Reversal Symmetry Criterion & Borda-Elimination
Steve Barney
barnes992001 at yahoo.com
Thu Nov 8 22:11:58 PST 2001
Rob LeGrand is correct. His example:
> 4:A>C>B
> 4:B>A>C
> 1:C>B>A
>
> and
>
> 4:B>C>A
> 4:C>A>B
> 1:A>B>C
does, indeed, prove that the Borda-Elimination method violates the reversal
symmetry criterion.
Name: Reversal Symmetry Criterion
Definition: If alternative X wins (excluding ties), and all rankings on all
ballots are reversed, then X must lose.
--"Electoral Methods: Standards and Criteria,"
http://www.fortunecity.com/meltingpot/harrow/124/criteria.html
Saari, too, has acknowledged his mistake. I should have caught this myself. All
runoff methods have problems with cyclic votes, such as those in Rob's example:
B>C>A
C>A>B
A>B>C.
Saari calls this a Condorcet triplet, where each candidate is in each position.
It only takes two of these voter types (such as the A>B>C and B>C>A types,
above) to create problems and disagreement among different voting methods.
According to Saari's analysis, ALL voting methods are susceptible to such
"Condorcet terms." In the case of the Borda Count method, it causes IIA
violations, and that is the source of the Borda-Elimination problem illustrated
by Rob's example. That is also the source of ALL the problems with the Borda
Count. All other methods have these problems, and more (again, according to
Saari's analysis).
Peace,
Steve Barney
PS: The plurality voting method violates this reversal symmetry criterion, too.
Again, Rob's example illustrates this:
> 4:A>C>B
> 4:B>A>C
> 1:C>B>A
Plurality Outcome: A~B>C
>
> and
>
> 4:B>C>A
> 4:C>A>B
> 1:A>B>C
Plurality Outcome: B~C>A
I see that the plurality method is not included in the webpage referenced
above, however. That seems like a glaring omission, to me.
=====
Steve Barney
See AR-NewsWI, a news service for Wisconsin animal advocates:
http://groups.yahoo.com/group/AR-NewsWI/
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