterion which ranks these four alternatives as suggested here. Defining social welfare under uncertainty is still a matter of bafflement. See Deschamps and Gevers (1977), Ben Porath, Gilboa and Schmeidler (1997), Fleurbaey (2010). Things are even more difficult when probabilities are subjective and individual beliefs may differ. Harsanyi’s aggregation theorem then transforms into an impossibility theorem. On this, see in particular Hylland and Zeckhauser (1979), Mongin (1995), Bradley (2005), Mongin and Pivato (2016, 2020), Dietrich (2021).