Arrow and Gibbard-Satterthwaite

Steve Eppley SEppley at
Sat Oct 4 15:25:46 PDT 1997

Markus wrote:
> Example 2:
>    Do election methods with additional ballotings
>    fail to meet IIAC? Does Approval voting fail to
>    meet IIAC?
>    Arrow doesn't say, that these methods necessarily
>    fail to meet IIAC, because these election methods
>    are no preferential social welfare functions.

It seems to me that Approval *is* a preference order method.
There are three possible votes on a candidate: Yes, No, and 
Abstain.   The Yes votes map to "first preference."  The No 
votes map to "third preference."  Abstentions map to "second 

Approval's preference orders aren't tallied pairwise, obviously.
But the same can be said of some other preference order methods 
like Borda and Instant Runoff.

Allowing only three preference positions violates Arrow's
"universal domain" axiom which asserts that all preference
orders must be allowed.  Practical ballot considerations may 
limit any preference order method, but probably not so severely 
as only three positions.  I speculate that a practical ballot 
might offer about 5 to 7 preference positions without the 
outcome being significantly affected:

           <-- more preferred     less preferred -->
    A               ( )  ( )  ( )  ( )  ( )
    B               (X)  ( )  ( )  ( )  ( )
    C               ( )  (X)  ( )  ( )  ( )
    ...             ( )  ( )  ( )  ( )  ( )

Someday when voting booths use a computerized voter interface
instead of fixed size paper, unlimited positions will become
practical, since the voter will be able to drag choices up and 
down in the sort order.

> I didn't want to anger Steve Eppley.
> I wanted to anger Mike Saari.

I wasn't angered.  :-)  It's been awhile since I wrote that 
message, but I don't recall disagreeing with Markus' points.

Good luck trying to irritate Saari; he seems quite serene. 
I'll try, though...  :-)    Some of his thinking is knuckle-
headed, such as his attempt to show that voters won't have a 
rational reason to strategize if his favorite voting method is 
used.  His example depended on a scenario where all voters but 
one had their votes conveniently cancel out, letting the one 
remaining voter vote with no benefit gained from strategizing.  
Such a scenario is implausible; there's no reason to expect all 
the other ballots to exactly tie.

---Steve     (Steve Eppley    seppley at

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