EC11CH01_Maskin ARjats.cls July 15, 2019 14:54
1. INTRODUCTION
Kenneth Arrow is a giant among economists. In the latter half of the twentieth century, only Paul
Samuelson had a comparable effect on the economics profession.1 Arrow created modern social
choice theory, established most of the major results in general equilibrium theory, pioneered con-
ceptual tools for studying asymmetric information and risk, and laid foundations for endogenous
growth theory, among many other contributions to economics.
His papers are frequently abstract and technically difficult. However, the abstractions enable
readers to see the essentials of a complicated issue. Indeed, his work, although highly theoretical,
has had significant repercussions for much applied research (e.g., computable general equilibrium
and health care economics) and for many fields outside economics, including political science,
philosophy, mathematics, operations research, and ecology.
Arrow’s academic output was enormous (on the order of 300 research papers and 22 books),
and his work has been exposited many times before (see, for example, Shoven 2009). Thus, I am
highly selective in my choice of articles and books to discuss in this review; indeed, I concentrate
primarily on the work for which he was awarded the Nobel Memorial Prize: social choice and
general equilibrium. By presenting the major results in some detail, I hope to make up for in
depth what this review lacks in breadth.
I begin in Section 2 with a short biographical sketch. Section 3 then discusses social choice,
and Section 4 treats general equilibrium. I briefly mention Arrow’s most influential other work
in Section 5. I conclude in Section 6 with a discussion of his contributions beyond research. In
Sections 3 and 4, I first present the material nontechnically and then, in most cases, offer a more
formal treatment in the asterisked version of that section.2 Readers uninterested in technicalities,
however, can safely skip over the asterisked sections.
2. BIOGRAPHICAL SKETCH
Kenneth Joseph Arrow was born on August 23, 1921, in New York City. His mother and father
were Jewish emigrants from Romania who, although poor, prized education and learning. Accord-
ing to Ken’s sister Anita,3 his parents willingly cut back on meat to afford the 10-cent daily subway
fare when he was admitted to Townsend Harris High School, a magnet school in Queens. Growing
up during the Great Depression was a deeply formative experience for Ken. It fostered his interest
in social welfare and even led him to give socialism careful consideration (see, for example, Arrow
1978).
Arrow got his bachelor’s degree at the City College of New York (then considered the poor
man’s Ivy League) in 1940. The degree was in social science with a major in mathematics—
foreshadowing his later preoccupations. His original goal had been to become a high school
math teacher, but the queue for jobs was so long that he decided instead to go to graduate school
at Columbia University in statistics (housed in the mathematics department); he was thinking
of a career as a life insurance actuary. He received his MA in 1941 and planned to work under
Harold Hotelling on his PhD research. Needing a fellowship, he asked Hotelling for a letter of
1Milton Friedman was better known to the public than either Arrow or Samuelson, but his scholarly work did
not rival theirs for influence.
2In this division, I emulate the expositional device of Amartya Sen in his classic monograph Collective Choice
and Welfare (Sen 1970, 2017). 3She made these comments in a talk given at the Arrow Memorial Symposium, Stanford University, October
9, 2017. Economics seems to have been in the family DNA. Anita became an economics professor herself and
married economist Robert Summers, a brother of Paul Samuelson. One of Anita’s and Bob’s sons is economist recommendation. However, Hotelling had little influence in the math department and persuaded
Arrow to switch to economics (Hotelling’s primary affiliation), where arranging for financial
support would not be difficult. As Ken liked to say, he went into economics because he was
bought.
World War II interrupted Arrow’s doctoral studies. From 1942 to 1946, he was a weather
officer in the Army Air Corps, which led to his first published paper (Arrow 1949). Afterwards,
he returned to Columbia for a year. However, unable to generate a thesis topic that he was happy
with, he moved in 1947 to a research position at the Cowles Commission, a research institute
at the University of Chicago devoted to mathematical economics and econometrics. Finally, in
the summer of 1949 (spent at the RAND Corporation), he found the big question that he had
hoped for—and quickly developed his Impossibility Theorem in social choice theory. This work
ultimately became his Columbia dissertation in 1951 (although the faculty there at first doubted
that the subject matter was truly part of economics).
Arrow found Cowles and Chicago to be a highly stimulating intellectual environment. On the
personal side, he met and married Selma Schweitzer there, a marriage that was to last 67 years, until
her death in 2015. However, partly because of Milton Friedman’s arrival in Chicago (Friedman
was quite hostile toward Cowles) and partly because of Stanford University’s attractions, Ken and
Selma moved to Palo Alto in 1949. They stayed until 1968, when Ken accepted a professorship at
Harvard University. Yet the Arrow family (by that time including sons David and Andy) left their
hearts in California and returned to Stanford every summer; they moved back for good in 1979.
Ken formally retired in 1991 but remained active in research, teaching, and public service to the
end of his life. He died at the age of 95 on February 21, 2017.
