[EM] Yee/B.Olson Diagram Remarks

[fsimmons at pcc.edu]

My last message under this title got messed up in the transmission.  Here's
another try:

In Yee/B.Olson Diagrams there is a rough correspondence between certain
geometric properties of the win regions with certain compliances of the method.

Convexity is a kind of geometric consistency that corresponds roughly with
traditional Consistency.  Condorcet methods have this kind of consistency: if
the ballots are divided into two subsets, and C is the CW of both subsets, then
C will be the CW of the entire ballot set.

Starlike  w.r.t. the candidate positions is a kind of monotonicity that
corresponds roughly (but not exactly) with ballot monotonicity.

Candidates squeezed out of their own win regions corresponds roughly to pushover
vulnerability.

In methods, like IRV, that proceed by elimination, if there are more than two
candidates, at some point all but three of the candidates have been eliminated.
 So three candidate diagrams are totally relevant to these methods.

Every triangle of candidates except an equilateral triangle suffers from the
squeeze effect under IRV for sufficiently large standard deviations of the voter
distributions.

Every obtuse triangle of candidates suffers from the non-starlike condition for
some intermediate range of sigma's.In this regard note that a randomly chosen
triangle is obtuse (more likely than not).

A randomly chosen Yee/B.Olson diagram (for IRV) will almost surely have one or
more non-convex win regions.

These remarks give us some idea of the extent of IRV's vulnerability to
Pushover, IRV's non-compliance with Monotonicity, and IRV's non-compliance with
Consistency. 

Remember that these observed pathologies are in an environment where compliance
is so easy that even  some fairly horrible methods look good!  If Yee/B.Olson
says you're bad, then you're bad.  The converse is not true.  If the electoscope
does not say you are bad, that doesn't mean you are good.

Borda doesn't look bad under this electoscope, because Borda complies with
Consistency and Monotonicity, but Borda is worse than IRV.  Borda is like the
little boy that is always nice in front of the teacher, but gets mean when the
teacher is not around.  But at least careful attention to the electoscope shows
Borda's Clone Loser problem as clones are added to a loser.

It is interesting that the electoscope is not sensitive enough to reveal
Copeland's Clone Loser problem.