[EM] Tideman and GMC

[Markus Schulze]

Dear Steve,

you wrote (9 May 2000):
> For reference, here's the Feb 3 example:
>
>    26 voters vote:  C>A>B>D
>    20 voters vote:  B>D>A>C
>    18 voters vote:  A>D>C>B
>    14 voters vote:  C>B>A>D
>     8 voters vote:  B>D>C>A
>     7 voters vote:  D>A>C>B
>     7 voters vote:  B>D>(A=C)
>        ==>  Majorities:  BD75, CB65, DC60, AD58, AB51, CA48
>
>    Tideman locks BD75, locks CB65, skips DC60 (because DC would
>    cycle with the locked majorities), locks AD58, locks AB51, 
>    and locks CA48.  The locked majorities imply that the 
>    "best ranking" is C>A>B>D, so Tideman elects C.
>
>    Schulze compares strongest opposing beatpaths:
>    ADCB58 > BDCA48     implies A finishes ahead of B.
>    ADC58  > CA48       implies A finishes ahead of C.
>    AD58   > DCA48      implies A finishes ahead of D.
>    CB65   > BDC60      implies C finishes ahead of B.
>    BD75   > DCB60      implies B finishes ahead of D.
>    CBD65  > DC60       implies C finishes ahead of D.
>    Schulze elects A.  (The implicit Schulze ranking is A>C>B>D,
>    though I'm not sure if Markus will acknowledge this.)
>
>    (For the record, BCM is indecisive between A&C.  IBCM,
>    after eliminating all but A&C, elects C due to CA48.)
>
> Markus claimed, if I haven't misinterpreted him, that the 
> example shows Tideman is more manipulable than Schulze, by 
> pointing out that some of the voters who prefer B to D and A to 
> C (the 20 "B>D>A>C" voters) can change the Tideman outcome from 
> C to A by not ranking B ahead of D (since that would reduce the 
> BD majority to 59 or smaller so DC60 would get locked in).
>
> What Markus apparently neglected to notice is that voters can 
> easily manipulate Schulze in this scenario too, to change the 
> Schulze winner from A to C.  Some of the voters who prefer C 
> to A and A to D (the 26 "C>A>B>D" voters and the 14 "C>B>A>D" 
> voters) can uprank D ahead of A or downrank A behind D, which 
> would reverse the A>D majority to D>A.  Since the ADC58 beatpath 
> depends on the A>D majority, the ADC58 beatpath would be 
> destroyed and Schulze would elect C.

What I criticize is that Tideman depends unnecessarily on the
strengths of too many pairwise defeats. In the example above,
whether candidate A or candidate C is the Tideman winner depends
on how many voters prefer candidate B to candidate D. To my
opinion, the strength of the pairwise defeat B:D doesn't contain
any information about whether candidate A or candidate C is
better. Therefore, to my opinion, whether candidate A or
candidate C is elected shouldn't depend unnecessarily on the
strength of the pairwise defeat B:D.

You now say that the voters can change the Schulze winner from
candidate A to candidate C by ranking candidate A behind D.
But it is a difference, whether you can -in the Schulze
method- make a candidate lose the elections by ranking this
candidate insincerely low resp. make a candidate win the
elections by ranking this candidate insincerely high or
whether you can -in the Tideman method- change the winner
from one candidate to another candidate by ranking two
completely different candidates insincerely.

The difference is:

(a) _Every_ Condorcet method can be manipulated by burying
[=lower a candidate with respect to sincere placement in the
hopes of defeating it] and compromising [=raise a candidate
with respect to sincere placement in the hopes of electing
it]. Actually, the Condorcet methods don't differ in how
much they are vulnerable by burying and compromising.

But the Condorcet methods differ extremely in how much they
are vulnerable by the other mentioned strategy [=changing
the winner from candidate X to Y by ranking two completely
different candidates V:W insincerely.] Example: The MinMax
method isn't vulnerable at all by this strategy.

Therefore, I proposed that the manipulability (by voting
insincerely) of a Condorcet method should be measured by
how much it is vulnerable by the other mentioned strategy.

(b) It can be argued that -in the Schulze method- if some
voters uprank D ahead of A or downrank A behind D then
this means that candidate A becomes less popular and that
it is therefore legitimate when candidate A loses the
elections.

But it is more difficult to argue why -in the Tideman
method- the winner should be changed from candidate C to
candidate A when some voters uprank B ahead of D or
downrank D behind B.

Markus Schulze
schulze at sol.physik.tu-berlin.de
schulze at math.tu-berlin.de
markusschulze at planet-interkom.de