[EM] Arrow's Theorem.
Adam Haas Tarr
atarr at ecn.purdue.edu
Mon Jul 14 11:18:04 PDT 2003
Wow, Eric went to the source and got the answer. Good work.
So, Arrow's original approach to the theorem could be summed up like this:
1) monotonicity + IIA => Pareto Efficiency.
2) IIA + Pareto Efficiency => Dictatorship
And you could skip the first step if you like. Alex's interpretation
(which is a very reasonable way to look at the theorem, in my opinion)
goes:
1) "reasonable method" = Pareto efficiency and non-dictatorship.
2) IIA + "reasonable method" => impossible.
So, no "reasonable method" can satisfy Independence of Irrelevant
Alternatives. This is not surprising to me, for I don't consider this
criterion a reasonable one. I'll explain. In the following archived
message, I provided a simple geometric argument why there could be a
sincere A beats B beats C beats A cyclic preference in the electorate:
http://groups.yahoo.com/group/election-methods-list/message/9857
There's certainly some implicit assumptions in there about the way voters
pick candidates, but they seem to be reasonable ones. So, given those
arguments, we can imagine an electorate that would prefer A to B in a
two-way race, B to C in a two-way race, and C to A in a two-way race.
I'll assume that the number of voters in each faction are not exactly
equal.
Now, imagine for a moment that we are considering some election method for
this election. We would expect that the method must declare some winner,
otherwise it is fairly useless. So, without loss of generality, I'll
assume that the method declares A the winner.
Now, imagine that B wasn't running, so the election only has C and A in
it. C beats A in any reasonable election method (only two candidates, and
a majority prefer C to A). Now, imagine that B enters the race.
According to IIA, only B or C should win the race. But A wins the race,
as we already stated.
So, given only some very reasonable assumptions about the nature of voter
preferences and election methods, I have demonstrated why no reasonable
method can be expected to pass IIA. This is why I think Arrow's theorem,
while perhaps elegant, is of little value as a practical guide to picking
an election method.
-Adam
>At 10:20 AM -0400 7/14/03, Eric Gorr wrote:
[quoting Arrow]
>Both statements are correct. The "monotonicity" condition together
>with Independence of Irrelevant Alternatives, implies the Pareto
>condition, which is the sufficient condition used in the "Vanderbilt"
>version. Actually, the monotonicity condition is used in the first
>statement of the theorem (first edition of my book, SOCIAL CHOICE AND
>INDIVIDUAL VALUES, 1951), while I used the Pareto condition in the
>second edition (1963). If one looks at the proof of the theorem in
>the first edition, I showed that the monotonicity condition implied
>the Pareto condition and then, in effect, derived the theorem from
>the Pareto condition. The difference is, therefore, not very large.
>--
>So, it would appear that in all cases monotonicity is there even if
>it is not mentioned explicitly.
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