[EM] "Leader of the Free World" Project

[Markus Schulze]

Hallo,

there was a discussion between Arto Tengvall (the specialist
on election methods of the "Leader of the Free World"
project) and me about which election method should be used
at the LFW project. I posted my paper "A New Monotonic
and Clone-Independent Single-Winner Election Method"
(http://groups.yahoo.com/group/election-methods-list/files/nmciswem.pdf)
to him. Arto Tengvall replied (10 Oct 2003):

> About the Leader of the World - project.
>
> There will be one official electoral result, counted by the very
> best method known so far. Today, it is my method, but if someone
> can show that there is another method still better (judged by
> essential mathematical and logical criteria), I would have no
> objection against changing the method. Here is a short and easy
> definition of my method. As you can see, it also satisfies the
> criteria you have used yourself. It satisfies some extra criteria,
> too, which I have no time to write now.
>
> 1. Count votes in all pairwise comparisons, according to this:
> Count a vote to X from every ballot where X is given better or
> equal preference than Y, and to Y from every ballot where Y is
> given better or equal preference than X. X beats Y if the former
> score is bigger than the latter, and vice-versa.
>
> 2. The vote minimum of each X is the smallest score of X in any
> of X:s comparisons.
>
> 3. There is a beatpath from X to Y if X beats Y (in pairwise
> comparison between these two) or X beats some Z that beats Y.
>
> 4. Eliminate every candidate X such that there is a beatpath
> from some Y to X but not vice-versa.
>
> 5. Check the vote minimum of each non-eliminated. The winner
> is the one with the biggest vote minimum. (A tie is possible.)

In short: The Tengvall winner is that Schwartz winner A whose
worst pairwise comparison with another candidate B (measured
by the absolute number of voters who strictly prefer candidate B
to candidate A) is minimal.

When every voter casts a complete ranking then Tengvall's method
is identical with Schwartz//MinMax.

On first view, Tengvall's method seems to be identical with
Schwartz//MinMax(VotesAgainst). But when the strength of a
pairwise comparison is measured by the absolute number of
voters who strictly prefer candidate B to candidate A then
it isn't guaranteed that the worst pairwise comparison
of the Schwartz winner A with another candidate is necessarily
with another Schwartz winner. Therefore, Tengvall's method
isn't quite Schwartz//MinMax(VotesAgainst).

I wrote to Tengvall, that his method violates independence of
clones and reversal symmetry. And I asked him which criteria
he uses to conclude that his method is the very best method
known so far judged by essential mathematical and logical
criteria. Arto Tengvall replied (13 Oct 2003):

> My method has similarities with Schwartz//Minmax, however one
> defines that method. I have seen many different definitions
> but my method is a little bit different. Maybe you should
> look again, without thinking Schwartz method, so without any
> pre-opinions. Then you can see, withaout any deeper analysis,
> that my method easily fullfills clone-independency and reversal
> symmetry.

Can somebody tell me where I have made a mistake?

Markus Schulze