[EM] Condorcet Loser, and equivalents

John john.r.moser at gmail.com
Tue Jun 4 11:50:48 PDT 2019

[Not subscribed, please CC me]

Andy Montroll beats Bob Kiss.
Andy Montroll beats Kurt Wright.
Bob Kiss beats Kurt Wright.
Montroll, Kiss, and Wright beat Dan Smith.
Montroll, Kiss, Wright, and Smith beat James Simpson.

James Simpson is the Condorcet Loser.

…is that right?

In the race {Montroll, Kiss, Wright}, Kurt Wright is the Condorcet Loser.

Let's say candidate D appears.  D signs up, becomes a candidate, is on the
ballot, and never campaigns.

D loses horribly, of course.

D is now the Condorcet Loser.  Kurt Wright isn't.

Does this change the nature of the candidate, Kurt Wright, or the social
choice made?

Think about it.  Without Simpson, Smith is the Condorcet Loser.  Without
Smith, Wright is the Condorcet Loser.  You have a chain of absolute
Condorcet Loser until you have a tie or a strongly connected component
containing more than one candidate which is not part of the Smith or
Schwartz set.

This is important.

We say Instant Runoff Voting passes the Condorcet Loser criterion, but can
elect the second-place Condorcet Loser.  That means Kurt Wright is the
Condorcet Loser and IRV can't elect Wright; but in theory, you can add
Candidate D and get Kurt Wright elected.

In practice, between two candidates, the loser is the Condorcet loser.
Montroll beats Kiss, so Kiss is the Condorcet Loser.  By adding Kurt
Wright, you have a new Condorcet Loser.  This eliminates Montroll and,
being that Kurt Wright is now the Condorcet Loser and is one-on-one with
another candidate, Bob Kiss wins.

It seems to me the Condorcet Loser criterion is incomplete and inexact:  a
single Condorcet Loser is meaningless.  The proper criterion should be ALL
Condorcet Losers, such that eliminating the single Condorcet Loser leaves
you with exactly one Condorcet Loser, thus both of them are the
least-optimal set.

I suppose we can call this the Generalized Least-Optimal Alternative
Theorem, unless somebody else (probably Markus Schulze) came up with it
before I did.  It's the property I systematically manipulate when breaking

Thoughts?  Has this been done before?  Does this generalize not just to
IRV, but to all systems which specifically pass the Condorcet Loser
criterion proper (i.e. they have no special property like Smith-efficiency
that implies Condorcet Loser criterion, but CAN elect the second-place
Condorcet loser)?  That last one seems like it must be true.
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