[EM] The challenge: reasoning from Berlin solving 2 candidate elections

[Craig Carey]

This message is minor: it says that Mr SChulzse ought not copy from
economists, adn concludes with a request for Mr Schulze to solve the
2 candidate elections (with the cases: 0 winners, 1 winner, and
2 winners).


At 2005-01-03 18:01 +0100 Monday, Markus Schulze wrote:
...
>However, Hylland proved that when there are only two candidates and
>the used single-winner election method is strategyproof then the
>result depends only on whether the individual voter strictly prefers
>candidate A to candidate B, strictly prefers candidate B to candidate A
>or is indifferent between candidate A and candidate B (Aanund Hylland,
>"Strategy Proofness of Voting Procedures with Lotteries as Outcomes
>and Infinite Sets of Strategies," University of Oslo, 1980).
>
>Therefore, I interpret May's theorem in connection with Hylland's
>theorem as follows:
>
>   When there are only two candidates then the unique anonymous,
>   neutral, decisive, and strategyproof single-winner election
>   method is FPP. Therefore, every single-winner election method
>   should satisfy the following criterion:
>
>   When there are only two candidates and the number of voters who
>   strictly prefer candidate A to candidate B is strictly larger
>   than the number of voters who strictly prefer candidate B to
>   candidate A, then candidate A should be elected with certainty.
>


When I look at that and try to get a meaning out of it, I think
have these ideas/thoughts:

(1) Mr Schulze avoided getting it wrong in the zero winner case.
   These words do that:

      "Therefore, every single-winner election method should...".

(2) The last paragraph lacks a meaning since subscribers do not know
    what meaning is allocated for the words:

                  "strictly prefer"
    If Mr Schulze believes that we have the meaning, then he is
    making a mistake.

(3) Mr Schulze copied in the word "strategyproof"..

   Maybe that is not serious enough. The rest of my e-mail is over
   the idea that to copy from most Soc Choice economists is not
   sufficiently serious. It could be light headed; however I ask
   a question of Mr SCHULZE:

   For example, after reading Mr Schulze's text, we can't tell if
   it is the "stategy=-proofness" rule/aim that requires the
   variable, "t", to be equal to 1, in this parameterized solution
   to the 2 candidate 1 winner problem:

               a0 (A)
               ab (AB)
               b0 (B)
               ba (BA)
               z0 ()

Equations finding the winner:
      [ 0 < a0-b0 + t*(ab-ba)     ] implies (A wins).
          [ a0-b0 + t*(ab-ba) < 0 ] implies (B wins).

I did a quick browse around around the Internet for words on the
strategy proof rule.

I succeed in getting to see yet another paper that successfully
mentioned:

                  Gibbard (1973) and Satterthwaite

mentioned in the first or second page. (The PDF page containing the
abstract is defined to be "Page Number 1"). I assume that there is
some economist on the surface of the globe that complains and says
that putting a reference on page 3 is a reference that is positioned
too late.

Suppose Kenneth Arrow did not actually define the term "dictatorial".
Would it be cotrrect for Mr Schulze to copy the word in ?.
The topic is about polytopes and OSSIPOFF and Schulze don't mention
that.

Can Mr Schulze derived the 1 winner 2 candidate solution using
the 5 symbols (a0,ab,b0,ba,z0) to represent the counts.

Since Mr Schulze is unlikely to be making progress if/since not using
the symbolism of algebra, I make the request only ask for 2 candidate
solutions.

Mr Schulze could be the 1st subscriber here (excluding me) that
proves that the 2 candidate election is not too difficult for that
individual.

To Mr Schulze: Write down the algebraic equation that defines
strategy-proofness. Add some restrictions on papers as desired.

The task set by me does not ask for some meaningless wording,
e.g. "strictly prefer".

I recall that I asked MIKE OSSIPOFF to solve the 2 candidate
election and he never actually did it.

I genuinely don't know if Mr Schulze can quickly and seriously
produce a solution. The strategy-proof idea can't be used since
Mr Schulze so far has not defined that term. Also decisive has not
been redefined or defined.

I used Symmetric Completion

  since when 4 candidates, that deletes the 4th preference and
  reduces the number of papers from 65 to 41.

However Mr Schulze might want to use a QE logic equation that
defines "strategy-proofness".

If Mr Schulze fails, then Mr Venkze could have an attempt.
Much of the problem is to select axioms that make t equal 1.

I can imagine that EML subscribrs are creating a vicious gauntlet
for the man of Berlin to run through.



Craig Carey