the Nobel Prize in Economic Science in 1972, was that no transitive vot-
ing method can satisfy all three of the above conditions as long as there
are at least two voters considering at least three different alternatives.The Present State of Arrow’s Theorem
The noted evolutionary biologist Stephen Jay Gould once proposed a the-
ory he called “punctuated equilibrium,”^10 in which the evolution of spe-
cies might undergo dramatic changes in a short period of time, and then
settle down for an extended period of quiescence. While this theory has
yet to be demonstrated to the complete satisfaction of evolutionary biol-
ogists, it makes for an accurate description of progress in science and
mathematics. This is precisely what happened here. Arrow’s theorem,
which was undeniably a significant advance, has been followed by a long
period during which the result has been extended by relatively small
amounts. There have been important developments in related problems,
which will be discussed in the next chapter, but Arrow’s theorem itself is
essentially still the state of the art.
Because Arrow’s theorem is a mathematical result, it is interesting to
see how mathematicians work with it. The most obvious place to start is
to examine the five hypotheses that appear in our formulation of Arrow’s
theorem. These are: rank ordering of individual and group selections,
transitivity of the group selection algorithm, absence of a dictator, una-
nimity, and the requirement that the death of a loser should not change
the outcome of the election.
One of the most intriguing facts about Arrow’s theorem is that the five
components are independent of each other, but together they are incom-
patible. One of the first questions a mathematician will ask upon seeing
an interesting theorem is, are all the hypotheses required to prove the re-
sult? It has been shown that if any one of the components of Arrow’s
theorem is removed, the remaining four are compatible, in the sense that
it is possible to construct voting methods satisfying the four remaining
conditions.^11 For instance, if we allow the society to have a dictator (a voter
whose ballot is universally adopted), the four remaining conditions are
automatically satisfied.
Since any individual’s selection is transitive, and the group selection
process is simply to adopt the dictator’s ballot, the group selection process
will be transitive as well. The unanimity requirement will also be satis-
fied; if everyone agreed that Candidate A was preferable to Candidate B,
the dictator also felt this way, and since the dictator’s ballot is adopted, the
group selection process prefers Candidate A to Candidate B. In practice,
Cracks in the Foundation 217