Arrow and Gibbard-Satterthwaite
schulze at sol.physik.tu-berlin.de
schulze at sol.physik.tu-berlin.de
Fri Sep 12 06:24:43 PDT 1997
Dear Steve,
the intention of my last e-mail was not to criticize
an election method, but to criticize Arrow's theorem.
Arrow demonstrated, that there is no preferential
social welfare function, that meets Pareto,
Non-Dictatorship, and IIAC simultaneously.
On the other side, I demonstrated, that there is no
election method, that meets PMC and IIAC simultaneously.
An election method meets PMC, if & only if:
If there were only two candidates, then that candidate,
who is prefered by more voters, is elected.
What is the difference between Arrow's version and
my version of the impossibility theorem?
Example 1:
Some of the election methods, which were discussed in
the election methods list, fail to meet Pareto (e.g.
Smith//Condorcet[EM] with the subcycle rule,
Smith//Random).
Arrow doesn't say, that these methods necessarily
fail to meet IIAC, because these election methods
fail to meet Pareto.
On the other side, I say, that these methods
fail to meet IIAC, because they meet PMC.
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.
On the other side, I say, that these methods
fail to meet IIAC at least as long as they meet PMC.
Example 3:
Suppose, the following election method is used:
If there is a candidate, who is prefered to every
other candidate by every voter, then this
candidate is elected. Otherwise, the winner is
chosen by lot and each candidate gets as many lots
as first preferences.
Obviously, this election method meets IIAC, Pareto,
and Non-Dictatorship. Due to Arrow, this is not
possible.
The problem of Arrow's theorem is the fact, that
Arrow discusses only "social welfare functions".
And social welfare functions are always deterministic.
On the other side, this election methods fails to
meet PMC and thus there is no problem with my
definition of the impossibility theorem.
The intention of my last e-mail was to demonstrate,
that the incompatibility of PMC and IIAC is a problem
of every election method.
In other words: IIAC is not a problem only of preferential
election methods. IIAC is a problem of every election
method, that meets majority rule.
I didn't want to anger Steve Eppley.
I wanted to anger Mike Saari.
Markus Schulze
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