[EM] Approval splitting pathology

[Chris Benham]

Ted,

> As a long time fan of Margin Sorted Approval (AKA Approval Sorted 
> Margins <https://electowiki.org/wiki/Approval_Sorted_Margins>), I've 
> been thinking about various ways to improve it.

To solve what "problem"?

>
> I started thinking about how one might implement an automatic approval 
> cutoff, designed to give each ballot its own strategically optimal cutoff.

Not possible with just ranking information. To revisit the classical 
example:

49 A        (presumably sincere, but maybe sincere is A>C)
24 B        (sincere may be B>C, that is what the C supporters believe)
27 C>B    (sincere, but could be C>B>>>A or C>>>B>A)

If we claim that we are trying to elect the sincere CW and therefore 
avoid giving the voters a strong truncation incentive then the approval 
cutoffs need to be explicit and manual so as to squash any post election 
complaints from losers.

>
> One reasonable strategy in Approval Voting generally is to determine, 
> for each ballot, the highest ranked candidate(s) who are in the Smith 
> Set, and then put the Approval Cutoff below that rank.

In the above example that would mean we interpret the C>B ballots as 
giving approval to both C and B, giving the win to B (as the sole winner 
by Double Defeat).  But then the C supporters could have a 
semi-reasonable complaint: "We didn't mean to really approve of B. We 
only expressed our weak B>A preference because we assumed (or were led 
to believe) that the B supporters would return the favour by voting 
B>C.  We object to them being rewarded for their treacherous defection."

But if the approval cutoffs are explicit any complaint like that gets a 
crushing rejoinder.  If they voted C>B|>A giving the win to B, it is 
"Your complaining about the election of a candidate you explicitly 
approved of? You could have prevented B from being elected by not 
approving B. You cannot reasonably expect the voting method to read 
minds and call C the winner.  C was Doubly Defeated  by A."

Or if they voted C|>B  causing A to win and their complaint is "But we 
prefer B to A and B pairwise beat A" then the rejoinder is "If you 
really wanted B to defeat A why didn't you approve of B? If you had, 
then B would have won."

With an explicit approval cutoff there is no clone issue because when 
the method has a ratings element candidates have to have the same 
ratings (here being on the same side of the approval cutoff) to qualify 
as clones.

So either we say to the voters "This is an Approval-ish high SU method, 
so we are not interested in your rankings among unapproved candidates" 
or we say "We are interested in whatever ranking you care to express and 
if there is a top cycle then your approval cutoffs will help determine 
the winner".

I am open to both but prefer the second because it allows greater voter 
expression.  In the first case we need a method that meets Double Defeat 
(Implicit) and in the second one that meets Double Defeat(Explicit).

My favourites are MSA(implicit) and MSA(explicit),  but the two versions 
of Smith//Approval also fill the bill and are not bad.

Chris Benham




On 22/10/2024 9:02 am, Ted Stern wrote:
> As a long time fan of Margin Sorted Approval (AKA Approval Sorted 
> Margins <https://electowiki.org/wiki/Approval_Sorted_Margins>), I've 
> been thinking about various ways to improve it.
>
> I started thinking about how one might implement an automatic approval 
> cutoff, designed to give each ballot its own strategically optimal cutoff.
>
> One reasonable strategy in Approval Voting generally is to determine, 
> for each ballot, the highest ranked candidate(s) who are in the Smith 
> Set, and then put the Approval Cutoff below that rank.
>
> But what happens if there is a Condorcet cycle, and the candidate who 
> would ordinarily win in Approval voting ("A") using that strategy is 
> cloned ("A1, A2, A3"). and there is a cycle in that clone set? With 
> the strategy.above, the approval cutoff will be placed just below the 
> top clone on the ballot. This would cause a split of approval for the 
> A# candidates, affecting not only Margin Sorted Approval but also 
> Approval Voting more generally.
>
> This special case seems highly unlikely, but it points to a cloning 
> resistance issue with any Approval augmented Condorcet method.
>
> Thoughts?
>
> ----
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