[EM] Arrow's Theorem.

Alex Small asmall at physics.ucsb.edu
Tue Jul 15 13:44:02 PDT 2003

> It seems to me that "c1=c2=...cn>cn+1=cn+2=...cz"  is exactly the
> definition  of approval, from the voters' view. c1,c2... cn all get 1
> vote and cn+1,  cn+2...cz all get zero.

Yes, that is how approval has each voter treat the candidates, but that is
not necessarily each voter's sincere preference.

To re-iterate, a preferential or ranked voting method does the following:

1)  Each voter is ALLOWED to rank the candidates in whatever transitive
order he or she deems appropriate.  If the preference is binary, so be it.

2)  The method uses the actual rankings reported by the voters to
determine a winner.  It uses no other informations (approval cut-offs,
cardinal utilities, a 5-4 Supreme Court ruling, or Robert Mugabe's whims)
to determine the winner, and the winner is uniquely determined from the
rankings submitted.

3)  Most people also impose the condition that the method is deterministic
(except perhaps when there's a tie or Katherine Harris is doing the
counting), i.e. no "random ballot" lottery.

At the risk of sounding like a curmudgeon, I'm genuinely surprised at how
tough it is to impart that Approval Voting is not equivalent to a ranked
method, except in the rare case when EVERY voter CHOOSES to indicate on
the ballot a binary preference order (that may not be a sincere preference
order, but the election method only knows what the voters tell it).

> Go to two ballots:
> 1. c1=c2=...cn>cn+1=cn+2=...cz
> 2: c1=c2=...cn=cn+9>cn+1=Cn+2=....cn+8=cn+10=... cz.
> Who is the winner when voter approves cn+9?  It is cn+9, and that voter
> is  Arrow's Dictator... If there are N voters, and M of the N played
> ballot 2, we  have cn+9 getting (N-M+M) approvals and c1 through cn
> getting N-M approvals,  while cn+1 through cz (excepting cn+9) all got 0
> approvals.
> Three ballots:
> 1. c1=c2=...cn>cn+1=cn+2=...cz
> 2: c1=c2=...cn,cn=cn+9>cn+1=Cn+2=....cn+8=cn+10=... cz.
> 3. cn+1=cn+2=...cz>c1=c2=...cn
> If there are exactly 3 voters cn+9 wins the approval election, 2-1.
> Extend to  4 voters:
> If two people select ballot 1, 1 selects ballot 2, and the fourth
> selects  balllot 3, there's a tie between cn+9 and c1...cn. If 1 person
> selects ballot 1  and 2 select ballot 2 and 1 selects ballot 3, then
> cn+9 wins 3-1 over everybody  else by approval. cn+9 also wins 3-1 if 1
> of the four voters chooses ballot  1, 1 chooses ballot 2, and 2 choose
> ballot 3.
> I don't pretend this is a rigourous proof, but it is clear in the
> 3-voter  example that one of the three voters holds dictatorial power
> over the outcome of  an approval election, so approval does not satisfy
> Arrow's criteria.

A voting method is dictatorial if there is a single voter who,
irrespective of how everybody else votes, always determines the outcome. 
Your examples depend on the election being close, so that any individual
voter can change the outcome by changing how he votes.  That isn't
dictatorship.  Dictatorship is when the favorite (let's call him George)
of one voter (let's call her Sandra) ALWAYS wins, regardless of how the
other people vote.

So, as I've kept stating:

1)  Approval is non-dictatorial.
2)  Approval is Pareto efficient (except perhaps in the case of a tie)
conclusions don't apply to approval.
4)  The main interesting conclusion of Arrow is that IIA cannot be
satisfied by any "reasonable" (pareto efficient and non-dictatorial)
ranked method when there are 3 or more candidates and at least some of the
voters have non-binary preferences (or at least indicate non-binary
preferences on their ballots).


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