important elections in the United States simply require the voter to make
a single selection, so in practice the Arrow’s theorem complications do
not arise (although, as will be seen in the next chapter, other complica-
tions do). However, the advantages of rank ordering that have been cited
(prioritizing and the avoidance of runoff elections) are sufficiently valua-
ble to persuade social scientists (a more pragmatic group than pure math-
ematicians) to continue to study voting methods using rank ordering.
The dead loser condition is probably the one that most frequently ap-
pears in versions of Arrow’s theorem—although Arrow himself thought
that the dead loser condition was the most pragmatically dispensable of
the five components. This raises the question of how we might mathe-
matically assess the value of a voting method—which is itself a subject
that is currently being pursued. Just as the first law of thermodynamics
compelled us to abandon the quest for free energy from the universe and
directed our search toward the maximization of efficiency, Arrow’s theo-
rem forces us to search for criteria by which to evaluate voting systems, as
there can be no perfect voting system.The Future of Arrow’s Theorem
Niels Bohr’s oft-quoted observation, “Prediction is difficult—especially of
the future,”^12 is applicable to developments in most scientific endeavors.
Even though it is impossible to predict what will happen, there are three
possible directions for future results that would raise eyebrows—and pos-
sibly win Nobel Prizes if the result were spectacular enough.
As has been observed, most of the results related to Arrow’s theorem
involve conditions quite similar to Arrow’s original ones. Finding an im-
possibility theorem with a significantly different set of conditions would
be extremely interesting, and is a direction which is quite probably being
pursued at the moment by social scientists wishing to make a name for
themselves. However, just as Arrow found that there was no social prefer-
ence ranking method satisfying all five conditions, future mathemati-
cians might discover that these conditions, or simple variants of them,
might be the only ones that yield an impossibility theorem. The result
that it is impossible to find a significantly different impossibility theorem
might be even more startling than Arrow’s original result.
Finally, one of the ways in which mathematics has always engendered
surprises is the variety of environments to which its results are applied.
Just as Einstein’s theory of relativity proved to be an unexpected and sig-
nificant application of differential geometry, there may be significant
applications of Arrow’s theorem (which is, at its core, a result in pure
Cracks in the Foundation 219