One of my former students at GMU Law e-mailed to ask me for a short verbal summary of Kenneth Arrow’s Impossibility Theorem. Arrow himself, on pages 24-25 of his 1974 book The Limits of Organization, provides a good summary:
The root facts here are the incommensurability and incomplete communicability of human wants and values. George Bernard Shaw long ago observed, “Do not do unto others as you would have them do unto you. They may have different tastes.” Social good, as in the determination of a just income distribution, is an abstraction of some kind from the individual values of the members of the society. But this abstraction can only be based on interpersonally observed behavior, as in market purchases or voting, not on the full range of an individual’s feelings. As is by now well known, attempts to form social judgments by aggregating individual expressed preferences always lead to the possibility of paradox.
Thus there cannot be a completely consistent meaning to collective rationality.