carnes@gargoyle.UUCP (Richard Carnes) (10/12/85)
Arrow's Theorem is one of the most striking results ever obtained by
an economist. The theorem originally appeared in Kenneth Arrow's
book *Social Choice and Individual Values*. There is a short and
apparently nontechnical book on the theorem by Alfred MacKay. Arrow
also discusses the theorem in a paper, "Values and Collective
Decision Making," which is reprinted in his collected papers. I will
send a copy of this paper to anyone who requests it. Arrow's Theorem
goes as follows:
The general problem is how to make a social choice on the basis of
individual values, i.e., preference orderings. First, a definition:
A CONSTITUTION is a rule which assigns to any set of individual
preference orderings a rule for making society's choices among
alternative social actions in any possible environment.
Next Arrow imposes four conditions on this constitution:
COLLECTIVE RATIONALITY: For any given set of orderings, the social
choice function is derivable from an ordering. This is what
economists usually mean by rationality in individuals, and Arrow is
applying the same condition to the society. This means that the
constitution must result in a consistent value system.
PARETO PRINCIPLE: If alternative X is preferred to alternative Y by
every single individual according to his ordering, then the social
ordering also ranks X above Y.
INDEPENDENCE OF IRRELEVANT ALTERNATIVES: The social choice made from
any environment depends only on the orderings of individuals with
respect to the alternatives in that environment. In other words,
unavailable alternatives have no bearing on the choice to be made.
NONDICTATORSHIP: There is no individual whose preferences are
automatically society's preferences independent of the preferences of
all other individuals.
Arrow's Theorem simply states that there can be no constitution
simultaneously satisfying all four conditions. The proof is not very
long or difficult, but I will refrain from typing it in.
An illustration is Condorcet's voting paradox. Consider a choice
among three social outcomes, say who is to be the next President, A,
B, or C. The population is divided into thirds that rank their
preferences as follows:
A B C -- first group
B C A -- second group
C A B -- third group
Thus there is a majority that prefers A to B, a majority that prefers
B to C, and a majority that prefers C to A. Arrow's Theorem shows
that there is no rule for choosing the President that satisfies the
four conditions.
--
Richard Carnes, ihnp4!gargoyle!carnes