Bachelor of Science (Mathematics) City College, New York, 1938. Master of Arts (Mathematics) New York University, NYC, New York, USA, 1939. Doctor of Philosophy (Mathematics) University N. Carolina, 1948.
Instructor Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., USA, 1948-1949. Assistant Professor Mathematics, Pennsylvania State College,
1949-1952. Research Mathematician, Mental Health Research Institute, Institution, University Michigan, 1963.
Co-dir., Social Science Research Council, United Kingdom or United States of America Research Training Institute, Institution, Stanford University, 1964. Professor Mathematics, Antioch College, Yellow Springs, Ohio, since 1952. Editorial Board, Social Choice and Welfare.
I was attracted to economics by Arrow’s book Social Choice and Individual Values, with its interplay of social theory and logic. I corrected the hypothesis of Arrow’s General Possibility Theorem. Using a potentially powerful technique invented by Arrow in his proof of the case of exactly three alternatives, and ideas of my own, I extended the validity of the theorem to any larger set of alternatives.
Later, originally for pedagogical reasons, I replaced my ponderous barnacled proof by a clearer, more direct argument. I was especially interested in Arrow’s condition of independence of irrelevant alternatives, which gives the subject its form, and weakened it so that in general only a small minority of sets were assumed independent. This minority sufficed for Arrow’s Theorem.
Used this idea also in weakening Sen’s liberalism and attempted to resolve the Liberal Paradox for the case of two people. I studied enlarging the range of the social preference and, for semiorder and many weaker variants, proved that Arrow’s conclusion extends to these cases. For acyclic social preferences, I proved (with Deb) the existence of a vetoer if there are enough alternatives, and also the existence of a group veto in a partition of society.
My interest in social choice led me to consider proportional representation and variants mandated by actual constitutions. I characterised these axiomatically in terms of coalitional invariance and included the case of an infinite number of ‘political positions’.