2. Current Analysis of Social Welfare Based on
Rankings. After a long but exiguous history, the gen-
eral theory of elections suddenly became a lively
subject of research beginning with the papers of Black
published in 1948 and 1949 and Arrow's 1951 mono-
graph. Since then there has been an uninterrupted
spate of discussion, which is still continuing. It is per-
haps not easy to see exactly why the interest has
changed so markedly. Neither Black nor Arrow were
aware at the time they first wrote of any of the preced-
ing literature, though it is hard to exclude the possi-
bility that some of this knowledge was in a vague sense
common property. Arrow has noted (Social Choice and
Individual Values, p. 93) that when he first hit upon
the paradox of voting, he felt sure that it was known,
though he was unable to recall any source.
Both Black and Arrow are economists, and some
historical tendencies in economics, in addition to the
general theory of marginal utility, played their role.
(1) A number of marginal utility theorists, such as
Marshall and Wicksteed, had tried to demonstrate that
their theories were, as Bentham had originally held,
applicable in fields wider than the purely economic.
(2) In particular, economists in the field of public
finance were forced to recognize that public expendi-
tures, which are plainly a form of economic activity,
were in principle regulated by voters. A voter who
was also a taxpayer could usefully be thought of as
making a choice between public and private goods;
the actual outcome would depend upon the voting
process. Problems of this type were studied by Knut
Wicksell in 1896, Erik Lindahl in 1919, and Howard
Bowen in 1943. These works tend in a general way
to a combined theory of political-economic choice. (3)
Other economists, particularly Harold Hotelling in
1929, and Joseph Schumpeter in his 1942 book Social-
ism, Capitalism, and Democracy, suggested models of
the political process analogous to that of the economic
system, with voters taking the place of consumers and
politicians that of entrepreneurs. (4) Marginal utility
theorists, e.g., Edgeworth in 1881, and the Austrians,
Carl Menger and Eugen von Böhm-Bawerk, about the
same time, had been concerned with problems of
bargaining, where one buyer meets one seller, rather
than the more usual competitive assumptions of many
buyers and sellers. The development of game theory by
von Neumann and Morgenstern was intended to meet
this problem, but the formulation took on such general
proportions that it suggested the possibility of a very
general theory of social behavior based on the founda-
tion of individual behavior as governed by utility func-
tions. (5) The ideas of Pareto and Bergson were now
widespread and raised demands for clarification.
Most of these topics could be interpreted both
descriptively and normatively, and some of this duality
has persisted in the current literature. There are two
main themes in the literature, associated with the
names of Black and Arrow, respectively: (1) demon-
stration that if the preference scales of individuals are
not arbitrary but satisfy certain hypotheses, then ma-
jority voting is transitive; (2) formulation of sets of
reasonable conditions for aggregating individual pref-
erences through a kind of generalized voting and
examining the consequences; if the set of conditions
is strong enough, there can be no system of voting
consistent with all of them.
Suppose that all the alternative decisions can be
imagined arrayed in a certain order in such a way that
each individual's preferences are single-peaked, i.e., of
any two alternatives to the left of the most preferred
(by an individual), he prefers the one nearest to it, and
similarly with two alternatives to the right. This would
be the case if the “Left-Right” ordering of political
parties were a valid empirical description. Black dem-
onstrated that if preferences are single-peaked then no
paradox of voting can arise. Put another way, the
relation, “alternative preferred by a majority to alter-
native B,” is an ordering and in particular is transitive.
Current work, particularly that of Amartya Sen and
Gordon Tullock, has developed generalizations of the
single-peaked preference condition in different direc-
tions. The conditions are too technical for brief pres-
entation, but, like single-peakedness, they imply cer-
tain types of similarity among the preference scales
of all individuals.
Arrow stated formally a set of apparently reasonable
criteria for social choice and demonstrated that they
were mutually inconsistent. The study arose as an
attempt to give operational content to Bergson's con-
cept of a social welfare function. The conditions on
the social decision procedure follow: (1) for any possi-
ble set of individual preference orderings, there should
be defined a social preference ordering (connected and
transitive) which governs social choices; (2) if every-
body prefers alternative A to alternative B, then society
must have the same preference (Parento-optimality);
(3) the social choice made from any set of available
alternatives should depend only on the orderings of
individuals with respect to those alternatives; (4) the
social decision procedure should not be dictatorial, in
the sense that there is one whose preferences prevail
regardless of the preferences of all others.
Condition (3) in effect restricts social decision pro-
cedures (or social welfare criteria) to generalized forms
of voting; only preferences among the available candi-
dates are used in deciding an election. The inconsist-
ency of these conditions is in fact a generalized form
of the paradox of voting; no system of voting, no matter
how complicated, can avoid a form of the paradox.
As in the original Condorcet case of simple majority
voting, all that is meant by the paradox is that it could
arise for certain sets of individual preference orderings.
If individual preference orderings were restricted to
a set for which the conditions of Black, Sen, or Tullock
hold, then majority voting and many other methods
would satisfy conditions (2-4).
The evaluation of the Arrow paradox has led to
considerable controversy, still persisting.
In one version of Arrow's system, condition (2) was
replaced by another which, loosely speaking, stated
that a change of individuals' preferences in favor of
a particular alternative A would raise its social prefer-
ence, if possible. The existence of the paradox is not
altered by this substitution. Recent work by Kenneth
May and later Yasusuke Murakami showed that this
condition, together with condition (3), had powerful
implications for the nature of the social decision proc-
ess. Specifically, it followed that the choice from any
pair of alternatives is made by a sequence of majority
votes, where outcomes of the vote at one step can enter
as a vote at a later step. In general, some individuals
may vote more than once, and some votes may be
prescribed in advance. If however it is assumed in
addition that all individuals should enter symmetrically
into the procedure and also that the voting rule should
be the same for all pairs of alternatives, then the only
possible voting rule is pairwise majority decision, i.e.,
the Condorcet criterion.
