[Election-Methods] YN model - simple voting model in which range optimal, others not

[Warren Smith]

>  YN is made to appear as a comparison of methods, whereas it is tailored to
>  make Range look good.

 --
 1. it does make range look good, but
 2. but no - I disagree it was "tailored" for that purpose.

 It is a natural thing to explore.   There are issues.  Candidates
 have different stances on them.  The usual voting system "properties"
 you see explored, completely ignore those facts.  As soon as you put
 those facts in the picture in what seems the most simple+natural
 possible way, you get the YN model.


>  For the type of Plurality demonstration desired here a biased collection
>  of voters was created.

--you are confused.    Within the YN model, for ANY distribution of
 voters, no matter how "biased" and no matter who creates it,
 range voting always behaves optimally.

 But the other usual voting systems (Borda, Condorcet, Plurality etc) do not.
 So then, once you see that, it is merely a matter of trying to find
 some voter distribution, for each of these methods, to illustrate that fact
 maximally dramatically.

--I would like to move on. As opposed to incessantly trying to question or
 somehow pour cold water on the YN model for apparent partisan purposes,
 while never actually investigating it ...  could we, like, actually
make further
 investigations of the YN model to find new truths?

 Here are some interesting example questions.
 Choose a completely RANDOM distribution of voters.  By which , I mean,
 each voter flips a fair coin m times, to decide her stances on the m issues.

 Once the voters have finished flipping all their coins and have
 decided all their
 stances, then NEGATE THE SIGNS of all the issues on which
 the electorate voted majority-N.  (This is simply a renaming which
 causes every issue to get a majority-Y vote.  It does not really change
 the random situation at all, it just makes it canonically-named.)

 OK.  Within these canonical-random YN-model scenarios:
 how likely is it that Condorcet, Borda, etc return nonbest winners
(i.e. winners
 other than YYYYY?) And how likely is it that they return winners
 with at least f percent Ns in their name?

 I conjecture the following in the m-->large limit with 2^m or more voters:
 1. Plurality voting: probability YYYY loses --> 100%.
    Probability that the winner has over 50% N's in its name --> at
 least some positive constant.
 2. IRV voting: probability YYYY loses --> at least some positive constant.
    (Which I suspect also is 100%.)
 3. Condorcet: the probability that a "beats-all Condorcet winner" exists,
   goes to 0.

There are a ton of interesting conjectures like this you can make and
try to prove or disprove.

For the present class of problem, you can investigate them by computer
simulation too.


-- 
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking "endorse"
as 1st step)
and
math.temple.edu/~wds/homepage/works.html