[EM] UncAAO

[Chris Benham]

Forest W Simmons wrote:

>A candidate X covers a 
>candidate Y if and only if X (pairwise) defeats both Y and each 
>candidate that Y defeats.
>
>So if X covers Y, then in a pairwise sense X dominates Y.
>
>Now for UncAAO:
>
>1. Write abbreviations for all of the candidate names on a big sheet of 
>butcher paper.
>
>2. For each candidate X, if X is covered by some candidate, then draw 
>an arrow from X to the candidate (among those that cover X)
>against which X has the least approval opposition.
>
>3. If X is not covered by any candidate, then do not draw any arrow 
>from X to another candidate.
>
>4. Once all of the arrows have been drawn, start at the candidate A 
>with the most approval, and follow the arrows until you reach (the name 
>of) an uncovered candidate.  This candidate is the winner.
>
>When there are only three candidates, UncAAO is the same as Smith 
>Approval.
>

Forest,
How is this supposed to be better than  ASM and DMC?

In April 2002 Adam Tarr discussed Condorcet//Approval (which he calls 
"ACC", Approval-Completed
Condorcet). Since his examples all have three candidates, it applies 
equally to  Smith//Approval and UncAOO.

http://lists.electorama.com/pipermail/election-methods-electorama.com/2002-April/008013.html

> I'm now going to compare ACC to margins and winning votes Condorcet
> methods, using the example that has become my signature example on this
> list.  The following are the sincere preferences of my example electorate:
>
> 49: Bush>Gore>Nader
> 12: Gore>Bush>Nader
> 12: Gore>Nader>Bush
> 27: Nader>Gore>Bush
>
> If everyone votes sincerely, then Gore is the Condorcet winner.  The
> problem arises when the Bush voters swap Nader and Gore on their ballots
> (in margins they can achieve the same effect by truncating, but I'll 
> ignore
> that for this analysis).  So the new "preferences" are
>
> 49: Bush>Nader>Gore
> 12: Gore>Bush>Nader
> 12: Gore>Nader>Bush
> 27: Nader>Gore>Bush
>
> In margins-based methods, the only way for Gore to still win the election
> is for the Nader voters to bury Nader behind Gore.  The stable 
> equilibrium
> ballots become:
>
> 49: Bush>Nader>Gore
> 12: Gore>Bush>Nader
> 39: Gore>Nader>Bush
>
> And this allows Gore to still carry the election.  This sort of 
> equilibrium
> is what Mike is talking about when he says that margins methods are
> "falsifying".
>
> In winning votes methods, the Nader camp can vote equal first-place
> rankings rather than swap Gore and Nader entirely.  The stable result is
> therefore:
>
> 49: Bush>Nader>Gore
> 12: Gore>Bush>Nader
> 12: Gore>Nader>Bush
> 27: Nader=Gore>Bush
>
> In ACC... we first have to define where the approval cutoffs on the 
> ballots
> are.  Since the approval tally is only used to break cyclic ties, clearly
> the Bush camp has no incentive to Approve of anyone except Bush.  I'm 
> going
> to make the assumption that since Gore and Bush are the apparent front
> runners in this race (the only two with a decent shot at election), every
> voter will approve one and not the other.  This is the logical approval
> cutoff to use, based on the approval strategy threads that have been
> circulating on the list of late.  So the ballots could look something 
> like
> this:  (>> denotes approval cutoff)
>
> 49: Bush>>Nader>Gore
> 12: Gore>>Bush>Nader
> 6: Gore>>Nader>Bush
> 6: Gore>Nader>>Bush
> 27: Nader>Gore>>Bush
>
> In this case, Gore wins the approval runoff 51-49-33.  So not only did 
> ACC
> avoid the need for defensive order-reversal like margins methods, but it
> avoided the need for defensive equal-ranking like winning votes
> methods.  This is a super result: totally strategy-free voting for the
> majority side.
>
> There is a dark side to this result, though.  Say that some of the
> Gore>Bush>Nader voters were extremely non-strategic and decided to 
> approve
> both Bush and Gore.  So the votes now look like:
>
> 49: Bush>>Nader>Gore
> 6: Gore>Bush>>Nader
> 6: Gore>>Bush>Nader
> 6: Gore>>Nader>Bush
> 6: Gore>Nader>>Bush
> 27: Nader>Gore>>Bush
>
> Now, Bush wins the approval runoff 55-51-33.  This is where ACC's 
> favorite
> betrayal scenario comes in.  Since Bush wins the approval vote, the only
> way the majority can guarantee a Gore win is to make Gore the initial
> Condorcet winner, which requires that the Nader camp vote Gore in 
> first place:
>
> 49: Bush>>Nader>Gore
> 6: Gore>Bush>>Nader
> 6: Gore>>Bush>Nader
> 6: Gore>>Nader>Bush
> 33: Gore>Nader>>Bush
>
> So this is more or less the same as the margins method equilibrium.
>
> In summary, if the voters are fairly logical in the placement of their
> approval cutoff, then ACC seems almost uniquely free of strategy
> considerations.  If the underlying approval votes do not back up the
> sincere Condorcet winner, however, then ACC becomes just as vulnerable to
> strategic manipulation as the margins methods are, if not more so.


Note that in his "dark side" example, ASM and DMC have no problem.

>Here's another classical example:
>
>49 C
>24 B>A
>27 A>B
>
>Under wv, this is not a Nash Equilibrium, because B can unilaterally 
>gain by truncating.
>
>But if the direct supporters of the CW strategically put their approval 
>cutoff just below A, then we end up with a Nash equilibrium, no matter 
>where the B faction puts its approval cutoff.
>
>49 C
>24 B>A
>27 A>>B
>
>As in wv, no defensive strategy is needed under zero info conditions.  
>But if you suspect that X is the CW, and you could live with X, then a 
>prudent move would be to approve X and above.
>

For what it's worth, this all applies at least as well to ASM and DMC. 
Of course some of the
sincere B>A preferrers have to at least truncate for A not to be alone 
in the Smith set.

When the ballot-style allows voters to rank among unapproved candidates 
ASM and DMC
are my co-equal favourites, and when it doesn't I prefer ASM.

http://wiki.electorama.com/wiki/Approval_Sorted_Margins

Chris Benham