The Possibility-Impossibility Boundary in Social Choice

Abstract

Kenneth Arrow’s impossibility theorem (Arrow, 1950, 1951, 1963) was a landmark in the history of ideas. It created new branches of logic and social studies, and it spawned an enormous literature.² It inspired other impossibility theorems which are themselves landmarks — most notably, Sen (1970a) — and, by teaching us how to formulate questions about the existence of decision-making mechanisms with specified properties, may claim the incentive theorems of Hurwicz (1972), Gibbard (1973) and Satterthwaite (1975) as offspring. Arrow’s theorem is extraordinarily robust, as forty years of social choice literature testifies. But it is time to turn from the production of impossibility theorems to the search for favourable trade-offs. In our view, a favourable trade-off is one that takes us far from dictatorship (or some other disastrous implication of a set of otherwise appealing axioms) without deviating much from the spirit of the criteria that we want reflected in the decision rule. We have taken some steps in this direction, and we highlight our own work on trade-offs in this paper. We have discovered that the prospects for favourable trade-offs are dim, except when one weakens Independence of Irrelevant Alternatives (IIA). Even a very small relaxation of IIA opens the door to rules that are far from dictatorial.

The authors gratefully acknowledge the valuable comments of Georges Bordes, Graciela Chichilnisky, Rajat Deb, Serge Kolm and Kevin Roberts on the first draft of this paper. We also thank John Weymark for extensive comments on Campbell and Kelly (l994a) which is a brief early version of this survey. Campbell’s research was financed by the National Science Foundation, grants SES 9007953 and SES 9209039.