# Condorect sub-cycle rule

David Marsay djmarsay at dra.hmg.gb
Mon Oct 13 05:33:38 PDT 1997

```Hugh Tobin has asked for some clarification on my previous posting.

Firstly, I don't have access to Condorcet's papers, and don't read
much French (unless it is to do with food). In translation, much of
what he says appears contradictory. It also seems to me that he may
have been attempting to over-simplify in places, and may have changed
his mind as he went on.

I referred to the following, but got the page wrong. Sorry.
I. McLean and F. Hewitt, Condorcet, Edward Elgar, 1994 p129
cites: "From the above remarks, we can obtain the following
general rule: each time we are forced to make a decision, we must
take all the propositions with plurality support in succession,
beginning with those that have most support. As soon as these
propositions give a result, we should take that to be the
decision, without taking into account the less probable
propositions which follow."

The quote goes on "If this method does not give us the result which
is least likely to be wrong, or a result which has a probability of
greater than one-half, and is made up of two propositions which are
both more probable than their contradictories, it at least provides
us with one which at least AVOIDS THE LEAST PROBABLE PROPOSITIONS and
which CAUSES THE LEAST INJUSTICE TO CANDIDATES, WHEN TAKEN TWO BY
TWO."

I commend the book. My tentative view is that the 'strength' of a
pair-wise majority should only depend on the preferences expressed
between the two: thus for X, Y one takes the number of ballots with X
above Y less the contrary ballots.

My modified Generalized Majority Criterion, 'GMC+', could be used
with any pair-wise notion of strength. For example, one could take
strength as the existence of a simple majority, so GMC+ reduces to
Smith.

Taking the pair-wise majority seems the obvious thing to do, and
seems to be what Condorcet intended. It also seems to be 'best' in
terms of the properties one gets.

Ideally, one would have a ranking method that was universally
understood, mature, stable, and relatively
simple. Thus could at least act as a baseline for discussing other
methods. I think Condorcet almost makes it.

Dave Marsay
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Sorry folks, but apparently I have to do this. :-(
The views expressed above are entirely those of the writer
and do not represent the views, policy or understanding of
any other person or official body.

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