near unanimity is not an uncommon occurrence in dictatorships. In pre-
war Iraq, Saddam Hussein’s choices were reported to have received 99.96
percent of the vote; recently, Bashar al-Assad was reelected president of
Syria with 97.62 percent of the vote. Finally, the death of a loser will not
affect the outcome of the election, for the removal of the loser from the
ballot will not change the dictator’s ordering of the remaining candidates,
and so the ordering of those other candidates by the society also will not
change. So a dictatorship is an example of a “voting method” that satis-
fies the other four conditions we have been considering.
The unanimity requirement has come in for some mathematical scru-
tiny that has real-world antecedents. As mentioned previously, in an elec-
tion with multiple choices, there are frequently options to which a voter
is indifferent. One modification of rank ordering, preference intensity,
has been discussed previously. At the time of this writing, there are ten
announced Republican candidates for president. Senator John McCain,
former governor Mitt Romney, and former mayor Rudy Giuliani have at-
tracted the lion’s share of the attention. A voter may have decided how to
rank these three candidates, but has no strong feelings about any of the
other. Some variations of Arrow’s theorem replace the condition if all vot-
ers prefer A to B, the voting method prefers A to B by something along the
lines of if no voter prefers B to A, then the voting method shall not prefer B to
A. This is clearly a modification of the unanimity requirement, to allow
for the possibility that a voter may see no reason to prefer A to B, or vice
versa, but the modification is not one that most people would regard as a
significant change.
One way to dispense with the dilemma presented by transitivity is to
simply require that the voting method be able to choose between two al-
ternatives, and not worry about whether the method is transitive or not.
Consider once again the situation we encountered with the Condorcet
paradox: the majority prefers A to B and B to C, yet prefers C to A. If the
voters were ignorant of the results of the A versus B race and the B ver-
sus C race, they would not be troubled with the verdict that the majority
prefers C to A. Ignorance in this case is bliss, for the subject of the Con-
dorcet paradox never arises. The Condorcet paradox is more likely to
trouble social scientists than actual voters, and by simply requiring that
the voting method be able to choose between two alternatives, the transi-
tivity problem is eliminated. The head-to head method certainly accom-
plishes this.
The two components of Arrow’s theorem that are most frequently cited
as the source of the incompatibility of the five conditions are rank order-
ing and the dead loser condition. As we have noted, many of the most
218 How Math Explains the World