[EM] Multiple Winners, Multiple Offices, and Proportional Approval Voting

[Alex Small]

Imagine an election which selects 2 winners, but one of the winners will
have more power than the other.  The candidate who performs best according
to our election method becomes chief executive.  The runner-up goes to the
legislature to act as the leader of the "Faithful Opposition".  You might
envision this as a unique type of "check and balance" on the power of the
executive.  Originally, the US Constitution did something like this:  The
VP was whoever came in #2 in the electoral vote, rather than the
running-mate of #1.


I was pondering the best way to handle this.  I have 2 opposing desires:

1)  Normally we like to see single-winner offices filled by a majoritarian
method, or at least a method that usually requires more support than the
34% required under many PR systems when there are 2 seats to fill.

Straight Approval Voting would meet this criterion.  Plurality voting
might not;  with 2 winners the "lesser-of-2-evils" problem is less severe.
 Vote-splitting may become more common, leading to an executive with less
support.  IRV (i.e. 1 winner, not STV for 2 winners) would work just fine,
as would Condorcet.

2)  We don't want a system that makes it easy for a single large faction
to elect both the executive and the opposition leader, unless perhaps that
faction constitues 2/3 or more of the voters.  Such a situation would
defeat the whole purpose of having both an executive and an opposition
leader.

Single-winner systems fare poorly here.  With ordinary Approval Voting,
51% of the voters could just approve 2 candidates, guaranteeing that both
of them win (although the other 49% would at least have a say in who takes
which office).  Or, suppose we used Condorcet or IRV, electing the winner
as executive and then running the election again without the first winner
on the ballot.  If 51% of the voters ranked A 1st and B 2nd, A would be
the executive.  With A gone from the ballots, B would be the "opposition"
leader.

PR systems meet this second criterion nicely, but they tend to fail the
first one.





One method that I can think of is Proportional Approval Voting.  The only
way for a single faction to guarantee the election of their favorite with
coordinated action is if they are at least 2/3 of the electorate.  And, if
the campaigning focuses on the more powerful executive office then we're
likely to get the cross-over voting needed to give the winner a majority
or something close to it.

Another method might be a modification of IRV or STV:  First use standard
IRV to elect the executive, eliminating candidates until somebody is at
the top of 50.0001% or more of the ballots.  Then, discard all ballots
listing the winner, and use standard IRV again, using only the remaining
ballots (so the quota is 50% of all ballots remaining, not 50% of all
ballots originally cast).

A modified Borda might "work" (to the extent that any form of Borda can be
desirable):  Give 2 votes to your favorite, 1 vote to your second
favorite, and no votes to any others.  A faction would need 2/3 to elect
BOTH of its candidates without any help from other voters.

I'm not sure about how to modify Condorcet or other methods.  Any thoughts?





Alex