[EM] Head-to-head: Schulze vs. MTM (majoritarian Tideman)

[Steve Eppley]

The following data, calculated by software written to simulate 
many voting methods, supports my contention that MTM dominates 
Schulze in the head-to-head comparison of whether voters would 
prefer one's winners more than the other's winners.

Norm Petry posted some of his own data a few days ago, but I 
believe his data is erroneous. (See my reply to Norm in a 
message being posted separately.)  In particular, Norm's data 
showed Schulze best when there are 3 alternatives, but my 
reckoning is that, when there are 3 alternatives, the Schulze 
winner never beats the MTM winner or the IBCM winner or the 
PC(wv) winner. (As Markus noted a year ago, when there are 3 
alternatives, the Schulze method chooses the same as PC(wv).)

Below the table are some notes which explain the elements of the 
table.

Some conclusions can be drawn from the table: 

1. With 3 (or fewer) alternatives, Schulze and MTM behave the 
same, in accordance with theory.

2. With 4 or more alternatives, MTM dominates Schulze, and the 
more alternatives there are, the greater is MTM's dominance of 
the "W1-W2" stat.

3. The secondary stat "V1-V2" (the number of voters preferring 
the MTM winner more than the Schulze winner minus the number of 
voters preferring the Schulze winner more than the MTM winner, 
averaged over the scenarios where the two methods are decisive 
and disagree) also favors MTM.

Note: Each reader must judge the relative importance of 
criteria.  The head-to-head comparison can serve to distinguish 
between methods which are not distinguished by any of the 
criteria the reader considers more important than the head-to-
head comparison.

   Method 1 = MTM (majoritarian Tideman)
   Method 2 = Schulze
   1000 voters
   1000 scenarios having no Condorcet winner.

   ~Prob  Alternatives   NoCW     1B2    2B1    W1-W2    V1-V2 
   -----  ------------  ------   -----  -----   -----    -----
    0           3         6.4%    0      0       0        0
    0           4        19.6%    5.4%   0.3%    1.0%     1.0%
    0           5        26.5%    0.7%   8.2%    2.0%     1.2%
    0           6        32.9%   11.1%   0.9%    3.4%     1.1%
    0          10        49.4%   16.7%   2.2%    7.2%     1.3%
    0          20        70.0%   22.1%   3.5%   13.0%     1.4%

    0.3         3         6.6%    0      0       0        0    
    0.3         4        13.9%    5.2%   0.4%    0.7%     1.2%
    0.3         5        19.1%   10.7%   0.6%    1.9%     1.9%
    0.3         6        25.4%   12.5%   0.9%    3.0%     1.7%
    0.3        10        43.0%   19.0%   2.4%    7.1%     1.7%
    0.3        20        63.9%   23.7%   2.7%   13.4%     1.7%

Remarks:
1. The "~Prob" (indifference probability) column shows a 
parameter used during the random generation of voters' rankings. 
For all adjacent alternatives in all rankings, it is the 
probability that the symbol between them is '~' (indifference) 
instead of '>' (strict preference). (Setting the probability to 
0 generates strict rankings.)  The results did not change 
significantly when ~Prob was changed from 0 to 0.3.

2. The "NoCW" column shows the percentage of scenarios having no 
Condorcet winner. (The two methods being tested here are 
decisive and agree when there is a Condorcet winner.)  The 
software generated enough scenarios that 1000 scenarios per 
trial had no Condorcet winner.

3. The "1B2" column shows the percentage of scenarios where the 
winner according to method 1 beat pairwise the winner according 
to method 2, in scenarios where both methods are decisive and 
disagree, relative to the number of scenarios having no 
Condorcet winner.

4. The "2B1" column shows the percentage of scenarios where the 
winner according to method 2 beat pairwise the winner according 
to method 1, in scenarios where both methods are decisive and 
disagree, relative to the number of scenarios having no 
Condorcet winner.

5. The "W1-W2" column is the primary statistic for the head-to-
head comparison.  Its formula is "W1-W2" = (NoCW x (1B2-2B1)).  
This is the overall edge for method 1 (if positive), as a 
percentage of total scenarios (not just scenarios having no 
Condorcet winner).  A negative value means that method 2 has the 
edge over method 1.

6. The "V1-V2" column is a secondary statistic for the head-to-
head comparison.  It shows the average margin of victory: the 
number of voters who rank the method 1 winner over the method 2 
winner minus the number of voters who rank the method 2 winner 
over the method 1 winner, averaged over the scenarios where both 
methods are decisive and disagree.


---Steve     (Steve Eppley    seppley at alumni.caltech.edu)