Arrow’s impossibility theorem is the most famous result in social choice theory, but its
significance for human affairs is still a matter for debate. Arrow considers the question of
whether it is possible to construct a social welfare function (henceforth SWF), which is a rule by
which the ordinal preferences of an arbitrary group of individuals can be aggregated to determine
a social ordering of the same alternatives, in accordance to the following superficially reasonable
axioms:
O. Ordering: the social ordering should have the same properties as the individual
orderings (e.g., completeness and transitivity) and should be determined only by
the individual orderings
U. Unrestricted domain: a social ordering should be determined for any logically
possible specifications of individual preferences
P. Pareto optimality: if everyone prefers x to y, then society should prefer x to y
D. Non-Dictatorship: there is no individual whose preferences always prevail over those
of all other individuals
I. Independence of irrelevant alternatives: the social ordering of x and y should depend
only on individual preferences between x and y, not preferences for any other
alternatives
Arrow’s theorem shows that this is impossible: no SWF can simultaneously satisfy all of these
axioms. The proof (following Vickrey) proceeds in several ingenious steps. First, the notion of
a decisive set is introduced. Define a set of individuals to be decisive for one alternative x over
another alternative y if, whenever they all prefer x over y, society does too, when all other
individuals have the oppositive preferences. Then:
(i)
Axioms O, U, I, and P imply that a set of individuals who are decisive for x over y are
also decisive for all other pairs of alternatives, as follows:
Let set D be decisive for x over y. Suppose everyone in set D has x > y > u while
everyone else has y > u > x. Then x > y must prevail, by the decisiveness of D.
Meanwhile everyone agrees y > u, so y > u must prevail by the Pareto rule, whence x > u
prevails by transitivity—even though only members of D have x > u as individuals!
Hence D is also decisive for x over u. Similarly:
z > x > u in D, u > z > x elsewhere D is decisive for z over u
z > u > w in D, u > w > z elsewhere D is decisive for z over w
(ii)
There is always at least one decisive set, namely the set of all individuals.
(iii)
Axioms O and U imply that any decisive set can be decomposed into two proper subsets,
at least one of which is itself decisive, which eventually leads down to a decisive set of
size 1—namely a dictator—in violation of axiom D, as follows:
Let D have proper subsets A and B, and let C be everyone else, with:
A: x > y > u,
B: y > u > x,
C: u > x > y
Since A∪B is decisive, y > u must prevail. If also x < y prevails, this must mean B is
decisive for x over y. But if x > y prevails, then x > u must prevail by transitivity, in
which case A is decisive. Gotcha!
In the 50 years since Arrow first proved this result, various authors have pointed out that it isn’t
as surprising or as dismal as it might have appeared at first glance. Objections can be raised
against most of the axioms, separately or in combination with each other, and against the whole
enterprise of searching for a universal social welfare function. First, consider axiom O, which
entails completeness and transitivity of both individual and social orderings. We have already
seen that completeness is an objectionable axiom even when applied to the preferences of an
individual, and its normative status is even more dubious when applied to a group of individuals
who are not of the same mind (unless they have had the opportunity to arbitrage-out their
differences of opinion—but that is another story!). In any case, do we really need a complete
social ordering of all the alternatives, or would it suffice to merely determine a “best” alternative
or even a “good” alternative for the problem at hand?
Would it be acceptable for society to occasionally be undecided, leaving some choices to be made by arbitrary or accidental tie-
breaking rules (hanging chad, etc.)?
The status of transitivity as a normative principle of
rationality for individuals has been questioned by Peter Fishburn and Robert Sudgen, among
others, and their arguments are even more compelling when applied to groups: if majority voting
sometimes leads to intransitive cycles in pairwise comparisons, so what?
Next, consider axiom
U. Why should a SWF be required to operate on completely arbitrary individual preferences, no
matter how perverse? Social norms, institutions, and evolutionary psychology may impose
constraints or symmetries on individual preferences that could facilitate preference aggregation
in some settings. (Arrow observed that if preferences are “single peaked,” a condition that
Duncan Black had used earlier to rationalize majority voting, it is possible to aggregate them in a
way that satisfies all the other axioms.) More generally, why should we let our social choices in this
world be governed by considerations of what might have happened in some weirdly different
hypothetical world?
Axiom I, despite its seductive and value-laden title, has often been
criticized for prohibiting the use of any data concerning intensities of preference between
alternatives, which might otherwise provide a basis for making rational tradeoffs between the
interests of different individuals. This axiom rules out otherwise-sensible preference aggregation
methods based on scoring systems (e.g., point totals or weighted voting) or measures of cardinal
utility (e.g., Harsanyi’s theorem). It not only requires the social ordering to be determined from
data on individual preferences: it requires the social ordering to be determined from low quality
data on individual preferences. Maybe we should not be surprised that this turns out to be
impossible. Even axiom D is not as uncontroversial as it might first appear: it is easy to
imagine situations in which one individual perhaps ought to be given dictatorial discretion over
some pairs of alternatives which affect her much more than they affect anyone else. Finally,
there is the question of how the axioms interact with each other. Each leverages the others, and
the key steps in the proof of the theorem use a combination of two or more axioms to produce an
extreme and surprising result—e.g., someone who has dictatorial discretion over any one pair of
alternatives must have dictatorial discretion over all alternatives. In his critique of Arrow’s
theorem, I.M.D. Little states: “The conclusion, to my mind, is that it is foolish to accept or
reject a set of ethical axioms one at a time. One must know the consequences before one can say
whether one finds the set acceptable—which sets a limit to the usefulness of deductive
techniques in ethics or in welfare economics.”
