If a social planner is indifferent between utility distributions (1,0) and (0,1), expected utility theory requires indifference between those distributions and an equal-probability lottery over them
Indifferent (indifference)(as used in decision theory)
Having no preference between two options—you don't care which one you get because they seem equally good to you.
Lottery (in probability/decision contexts)(as used in decision theory)
A random choice between different outcomes, where each outcome has a specific chance of happening—like a raffle where you might win one prize or another.
Social planner(as used in economics and ethics)
A person or decision-maker who is trying to figure out the best way to distribute resources or well-being among a group of people.
Utility distribution(as used in ethics and economics)
A way of dividing up satisfaction or well-being among different people—for example, giving person A a lot of happiness and person B none, or vice versa.
utility(Mill's qualification distinguishing his conception of utility from narrower hedonistic or preference-based interpretations.)
Utility in the largest sense, grounded on the permanent interests of man as a progressive being — not mere immediate pleasure or preference satisfaction.
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)