[EM] Three Tier, Dyadic via CR, etc.

[Forest Simmons]

Correction:

Re my IRV fixit method below that eliminates either the minmax loser or
the Copeland loser, whichever is weaker pairwise (if they are not the
same).

Copeland is susceptible to clones. Minmax violates reverse symmetry.

The combination seems to ameliorate if not totally eliminate these
defects.

In a three way contest, both methods pick the Ranked Pairs winner and
satisfiy reverse symmetry, so in the elimination method they agree on
which candidate should be eliminated.

So we need both only when there are four or more candidates.

Forest


On Sat, 25 Aug 2001, Forest Simmons wrote:


<big snip>

> Here's a brief offering in the IRV fixit category:
> 
> At each stage of the runoff eliminate either the candidate with the
> greatest number of pairwise losses or the candidate whose maximimum
> victory is minimal, whichever loses in a head-to-head comparison.
> 
> Rationale:
> 
> Both candidates for elimination reduce to the Condorcet Loser if there is
> one, otherwise they are two different kinds of approximations of what it
> means to be a Condorcet Loser.
> 
> Basically, we're pitting the Copeland loser against the minmax loser at
> each stage of the runoff. [Here I'm using method loser to mean method
> winner when the pairwise matrix is transposed, i.e. when preferences are
> reversed.]
> 
> Obviously, the CW is never eliminated.
> 
> In three way contests, the Ranked Pairs winner is not eliminated.
> 
> In the examples that I have worked out, clones do not upset the results.
> 
> Clarification:
> 
> If there are two or more candidates with the same number of losses, then
> the one nearest to having another loss is the one pitted head-to-head with
> the minmax loser. 
> 
> One way to interpret "nearest to having another loss" is having the
> smallest smallest winning margin.
> 
> Comment:
> 
> Minmax doesn't satisfy reverse symmetry. A minmax winner can also be the
> minmax loser (i.e. minmax winner when preferences are reversed).
> 
> That's one reason why we have to pit the minmax loser against the Copeland
> loser: i.e. to make sure we don't eliminate the CW if there is one.
> 
> Unfortunately Copeland elimination by itself is not good enough to
> guarantee the Ranked Pairs winner in a three way contest.