# What about a meta election?

Markus Schulze schulze at sol.physik.tu-berlin.de
Fri Oct 2 07:58:39 PDT 1998

```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

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

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