central to democratic elections: the votes of later voters are worth more
than the votes of earlier voters. So the question arises: Does there exist a
voting method that eliminates the possibility of insincere voting?
The Gibbard-Satterthwaite Theorem
Like the quest for the perfect system of translating individual preferences
into the preferences of the society, the quest for a voting method that
eliminates the possibility of insincere voting ends in failure (by this time,
you’re probably not too surprised that this would be the case). The Gib-
bard-Satterthwaite theorem^2 states that any voting method must satisfy at
least one of three conditions. The phrasing below is slightly different
from that in Arrow’s theorem, which was stated as “No voting method
exists which satisfies.. .”; it is a little easier to phrase the last condition of
the Gibbard-Satterthwaite theorem if we state it as “Every voting method
must satisfy one of the following conditions.” As a result, some of the
conditions look like negations of similar conditions in Arrow’s theorem.
- Some voter has dictatorial power. This is the negation of one of the
conditions in Arrow’s theorem. - Some candidate is unelectable. The Gibbard-Satterthwaite theorem
does not specify why the candidate is unelectable. It may possibly be
that he is extremely unpopular, or is running for an office for which
he is not eligible. Or, as has happened in American politics, he may
be dead. - Some voter with full knowledge of how the other voters will cast their
ballots can alter the outcome by switching his or her vote to ensure
the election of a different candidate.
The last condition is, of course, the critical one, as it is the essence of
insincere voting.
There are two important points to notice about the Gibbard-Satterthwaite
theorem. First, it is far more likely that insincere voting will inf luence the
outcome of an election if the number of ballots is relatively small, as it is
obviously unlikely that a voter casting a ballot for senator in California (or
even Wyoming) can inf luence the outcome of an election by changing his
or her vote. However, there are many elections in which relatively few bal-
lots are cast—chairmanship of committees and nominating conventions
are two such examples—and when one considers the possibility of indi-
vidual voters banding together as a bloc, the scope of the theorem widens
significantly.
226 How Math Explains the World