of this hypothetical election, and is asked whether he can come up with
something better. After studying the example, he might observe that
some of the difficulty was engendered by the fact that A is a candidate
that polarizes the electorate: eighteen voters prefer A to any other, but the
remaining thirty-seven voters rank A dead last. The obvious thing to do
would be to devise an algorithm that would help alleviate the problem
of polarizing candidates. This can certainly be done, but in doing so our
election consultant would undoubtedly notice that no matter what algo-
rithm he devised, other situations that would generally be considered
undesirable might arise. In fact, that’s exactly what happened to Kenneth
Arrow when he started his investigations.
The Impossibility Theorem
Kenneth Arrow was born in New York in 1921. His career was surpris-
ingly similar to Condorcet’s: Arrow, like Condorcet, started his career as
a mathematician. Like Condorcet, he detoured into economics; and, like
Condorcet, it brought him fame and fortune. Like Condorcet, Arrow’s
work has triggered an intensive investigation of the problem he first
brought to attention. One important exception is that as of this writing,
Arrow is alive and well, living happily in Palo Alto, and has not yet been
forced to f lee for his life because he offended the Jacobins or some other
political hierarchy. Arrow attended City College in New York, where he
majored in mathematics. He continued his mathematical education at
Columbia University, obtaining a master’s degree, but became interested
in economics as the result of meeting Harold Hotelling,^7 a prominent
economist and statistician, and decided to obtain a doctorate in econom-
ics. World War II intervened, and Arrow served as a weather officer in the
Army Air Corps. Arrow spent his tour of duty doing research, eventually
publishing a paper on the optimal use of winds in f light planning.
After the war, Arrow resumed his graduate studies, but also worked for
the RAND Corporation (one of the first of the think tanks) in Santa
Monica, California. He became interested in the problem of constructing
methods of translating individual preference rankings into preference
rankings for the society. Arrow decided that he would concentrate on
those societal ranking methods that were transitive, because transitivity
is a property that is easy to express mathematically. This fact readily al-
lows deductions to be made.
Arrow described his progress toward his most famous result, which re-
sembled the efforts of our hypothetical election consultant. At first he
tried to devise an algorithm that would eliminate some of the difficulties
214 How Math Explains the World