Short answers, pulled from the story.
Abram Bergson published the article A Reformulation of Certain Aspects of Welfare Economics in February 1938. This paper introduced the term social welfare function to state precise value judgments required for deriving conditions of maximum economic welfare.
Kenneth Arrow published Social Choice and Individual Values in 1951 with a second edition appearing in 1963. His work proved no perfect voting system exists satisfying all rationality axioms simultaneously across diverse populations.
Amartya Sen published Collective Choice and Social Welfare in 1970 containing chapter three discussing social welfare functions extensively. His 1973 proposal multiplied average per capita income by one minus relative inequality measured through Gini index values.
Kenneth Arrow demonstrated in his 1963 book Social Choice and Individual Values that dropping real-valued requirements made rational social behavior impossible. His impossibility theorem showed no ordinal social welfare function could satisfy independence of irrelevant alternatives alongside other standard axioms.
The utilitarian Benthamite function measures total societal welfare as the sum of individual utilities across all members. In contrast John Rawls proposed max-min functions focusing exclusively on improving conditions for the least advantaged person.