Arrow and Gibbard-Satterthwaite
schulze at sol.physik.tu-berlin.de
schulze at sol.physik.tu-berlin.de
Sun Sep 14 09:07:27 PDT 1997
Definition ("Pareto"):
A method meets Pareto if & only if:
An alternative B must not be elected, if there
is an alternative A such that every voter
prefers A to B.
Definition ("Smith//Random"):
The winner is chosen randomly from the members
of the Smith set. Each member of the Smith set
is chosen with the same probability.
Smith//Random fails to meet Pareto.
Example:
40 voters vote ABCD.
35 voters vote CDAB.
25 voters vote DABC.
A:B=100:0
A:C=65:35
A:D=40:60
B:C=65:35
B:D=40:60
C:D=75:25
The Smith set consists of A, B, C, and D, because
A > B > C > D > A.
Thus, B is elected with a probability of 25%, although
every voter prefers A to B.
Thus: A method, that meets the Condorcet Criterion
or the Smith Criterion, doesn't necessarily meet Pareto.
Markus Schulze (schulze at sol.physik.tu-berlin.de)
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