How Math Explains the World.pdf

loser simply results in his or her being removed from the ballot. Sup-

pose we have an election with the following collection of ballots.

Number of Ballots First Choice Second Choice Third Choice

40 A C B

35 C B A

25 B A C

This is a very interesting example, for as we shall see, no matter who
wins, the death of the “wrong” loser can alter the outcome of the elec-
tion.
Suppose our voting method results in A winning the election. Now sup-
pose that C dies. Once his name (or letter) is removed from the ballot, we
see that 35 25 60 people prefer B to A, so B would win.
Suppose instead that our voting method results in B winning, and A
dies. In this case, 40 35 75 people prefer C to B, so C would win.
Finally, suppose C wins, and B dies. In this scenario, 40 25  65 people
prefer A to C, again changing the outcome of the election.
No matter what voting method is used to determine the winner, the
death of the wrong loser changes the outcome of the election. This is
clearly very undesirable.
This condition also brings into focus another aspect of the election proc-
ess. It is a well-known political aphorism that the presence of a third can-
didate who has no real chance of winning can have a significant effect on
the outcome of an election, as did the presence of Ralph Nader in the
2000 election. This is simply the reverse of the “dead loser” condition
above; instead of a losing candidate dying, a candidate who cannot win
(and is destined to be a loser) enters an election and changes the outcome.
Obviously, we cannot be certain what would have happened in the 2000
election had Nader not been a candidate, but it is generally thought that
he drew the great majority of his votes from liberals who would have
voted for Al Gore rather than George Bush. Nader received 97,000 votes in
Florida, the pivotal state in the election. Bush eventually carried the state
by less than 1,000 votes, so Nader’s presence probably changed the out-
come of the election.
The previous example shows that no voting method can prevent the
death of a loser from changing the outcome in an election in which there
are three candidates and a hundred voters. But what would happen if
there were more candidates, or a different number of voters? What Arrow
showed in his famous impossibility theorem, for which he was awarded

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