[EM] Yee/B.Olson Diagram Remarks

[fsimmons at pcc.edu]

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.
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