Explaining Schulze

Blake Cretney bcretney at my-dejanews.com
Wed Sep 2 14:11:34 PDT 1998

Sometime back, I suggested a modification to Sequential Dropping.  
I think it's equivalent to Schulze, but I'm going to call it the Least Contradicted Majority Method (LCM) to differentiate it.  I see its
main purpose as a way to explain Schulze.

LCM can be defined as
1.	Sequentially drop victories of the form A > B from least
to greatest, skipping if there is any candidate with a one-way beat
path to B.  Of course, I'm refering to beat paths of non-dropped
victories.  The candidate who has all his defeats dropped first

Or equivalently

2.	Eliminate all candidates not in the Smith set.  Drop victories
(among non-eliminated candidates) from least to greatest, until there
is a new smaller Smith set.  In other words, until some subset has
its last external loss dropped.  Then repeat the process until only
one candidate is left.

I believe that these are both equivalent to Schulze (except that they
don't specify what to do about ties), and that #2 can provide a good
way to explain Schulze to the novice.

Here's an outline of a possible explanation
1.  We believe strongly in the principle of majority rule
2.  As a result, we deal with the majority's preference between
two candidates, instead of more, which would only yield a plurality

Now, give an example of a Condorcet winner.  Point out that to elect
anyone else would contradict a majority preference.  Point out that
electing this candidate doesn't contradict any majority preference.

Present an example where there is no Condorcet winner, but there
is a Condorcet Loser (D).  Point out that there are contradictory
majorities, explain how this might happen.

3.  We will only disregard a majority preference if it is contradicted
by greater majority preferences.

Point out that the majority decision over D is uncontradicted.  Suggest
that if we can divide the candidates into 2 groups, with the second
group having only losses to the first, that the second group can be
eliminated as contenders.

Show the result with D eliminated.  Refer to rule 3, and eliminate the
lowest majority.  Reduce to the Smith set and obtain a result.  Remark
on how the 2 steps (reducing to Smith and ignoring a victory) could be
repeated as many times as was necessary.
Of course, the preceding is not meant as a rigorous proof of the merits
of Schulze.  It is meant as a process with intuitive steps corresponding
to basic principles.

I think this is what attracts most people to IRO.  They start by explaining
why you cannot use plurality to pick a clear winner.  Then they go on to
give a reasonable sounding step-by-step process that ends in someone 
getting a majority.  Of course, what most people, myself included, do not
think of when we first here the explanation of IRO is, "why if we don't
think plurality can pick a clear winner, do we assume it can pick a clear
loser?"  I do not think my explanation of Schulze has any of these kind
of traps.

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