Arrow’s work did not lack for recognition. I mention just a few of his honors: In 1957, he re-
ceived the John Bates Clark Medal, awarded to an outstanding American economist under 40 (at
the medal ceremony, George Stigler urged him to begin his acceptance speech by saying “Sym-
bols fail me”). In 1972, he shared the Nobel Memorial Prize in Economics with John Hicks for
their (separate) work in general equilibrium and welfare theory (Arrow was then 51 and remains
the youngest ever recipient of that prize). He was awarded the National Medal of Science in
2004.
3. SOCIAL CHOICE
As I mention in Section 1, Kenneth Arrow created the modern field of social choice theory, the
study of how society should make collective decisions on the basis of individuals’ preferences.4
There had been scattered contributions to social choice before Arrow, going back (at least) to
Jean-Charles Borda (1781) and the Marquis de Condorcet (1785). However, most earlier writers
had exclusively focused on elections and voting. Indeed, they usually examined the properties of
particular voting rules [I ignore in this section the large literature on utilitarianism—following
Jeremy Bentham (1789)—which I touch on below]. Arrow’s approach, by contrast, encompassed
not only all possible voting rules (with some qualifications discussed below), but also the issue of
aggregating individuals’ preferences or welfares more generally.
Arrow’s first paper in this field was “A Difficulty in the Concept of Social Welfare” (Arrow
1950), which he then expanded into the celebrated monograph Social Choice and Individual Values
(Arrow 1951a, 1963a, 2012). His formulation starts with two things: (a) a society, which is a group
of individuals, and (b) a set of social al ternatives from which society must choose.
The interpretation of this setup depends on the context. For example, imagine a town that is
considering whether to build a bridge across the local river. In this case, society comprises the
citizens of the town, and the social alternatives are simply the options to build the bridge and not
to build it. We can also think of a situation involving pure distribution. Suppose that there is a jug
of milk and a plate of cookies to be divided among a group of children. In this case, the children
are society and the different ways to allocate the milk and cookies among them are the alternatives.
As a third example, think of a committee that must elect a chairperson. In this case, society is the
committee and the social alternatives are the various candidates for chair.
Those are just a few interpretations of the Arrow setup, and there is clearly an unlimited num-
ber of other possibilities. An important feature of the formulation is its generality.
Presumably, each member of society has preferences over the social alternatives. That means
that the individual can rank the alternatives from best to worst. Thus, in the bridge example,
a citizen might prefer building the bridge to not building it. A social welfare function (SWF),
according to Arrow, is a rule for going from the citizens’ rankings to social preferences (i.e., a
social ranking). Thus, social preferences are a function of citizens’ preferences.5 In the bridge
setting, one possible SWF is majority rule, meaning that, if a majority of citizens prefer build-
ing the bridge to not building it, then building is socially preferred—the town should build the
bridge.
Although highly permissive in some respects, this way of formulating a SWF still excludes some
important possibilities. First, it rules out making use of the intensities of individuals’ preferences
(or other cardinal information). For example, it disallows a procedure in which each individual
assigns a numerical utility (or grade) to every alternative (say, on a scale from 1 to 5), and alter-
natives are then ordered according to the median of utilities (for a recent approach along these
lines, see Balinski & Laraki 2010). Arrow’s rationale for excluding cardinality—following Robbins
(1932)—is that such information cannot be reliably obtained empirically unless individuals trade
off alternatives against some other good like money, in which case the set of alternatives that
we started with does not fully describe the possibilities (one sort of cardinal information that
can be obtained empirically is data about individuals’ risk preferences; I discuss this possibility in
Section 3∗).
A second (and closely related) omission is that the formulation does not allow for interpersonal
comparisons (for formulations that do permit such comparisons, see Sen 2017, chapter A3∗). For
example, there is no way of expressing the possibility that individual 1 might gain more than in-
dividual 2 loses if alternative a is replaced by alternative b. Thus, Arrow’s setup excludes classical
utilitarianism à la Bentham, according to which a is socially preferred to b if the sum of individ-
uals’ utilities for a is greater than that for b. The formulation also rules out a comparison such as
“Individual 1 is worse off with alternative a than individual 2 is with alternative b.” Thus, Rawls’s
(1971) maximin criterion (in which a is socially preferred to b if the worst-off individual with al-
ternative a is better off than the worst-off individual with alternative b) is also off the table. Arrow
avoids interpersonal comparisons because, again, he argues that they lack an empirical basis (he
doubts that there are experiments that we could perform to test claims such as “This hurts me
more than it hurts you” or “My welfare is lower than yours.”)
Table 1 Citizens’ rankings of three alternatives
35% 33% 32%
B T N
T N B
N B T
Abbreviations: B, in favor of building a bridge; N, in favor of doing nothing; T, in favor of building a tunnel.
