[EM] Approval ballots. Two to elect. Best method? (Andy Jennings)

[Forest Simmons]

This query has lead me to some interesting ideas:

Approval Ballots.  Two to elect:

For each pair of candidates {X1, X2}, let p1 and p2 be the respective
probabilities that X1 or X2 would be selected by a random approval ballot
drawing (restricted to our pair of candidates), and let p0 be the
probability that a random ballot would approve neither X1 nor X2.  Elect
the pair with the greatest value of min(p1, p2) - p0.

This actually gives two methods, since there are two natural ways of
selecting a candidate by random approval ballots.

The first way is to select ballots at random until the approval set for one
of them has non-empty intersection with the set from which we are to select
a winner.  The names of the candidates are drawn randomly from a hat.  The
first name drawn of a candidate in the intersection set is the name of the
winner.

The second way starts out as above, but once the first non-empty
intersection set is determined, additional ballots are drawn as needed to
narrow down the intersection to one candidate, the winner.

More later ...

From: Forest Simmons <fsimmons at pcc.edu>
> To: EM <election-methods at lists.electorama.com>,         Andy Jennings
>         <elections at jenningsstory.com>
>
>
> How about thie following ideas?
>
> Elect the pair that covers the most voters (i.e. that leaves the fewest
> voters with nobody that they approved elected).  In case of ties, among
> tied pairs elect the one whose weaker member has the most approval.
>
> Or this variant:  If no pair covers more than 70 percent of the voters,
> elect the pair that covers the greatest number of voters.  Otherwise
> consider all pairs that cover at least 70 percent of the voters to be
> tied.  Then among tied pairs, elect the one whose weaker member has the
> greatest approval.
>
>
> From: Andy Jennings <elections at jenningsstory.com>
> > To: Election Methods <election-methods at electorama.com>
> > Subject: [EM] Approval ballots. Two to elect. Best method?
> >
> > SPAV?
> > 1. Candidate with most approvals wins.
> > 2. That candidate's voters have their voting weight halved (or multiplied
> > by 1/3).
> > 3. Remaining candidate with most points wins.
> >
> > STV-like?
> > 1. Choose quota Q = one-third (or one half) of voters.
> > 2. Candidate with most approvals wins.  (T = # of approvals)
> > 3. That candidate's voters have their voting weight multiplied by
> > max(1-(Q/T), 0)
> > 4. Remaining candidate with most points wins.
> >
> > Monroe-like?
> > 1. For each pair of candidates, find the voter-assignment which maximizes
> > the number of voters assigned to a candidate they approved, such that no
> > more than half the voters are assigned to one candidate.
> > 2. Elect the pair which satisfies the most voters.
> >
> > Others?  Toby, what are your favorite PR methods at the moment?  Can you
> > give a short explanation of how Phragmen/Ebert would work with only two
> to
> > elect?
> >
> >
> >
> > Specifically, I'm worried that in practically every approval-ballot PR
> > method, if there is a candidate you really like, but are sure that she
> can
> > get elected without your vote, you gain an advantage by not approving
> > them.  Is there any method that minimizes that incentive?
> >
> > ~ Andy
> >
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