How Math Explains the World.pdf

encountered by existing methods, but after each proposed algorithm

eliminated one problem but introduced another, he began to consider the

question of whether it was impossible to achieve the desired result.

I started out with some examples. I had already discovered that

these led to some problems. The next thing that was reasonable

was to write down a condition that I could outlaw. Then I con-

structed another example, another method that seemed to meet

that problem, and something else didn’t seem very right about it.

I found I was having difficulty satisfying all of these properties

that I thought were desirable, and it occurred to me that they

couldn’t be satisfied.

After having formulated three or four conditions of this kind, I

kept on experimenting. And lo and behold, no matter what I did,

there was nothing that would satisfy these axioms. So after a few

days of this, I began to get the idea that maybe there was another

kind of theorem here, namely, that there was no voting method

that would satisfy all the conditions that I regarded as rational and

reasonable. It was at this point that I set out to prove it. And it ac-

tually turned out to be a matter of only a few days’ work.^8

What were the conditions Arrow had discovered that could not be simul-
taneously satisfied by any voting method? Arrow’s original formulation is
somewhat technical;^9 here is a slightly weaker version of Arrow’s condi-
tions that is a little more natural than the one that appears in his disserta-
tion.

1. No voter should have dictatorial powers. The first condition is some-
   thing we would certainly want a democracy to have. In other words,
   when any one individual casts a ballot, the rest of the voters can al-
   ways vote in such a way that the voting method overrides that indi-
   vidual’s preferences.


2. If every voter prefers candidate A to candidate B, then the voting method
   must prefer candidate A to candidate B. The second condition is una-
   nimity. This also seems like an obvious and natural condition for a
   reasonable voting method to satisfy: if everybody loves it, the society
   should do it.


3. The death of a loser should not change the outcome of the election. At first
   glance, this condition seems almost unnecessary. We all accept the
   fact that the death of a winner must necessarily alter the outcome of
   an election, but how can the death of a loser alter the outcome of an
   election? To see how this can happen, we’ll assume that the death of a

Cracks in the Foundation 215