[EM] Arrow's Theorem.

Alex Small asmall at physics.ucsb.edu
Tue Jul 15 17:46:20 PDT 2003

Forest Simmons said:
> In the case of Approval, if the original Approval ballots are the sole
> source of information, the ballots with a name deleted may well approve
> all the remaining candidates (or approve none) which would give these
> ballots zero influence on the outcome, an unlikely choice if the voters
> themselves could revise the ballots.
> So Approval may formally pass the IIAC, but only at the expense of some
> voters' ballots turning out to be dummy ballots.
> The same goes for plurality.

This argument works for the "deleting candidates" version of IIAC.  But
the "adding candidates" version of IIAC is indisputably flunked by
plurality, because the only way to make the addition of candidates
meaningful is to let the voters modify their ballots.

As for Approval, it stands a better chance of passing the "adding
candidates" version of IIAC, or at least of not flunking it egregiously. 
Removing the only candidate that a voter approved renders his ballot
ineffective.  But saying "Your approval or disapproval of every other
candidate can't change, but you have a choice on whether you'll approve
the new guy as well" doesn't render any person's ballot worthless.

We could say this for Approval and the "deleting candidates" version of
IIAC when voters are given a chance to revise their ballots:  Under very
reasonable assumptions about voter behavior, Approval passes the "deleting
candidates" version provided that there is no voter who approved every
candidate except the deleted one.

In any case, I still stand by my contention that Approval's compliance or
non-compliance with IIAC cannot be determined from Arrow's Theorem, while
every "reasonable" ranked method can be shown to flunk IIAC as a
consequence of Arrow's Theorem.


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