What about a meta election?

[Markus Schulze]

Dear Charles,

you wrote (02 Oct 1998):
> A system that picks the most popular candidate and a system
> that excludes the most unpopular candidate will give very
> different answers.

Donald G. Saari discusses this problem in his book "Geometry
of Voting" and in some other papers. He introduced the
"Reversal Symmetry Criterion." This criterion says:

    Suppose, that candidate A is elected. Then, if every
    voter inverts his preferences, candidate A must not be
    elected.

The aim of this criterion is to guarantee, that a voter
doesn't need to make strategical considerations on whether
he wants to vote _for_ a certain candidate or _against_
another certain candidate. This criterion says, that the
election result of those methods, that meet this criterion,
is independent from whether voters vote _for_ or _against_
candidates.

I wrote (04 Oct 1997):
> Another reason, why I prefer Tideman to Smith//Condorcet[EM]
> is the fact, that Smith//Condorcet[EM] is *strange*, while
> Tideman is not *strange*.
>
> Definition:
>
>    A method is *strange* if & only if it is possible, that
>    a unique *best* candidate is a unique *worst* candidate.
>
> Remarks:
>
> If a unique *best* candidate is a unique *worst* candidate,
> then this candidate is called *strange winner*.
>
> *Strangeness* is a generalization of the "Majority Loser
> Criterion" and the "Condorcet Loser Criterion."
>
> Example:
>
>    18 voters prefer A to B to C to D to E.
>    17 voters prefer E to D to A to B to C.
>    17 voters prefer B to C to A to D to E.
>    15 voters prefer D to E to C to A to B.
>    14 voters prefer E to D to B to C to A.
>    11 voters prefer E to C to A to B to D.
>    2 voters prefer C to A to B to D to E.
>    1 voter prefers A to C to B to D to E.
>    1 voter prefers C to B to E to A to D.
>    1 voter prefers C to E to B to A to D.
>    1 voter prefers E to B to C to D to A.
>    1 voter prefers E to C to D to A to B.
>    1 voter prefers D to E to B to C to A.
>
> Case 1: Suppose, the voters were asked, who the *best*
> candidate is. [Example: There is only one seat to be filled.]
>
>    18 voters would say ABCDE.
>    17 voters would say EDABC.
>    17 voters would say BCADE.
>    15 voters would say DECAB.
>    14 voters would say EDBCA.
>    11 voters would say ECABD.
>    2 voters would say CABDE.
>    1 voter would say ACBDE.
>    1 voter would say CBEAD.
>    1 voter would say CEBAD.
>    1 voter would say EBCDA.
>    1 voter would say ECDAB.
>    1 voter would say DEBCA.
>
>    A:B=65:35
>    A:C=36:64
>    A:D=51:49
>    A:E=38:62
>    B:C=68:32
>    B:D=52:48
>    B:E=39:61
>    C:D=53:47
>    C:E=40:60
>    D:E=54:46
>
> Suppose, Smith//Condorcet[EM] was used to determine the *best*
> candidate. Then, the unique *best* candidate would be D.
>
> Case 2: Suppose, the voters were asked, who the *worst*
> candidate is. [Example: There are 4 seats to be filled and
> 5 candidates. The voters are asked to eliminate one
> candidate.]
>
>    18 voters would say EDCBA.
>    17 voters would say CBADE.
>    17 voters would say EDACB.
>    15 voters would say BACED.
>    14 voters would say ACBDE.
>    11 voters would say DBACE.
>    2 voters would say EDBAC.
>    1 voter would say EDBCA.
>    1 voter would say DAEBC.
>    1 voter would say DABEC.
>    1 voter would say ADCBE.
>    1 voter would say BADCE.
>    1 voter would say ACBED.
>
>    A:B=35:65
>    A:C=64:36
>    A:D=49:51
>    A:E=62:38
>    B:C=32:68
>    B:D=48:52
>    B:E=61:39
>    C:D=47:53
>    C:E=60:40
>    D:E=46:54
>
> Suppose, Smith//Condorcet[EM] was used to determine the *worst*
> candidate. Then, the unique *worst* candidate would be D.
>
> Thus: The unique *best* candidate is the unique *worst* candidate!
>
> Tideman is not *strange*, because the order in which pairwise comparisons
> are locked resp. skipped does not depend on whether the voters vote for
> or against candidates.
>
> Case 1:
>
>    Lock B > C.
>    Lock A > B.
>    Skip C > A.
>    Lock E > A.
>    Lock E > B.
>    Lock E > C.
>    Lock D > E.
>    Skip C > D.
>    Skip B > D.
>    Skip A > D.
>
> D is the only candidate, who is not *dominated*. Thus: D is the unique
> *best* candidate.
>
> Case 2:
>
>    Lock B < C.
>    Lock A < B.
>    Skip C < A.
>    Lock E < A.
>    Lock E < B.
>    Lock E < C.
>    Lock D < E.
>    Skip C < D.
>    Skip B < D.
>    Skip A < D.
>
> C is the only candidate, who is not *dominated*. Thus: C is the unique
> *worst* candidate.

Markus Schulze