[EM] Arrow's Theorem - The Return (again)

[Alex Small]

OK, I stand corrected on the assertion that Arrow's Theorem assumes strict
preferences.

However, this whole discussion began over the question "Does Approval
Voting satisfy Arrow's assumptions?"  Approval Voting certain satisfies
non-dictatorship and Pareto.  Does it satisfy the assumption that the
winner is uniquely determined from the voters' preference orders?  If so,
then it cannot satisfy IIAC.  If it doesn't satisfy that, then we must
look elsewhere to see if Approval satisfies IIAC.

Arrow assumes that, if we know each voter's ranking of the candidates we
can uniquely determine who will win (leave aside the question of ties). 
We don't need to know cardinal utilities, we don't need to know an
Approval cut-off, we don't need a 5-4 Supreme Court ruling, and we don't
need input from Chicago graveyards.  All we need is the set of voter
preferences and we know the winner.

If I say to you "My preference is Libertarian>Democrat>Repubican", can you
predict with certainty what my Approval ballot will be?  Of course not. 
Now, if you knew my cardinal utilities, maybe (just maybe) you could use
one of the methods that Forest is developing.  If you had polling data,
you could assume that I'll vote for my favorite of the top 2.  But you
don't have that.  You CAN assume that I'll approve the Libertarian but not
the Republican if you make the assumption that I am a rational person (my
wife might argue with that...).  But you have no idea if I'll approve the
Democrat.

So, if we knew how many people hold each preference order in a 3-candidate
race, unless everybody happened to have binary preferences (e.g.
Libertarian=Democrat>Republican) we would have no clue who the Approval
winner is in general.

(There are exceptions.  If 45% of the electorate put candidate A in last
place, 45% put candidate B in last place, and 60% put C in first place,
then the most that A and B can get is 55%, while C is guaranteed at least
60%.  But these are exceptions.  In general, we can't predict the outcome
of an Approval election purely from the set of individual preferences.)

Anyway, because the outcome of an Approval Voting election is not uniquely
determined from the set of individual preferences, Approval Voting does
not satisfy one of Arrow's assumptions.  Therefore, it is not governed by
Arrow's conclusions, and we have to look elsewhere to figure out if
Approval satisfies IIAC.



Here's a good argument for why Approval flunks IIAC:

Say that the electorate is as follows:

30 (A>B)>C
30 (C)>A>B
40 (B)>A>C

The parentheses indicate which candidates a particular group of voters
approved.  In this case, B wins with 70 votes.

Now give all of the voters a ballot again, but don't include candidate C
on it.  If the people who approved both A and B the first time around are
rational they will only approve A this time.  If the people who only
approved C the first time around are rational they will only approve A
this time.  A now wins 60-40.

A mathematician at Northwestern published a paper on this premise a few
years back.  He used more sophisticated assumptions about voter behavior,
but the final conclusion was that Approval Voting flunks IIAC.  I wasn't
impressed when I ran across it.  If we can use a single counter-example to
prove that a statement is false, then a long proof is unnecessary.  But I
suspect he's from the Saari school of voting theory (Saari used to teach
at Northwestern), and really wanted to drive home that Approval Voting
isn't perfect.  (I'm sure everybody here, Donald especially, was
completely shocked to learn that Approval Voting isn't perfect.  ;)



Alex