Short answers, pulled from the story.
A social welfare function is a mathematical function that ranks a set of social states by their desirability, combining each person's preferences into a single judgment about which outcome is best for society as a whole. Abram Bergson introduced the term in a 1938 article. It can be viewed as formalizing Rousseau's idea of a general will.
Abram Bergson introduced the social welfare function in a 1938 article, with the stated intention of putting into precise form the value judgments required for the derivation of conditions of maximum economic welfare.
Arrow's impossibility theorem, demonstrated in Kenneth Arrow's 1963 book, shows that no ordinal social welfare function can satisfy the standard axiom of rational behavior called independence of irrelevant alternatives. This means it is impossible to construct a consistent, rational collective decision procedure based solely on ranked preferences.
The utilitarian social welfare function measures welfare as the total or average sum of all individual utilities, with no regard for how they are distributed. The Rawlsian max-min function, based on the philosophical work of John Rawls, measures social welfare solely on the welfare of the least well-off individual, maximizing the income of the poorest person regardless of outcomes for others.
Amartya Sen proposed a welfare function in 1973 that multiplies the average per capita income of a measured group by a factor derived from the Gini index, a relative measure of inequality. A more unequal distribution therefore reduces the welfare value even when average income remains constant.
Harsanyi's utilitarian theorem holds that if social choice must satisfy the axioms of rational choice under uncertainty, ruling out Dutch Books, the only coherent social welfare function is the utilitarian rule, where the weighting function equals the utility functions of each individual. Any non-utilitarian function will accept bets that every member of society unanimously opposes.