But why did Arrow stipulate #1? If you remove this requirement, does the conclusion that "there is no perfect voting system" still follow, and is CR an example of a "perfect" system according to Arrow's remaining criteria?Date: Mon, 01 Mar 2004 12:37:08 +0100 From: Markus Schulze <[EMAIL PROTECTED]> ... Arrow proved that there is no single-winner election method with the following four properties:
1) It is a rank method (= a ranked-preference method). 2) It satisfies Pareto. 3) It is non-dictatorial. 4) It satisfies IIA.
All four properties are needed to get an incompatibility. For example, RandomDictatorship is a paretian rank method that satisfies IIA, RandomCandidate is a non-dictatorial rank method that satisfies IIA, Approval Voting is a paretian non-dictatorial method that satisfies IIA, my beatpath method is a paretian non-dictatorial rank method.
(By the way, shouldn't the criteria also include transitivity, or does that follow from the other criteria?)
...So the ideal of the "perfect voting system" is unattainable in the real world because people exaggerate and misrepresent their preferences (i.e., they lie). Nevertheless, if CR would satisfy some reasonably defined standard of ideality in the absence of strategical voting, then I would think "sincere CR", though unattainable, would provide a useful standard by which other systems (including strategical CR) can be judged.
Even though the presumption that the used single-winner
election method is a rank method is necessary to prove
Arrow's Theorem, this presumption is not necessary to prove
the Gibbard-Satterthwaite Theorem. The Gibbard-Satterthwaite
Theorem says that there is no paretian non-dictatorial
method that isn't vulnerable to strategical voting.
In my view, the typical kinds of arguments that people make supporting one type of voting system or another are inconclusive because the premises lack sufficient information to say which alternative gives the more reasonable result. For example, Condorcet and Approval give conflicting results (A and B, resp) in the following scenario:
35 (A > B) > C
30 B > (A > C)
25 (A > C) > B
10 (C > B) > A
(Approved candidates are parenthesized.) Both voting systems incur significant loss of information (e.g., with Condorcet the ranking "A > B > C" tells you nothing about whether B would be approved; whereas the Approval rating "(A, B) > C" tells you nothing about which of A or B is preferred). Combining preference and approval ballots, as above, provides more information, but the situation is still ambiguous. For example, what does the first ranking "(A>B)" mean? Maybe the voters believe the future of the free world and civilization depends on A winning. Or maybe they have no particular preference between A and B, but they choose A because they like his mustache. (Given that the first 35 voters approved both A and B, the second interpretation is more likely, in which case the Condorcet result hinges on whether the voters like A's mustache.)
The interpretive ambiguities illustrated above could be resolved by presuming some distribution of sincere CR ratings and asking which method is more consistent with sincere CR. Of course, you can never know what people's sincere CR ratings are (they lie), but you could consider a statistical ensemble of all possible sincere CR rating profiles and ask, for example, which method selects a winner with the higher aggregate CR rating on average.
Ken Johnson
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