[EM] meditations

[fsimmons at pcc.edu]

My conjectures turned out to be true:

Lemma:  If range values are limited to k levels, and alternative X beats
alternative Y with a margin ratio greater than (k-1)/1, then alternative X has a
greater range score than alternative Y.

Proof:  Without loss in generality assume that the k possible ratings on each
ballot are 0, 1, ...(k-1). If there are x ballots on which X is rated above Y
for every y ballots on which Y is rated above X, then the least the difference
in the respective range scores could be is

    d = 1*x - (k-1)*y ,

since the least possible difference in ratings on any single ballot is one, and
the greatest possible difference in ratings on any ballot is (k-1).

But when the margin ratio  x/y is greater than (k-1)/1, the value of d is positive.

Therefore X has a greater total range score than Y.


Corollary 1.  If range values are limited to k levels, then there can be no beat
cycle where all of the defeats have margin ratios greater than (k-1)/1.

Corollary 2.  If range values are limited to k levels, then no beatpath with
margin ratio strength greater than (k-1)/1 can be longer than k times the number
of ballots, no matter how many alternatives are rated on the ballots.

Corollary 3.  In the case of ordinal ballots, if no ballot ranks candidates at
more than (k-1) levels, then the conclusions of Corollaries 2 and 3 still hold.

Corollary 4.  If there are only k candidates , then the conclusions of
Corollaries 2 and 3 still hold.

How can we put this information to good use?

Suppose that we are dealing with 3 slot ballots as in MCA, APV, MAFP, etc.   

It may not be too common for one candidate to have a wv score against another
candidate consisting of more than two thirds of the vote.  But that is not
needed here, only a margin ratio greater than two to one is needed. In other
words, if eleven percent of the voters prefer X over Y but only five percent of
the voters prefer Y over X, then we have a margin ratio that cannot be sustained
indefinitely in a beatpath, and (more to the point) cannot sustain any cycle no
matter how long or short.  So the losers in all such defeats can be eliminated
without fear of eliminating all of the candidates.  

Doing so would automatically eliminate all of the Pareto dominated candidates,
too, and make the method independent from Pareto dominated candidates.

Any other ideas on how to put these facts to use?

Forest