[EM] A Comparison of the Two Known Monotone, Clone Free Methods for Electing Uncovered Alternatives

[fsimmons at pcc.edu]

> Where's a good place to find out more about the Landau set? Is
> it really
> possible to have a monotone, clone free method that is
> independent of non-Landau
> alternatives?

It turns out that there are several versions of covering, depending on how ties
are treated.  All of them including the Landau set are the same when there are
no pairwise ties (except with self).

In July of this year I gave an example that shows that no decent deterministic
monotone method can be independent from covered alternatives.  The example
applies to the Landau version of uncovered.  So neither Ranked Pairs nor
Beatpath nor Range restricted to Landau can monotone.

Here's the example

Suppose that we have a method that satisfies independence from non-Landau
alternatives, and that gives 
greater winning probability to alternative B in this scenario

40 B>C>A
30 C>A>B
30 A>B>C

than in this scenario

40 D>B>C
30 B>C>D
30 C>D>B

as any decent method would.

Then consider the scenario

40 D>B>C>A
30 A>B>C>D
30 C>A>D>B

The Landau set is {A,B,C} and so by independence from non-Landau alternatives
the winner is chosen 
according to the first scenario above.

Now switch A and B in the second faction to get

40 D>B>C>A
30 B>A>C>D
30 C>A>D>B .

The Landau set becomes  {D,B,C}.and so by independence from non-Landau
alternatives the winner is 
chosen according to the second scenario above.

In summary, solely by raising B relative to A , the winning probability of B
decreased.

For a deterministic method the probability of B would go from one to zero.