[EM] The importance of being uncovered (improved layout)

[fsimmons at pcc.edu]

The format on that last message was so bad, that I'm going to re-send it:

In the previous installment on this topic I explained why I thought that it
would be much easier to defend an uncovered winner from complaints of
unfairness, than to defend a covered winner, no matter the wonderful strengths
of the pairwise victories.

Here I just want to relate the concept of covering to more familiar ideas.

In the first place, whenever there is a Condorcet Winner, the CW is the only
uncovered candidate, so any method that always elects an uncovered candidate
automatically satisfies the Condorcet Criterion, whether or not it mentions the
CW in its description.

Covering is a form of dominance like Pareto dominance.  If alternative Y covers
alternative X, then it is as good or better than X (with regard to pairwise
wins, losses, and ties) across the board, and strictly better in at least one case.

In terms of the Copeland win, loss, tie matrix this means that row Y dominates
row X if and only if alternative Y covers alternative X.

The Copeland matrix can be formed from the margins matrix by replacing each
entry with its sign, so that win, loss, and tie are represented by the
respective values of 1, -1, and 0.  To get the more traditional Copeland matrix,
add one to each value and then divide by two.  In this version win, loss, and
tie are represented respectively by 1, 0, and 1/2 .  These two versions of the
Copeland matrix have in common that there is a value W that represents a
pairwise win, and smaller value T that represents a pairwise tie, and a still
smaller value L that represents a pairwise loss.

The value (W, T, or L) that appears in row i and column j of the Copeland matrix
tells whether the pairwise comparison of alternatives i and j resulted in a win,
tie, or loss for alternative i.  It follows that the main diagonal of the
Copeland matrix consists entirely of T's, since each alternative is tied with
itself.

If each value in row i of the Copeland matrix is at least as large as the
corresponding value (i.e. in the same column) of row k, while at least one value
of row i is strictly greater than the corresponding value of row k, then row i
dominates row k, and alternative i covers alternative k.

This approach gives a slightly weaker definition of covering than the one I
prefer.  It is weaker because, the one I prefer requires the covering
alternative to strictly beat every alternative that the covered alternative
beats or ties.  The reason I prefer the stronger definition is that in our
application the covering relation is used to over-ride "approval" as judged by
the number of positive scores; if the "approval champion" X is covered, then the
method will elect some alternative that covers X.

The shortest definition of this stronger covering relation is this:

Alternative Y covers alternative X
          if and only if
every alternative that does not pairwise beat X (including X itself) is beaten
pairwise by Y.

In terms of the Copeland matrix this means that in any column where row X does
not have an L (i.e. does have a T or W), row Y will have a W in that column.

I hope that these remarks are helpful for understanding the concept of covering,
and for seeing its relevance to the Condorcet Criterion.

FWS



----- Original Message -----
From:
Date: Monday, June 13, 2011 1:13 pm
Subject: The importance of being uncovered (was C//A)
To: election-methods at lists.electorama.com,

>
> > From: Kathy Dopp
> > What does C/A stand for?
> > Condorcet/Approval?
> >
> Yes, and when the top approval candidate in the Smith set is
> uncovered, as it is whenever the Smith set
> has fewer than four candidates, the method I described elects
> the same candidate as Smith//Approval.
>
> The importance of electing an uncovered alternative when the
> pairwise defeats are public knowledge (as
> they should be) is shown by the following imaginary conversation
> between two basket ball coaches after
> a round robin tournament:
>
> A: How come your team gets the trophy, when my team beat yours?
>
> B: Well, we could almost beat ourselves, and we did beat team
> C, which defeated you guys.
>
> A: I guess you got a point there.
>
> In other words, team B defends its trophy against the claim of
> team A only by pointing out that A did not
> cover team B.
>
> If A had covered team B, imagine how hard it would be to defend
> the trophy decision:
>
> B: Yes, it is true that you beat us, but we beat teams C, D,
> and E.
>
> A: So what? We did too.
>
> B: But we beat team C by twenty points, and you only beat them
> by five.
>
> A: That's because we sent in our second string players when we
> saw how weak they were.
>
> In the election context, the corresponding statement would be,
> "We knew that we were going to beat
> him nationally, so we didn't waste time campaigning in the state
> where his base supporters live."
>
> In sum, there is no use trying to justify the election of a
> covered candidate by secondary considerations
> (like defeat strength), since the primary consideration in a
> pairwise based method is who beats whom
> pairwise, and that by itself is sufficient to eliminate a
> covered candidate.