Steve Eppley SEppley at alumni.caltech.edu
Fri Jun 2 18:12:51 PDT 2000

```The following data, calculated by software written to simulate
many voting methods, supports my contention that MTM dominates
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
between methods which are not distinguished by any of the

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

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)

```