[EM] UncAAO
Forest W Simmons
fsimmons at pcc.edu
Thu Mar 1 12:42:17 PST 2007
Jobst,
You've probably already figured this out, but here goes:
UncAAO fails IDPA to the same extent that Approval does, because it is
possible (however unlikely) for a Pareto Dominated alternative to get
as much or more approval than an alternative that dominates it.
But note that if Y' Pareto dominates Y, then Y' covers Y. Furthermore,
if Y' Pareto dominates Y, then the approval opposition of X over Y' is
no greater than the approval opposition of X over Y, unless Y is
(mistakenly) approved over Y' on some ballots.
So practically speaking, elimination of Pareto dominated alternatives
is extremely unlikely to have any effect, although UncAAO technically
fails the criterion.
Note that if approval is measured in relation to a virtual "approval
cutoff candidate," or if the approval order is automatically refined by
the rank order on each ballot to enforce a "sincere approval"
requirement, then the alternative Y could tie Y' but could not beat it.
If all such ties were broken by random ballot, then Y' would beat Y,
and that would make UncAAO Independent from Pareto Dominated
Alternatives.
If the approval winner A is a member of Smith, then A, f(A), f(f(A)),
etc. are covered only by other members of Smith, so in that case the
UncAAO winner is independent of non-Smith alternatives.
But if the approval winner A is not a member of Smith, then it is
possible (however unlikely) that the first member of the sequence A,
f(A), f(f(A)), ... that resides in Smith is not the highest approval
member of Smith, and that this sequence would lead to a different
UncAAO winner if the highest approval Smith member were taken as the
initial point.
So we see how the method can fail Independence from Non-Smith
Alternatives.
This doesn't bother me too much, because I think that the approval
winner should have some influence on the outcome, even when not a
member of Smith.
Is TACC monotone? It seems to me that the winner W could improve in
approval enough to overtake and surpass some W' in approval without
defeating W' pairwise, though W' covers W.
My Best,
Forest
Jobst Heitzig wrote:
>Dear Forest,
>
>you proposed UncAAO:
>> UncAAO stands for Uncovered, Approval, Approval Opposition. Here's
>> how it works:
>>
>> For each candidate X,
>>
>> if X is uncovered,
>>
>> then let f(X)=X,
>>
>> else let f(X) be the candidate against which X has the least approval
>> opposition, among those candidates that cover X.
>>
>> Start with the approval winner A and apply the function f repeatedly
>> until the output equals the input. This "fixed point" of f is the
>> method winner.
>>
>> ["Approval opposition" of X against Y is the number of ballots on
>> which X but not Y is approved.]
>
>It would be interesting to compare this to other monotonic and
>clone-free methods that always pick the winner from the uncovered
>options.
>
>For example "total approval chain climbing" (TACC), a method I proposed
>a long time ago: Construct a chain of options by starting with an empty
>chain, then processing the options in order of ascending approval
>score, and adding the option at hand to the chain iff that option
>pairwise beats all options already in the chain; the winner is the
>option added last to the chain.
>
>Like UncAAO, also TACC has many good properties but also a main
>disadvantage: Because we start with the least approved option, the
>result depends too much on "noise" options.
>
>It seems UncAAO is much more robust against adding noise candidates,
but
>can we prove that it fulfils related conditions like IPDA or ISDA etc.?
>Is the result even independent from options outside the Smith set?
>
>Yours, Jobst
>
>>
>> This method requires a tally of both pairwise approval and pairwise
>> ordinal information, but both are efficiently summable in N by N
>> matrices, where N is the number of candidates.
>>
>> This method (UncAAO) is monotone, clone free, and always picks from
>> the uncovered set, which is a subset of Smith.
>>
>> Zero info strategy is sincere.
>>
>> Even perfect info incentives for burial and betrayal are practically
>> nil.
>>
>> As near as I can tell, the only bad thing about the method is the
>> "tyranny of the majority" problem shared by most, if not all,
>> deterministic methods.
>>
>> Comments?
>>
>> Forest
>> ----
>> election-methods mailing list - see http://electorama.com/em for list
>> info
>
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