[EM] There's nothing wrong with Average Rating.

[Ken Johnson]

>Date: Mon, 01 Mar 2004 12:37:08 +0100
>From: Markus Schulze <markus.schulze at alumni.tu-berlin.de>
>...
>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.
>
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?

(By the way, shouldn't the criteria also include transitivity, or does 
that follow from the other criteria?)

> ...
>
>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.
>  
>
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.

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