[EM] [OT] Kenneth Arrow theory... anyone?

Alex Small asmall at physics.ucsb.edu
Thu Nov 20 17:30:05 PST 2003

That's the first time I've ever heard such a misinterpretation of Arrow's
Theorem.  It's amazing what things people will come up with.

Here's my more-or-less technical take on Arrow's Theorem.  Many voting
paradoxes seem to boil down to Condorcet's paradox.  I don't think I've
seen any proofs of Arrow's Theorem explicitly showing the connection to
the Condorcet paradox, but I've seen proofs of the Gibbard-Satterthwaite
Theorem explicitly connecting it to Condorcet's paradox.  I've also seen a
paper where Arrow's Theorem and the Gibbard-Satterthwaite Theorem are
proven side-by-side using almost (but not quite) identical methods and
assumptions.  Finally, one can devise methods that only violate Arrow's
conflicting criteria when Condorcet's paradox occurs.

So there seems to be some sort of connection between Arrow's conclusion (a
"perfect" voting system is impossible) and Condorcet's paradox. 
Condorcet's paradox is in some sense an inability to reach consensus on
the relative merits of 3 or more alternatives.  One could compare that
lack of consensus with a diverse society where the various viewpoints
don't all fall along a traditional one-dimensional ideological spectrum.

But the inability to achieve a majority consensus in a diverse society
doesn't mean that democracy is impossible.  OK, if you define democracy in
a very narrow sense (absolute rule of a single majority faction) then I
guess democracy is impossible in a diverse society.  But if you break down
the issues on an issue-by-issue basis, on any given question there will be
a majority opinion one way or another.  So a combination of referenda and
proportional representation in the legislature could produce a perfectly
functioning democracy.

(Somebody will probably bring up the problem of majority tyranny.  The
existence of a Condorcet cycle indicates that there is no monolithic
majority faction.  Every faction is a minority faction, and the odds are
good that most minority factions will be on the winning side of at least a
few of the issues.  So everybody, or almost everybody, gets something. 
Not a bad situation to be in.  Far better than America's winner-take-all
systems, where our main mechanism for preventing majority tyranny is the
arbitrary disproportionate representation of small states.)

> * Would that Nobel Prize be alive and ready to speak out and say that
> his view on the topic.

Several months back somebody on this list actually did contact Prof. Arrow
at Stanford.  We had a debate on this list regarding technical aspects of
Arrow's Theorem, and how to interpret it in regard to certain voting
systems.  I forget who contacted him, but Arrow replied.  I don't know
whether Arrow would feel like speaking out on this, but it can't hurt to
ask him.  The worst that can happen is he'll simply not reply.

Alex Small

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