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

[Dave Ketchum]

On  Fri, 28 Mar 2008 08:30:47 -0400 Warren Smith wrote:
>> 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.)

Building on those thoughts, let's try something with Plurality:
      Start with that collection of voters and issues.
      Invert all the issues so that a Y will attract the same voters as an 
N did, and an N will attract those who had gone for Y.
      Note that YYYY will now attract the same 5 voters who had gone for 
NNNN, and the new NNNN will get 0 votes.
      The collection of voters, while owning no claims to randomness, 
remain as legitimate as they had been.

Making corresponding changes in tests of other methods should make 
corresponding changes in their results.

DWK
> 
>  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.
-- 
  davek at clarityconnect.com    people.clarityconnect.com/webpages3/davek
  Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
            Do to no one what you would not want done to you.
                  If you want peace, work for justice.