[EM] Gibbard-Satterthwaite
Rob LeGrand
honky1998 at yahoo.com
Sun Feb 10 19:28:37 PST 2002
Mike wrote:
> Have you heard about the Gibbard-Satterthwaite theorem? I haven't
> personally seen it, but it has to do with the unavoidableness of
> strategy. I don't know what kinds of methods its application is
> limited to.
Peter Ordeshook's book Game Theory and Political Theory discusses and proves
the Gibbard-Satterthwaite theorem, which in my opinion is a much more important
result than Arrow's impossibility theorem. Informally, the G-S theorem says
that, given the voters' ordered preferences over at least three candidates, any
nonmanipulable social choice function that always yields one outcome is
equivalent to a dictatorship. (Note that finding a single voter able to
improve his outcome by voting insincerely given any single set of preferences
is enough to show that a method is manipulable.) Thomas Schwartz proved that
the theorem holds even if the social choice function isn't always decisive, as
long as the voters' preferences are strict. Ordeshook uses Arrow's result to
prove the G-S theorem, which applies to any social choice function that takes
ordered preferences as input. I believe the theorem also holds for Approval no
matter how one might define sincere Approval voting, as long as there were only
one way for a voter to vote sincerely. If any Approval vote that doesn't vote
a less-liked candidate over a more-liked candidate is called sincere, then of
course Approval would count as a "nonmanipulable" method.
=====
Rob LeGrand
honky98 at aggies.org
http://www.aggies.org/honky98/
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