Finally, the requirement that social preferences constitute a complete ranking6 may seem to
attribute a degree of rationality to society that is questionable (see, in particular, Buchanan 1954).
Arrow’s reason for positing a social ranking, however, is purely pragmatic: The social ranking
specifies what society ought to choose when the feasibility of the various alternatives is not known
in advance. Specifically, society should choose the top-ranked alternative, a, if a is feasible; should
choose the next best alternative, b, if a is infeasible, and so on.
Yet requiring a social ranking is a potential problem for the best-known way of determining
social preferences, majority rule. I note above that majority rule works fine in the bridge example,
where there are only two possible choices. Imagine, however, that there are three alternatives:
building a bridge (B), building a tunnel (T), and doing nothing (N ). Suppose, for example, that
35% of the citizens in the town prefer B to T and T to N, 33% prefer T to N and N to B, and 32%
prefer N to B and B to T (these preferences are summarized in Table 1).
In this case, under majority rule, N is socially preferred to B because a majority (33% + 32%)
prefer N. Furthermore, T is socially preferred to N because a majority (35% + 33%) prefer T.
However, B is socially preferred to T because a majority (35% + 32%) prefer B. Clearly, majority
rule does not give rise to a well-defined social ranking in this case.
As far as we know, it was Condorcet who first noted the possibility of a Condorcet cycle, in
which majorities prefer N to B, T to N, and B to T (although he was himself a strong proponent of
majority rule). Condorcet cycles were Arrow’s starting point in his thinking about social choice (see
Arrow 2014). Interestingly, he was, at that time, unaware of Condorcet’s work but rediscovered
the above problem with majority rule for himself. It led him to wonder whether there is some
other reasonable way of determining social preferences that does succeed as a SWF.
By “reasonable,” Arrow first required that the SWF should always work. That is, it should
determine the social ranking no matter what preferences individuals happen to have. This is called
the Unrestricted Domain (UD) condition. It is the UD condition that majority rule violates.
Second, he insisted that, if all individuals prefer alternative a to alternative b, then society should
rank a above b. After all, it would be quite perverse for society to choose b when everyone thinks
that a is better. This is called the Pareto (P) condition.
Third, Arrow required that the social preference between alternatives a and b should depend
only individuals’ preferences between a and b, and not on their views about some third alternative
c. He argued that c is irrelevant to society’s choice between a and b, and so that choice should be
independent of c. This is called the Independence of Irrelevant Alternatives (IIA) condition.7
These three conditions—UD, P, and IIA—all seem quite natural and, on the face of it, not
terribly demanding. However, remarkably, Arrow showed that the only SWF satisfying all three
conditions is a dictatorship, a highly extreme sort of SWF in which there is a single individual—
the dictator—who always gets his way: If he prefers alternative a to alternative b, then society
prefers a to b, regardless of other individuals’ preferences. Thus, if we introduce an additional
requirement—a Nondictatorship (ND) condition, which demands that the SWF should not have
a dictator—then we obtain Arrow’s Impossibility Theorem: With three or more possible social
alternatives, there is no SWF that satisfies UD, P, IIA, and ND.
The Impossibility Theorem—Arrow’s most famous discovery—is truly a landmark in twentieth
century thought. As a crude measure of its influence, I note that Social Choice and Individual Values
has close to 19,000 Google Scholar citations (as of July 2018).
3∗. SOCIAL CHOICE—FORMALITIES
3.1∗. Basic Formulation
Let us consider a society consisting of n individuals (indexed i = 1, ... , n) and a set of social alter-
natives A. For each individual i, let i be a set of possible orderings8 of A for individual i, then a
SWF F is a mapping
F : 1 ×···×n → ,
where is also a set of orderings.
The Arrow conditions on SWFs are:
1. Unrestricted Domain (UD): The SWF must determine social preferences for all possible
preferences that individuals might have. Formally, for all i = 1, ... , n, i consists of all or-
derings of A.
2. Pareto property (P):9 If all individuals strictly prefer a to b, then a must be strictly socially
preferred. Formally, for all (-
1, ... , -
n ) ∈ 1 ×···×n and for all a, b ∈ A, if aib10 for all
i, then asb, where -
s= F(-
1, ... , -
n ).
3. Independence of Irrelevant Alternatives (IIA): Social preferences between a and b should
depend only on individuals’ preferences between a and b, and not on their preferences con-
cerning some third alternative. Formally, for all (-
1, ... , -
n ), (-
1, ... , -
n ) ∈ 1 ×···×n
and all a, b ∈ A, if, for all i, -
i ranks a and b the same way that -
i does, then -
s ranks a and
b the same way that -
s does, where -
s= F(-
1, ... , -
n ) and -
s
= F(-
1, ... , -
n ).
4. Nondictatorship (ND): There exists no individual who always gets his way in the sense that,
if he prefers a to b, then a must be socially preferred to b, regardless of others’ preferences.
End😇