David Catchpole s349436 at student.uq.edu.au
Sun Jun 4 00:09:20 PDT 2000

```On Fri, 2 Jun 2000, Steve Eppley wrote:

> 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.
>

Wouldn't it be obvious that the best method against that test would be a
simple biggest-pairwise-majority system?

> 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
> 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)
>
>

-------------------------------------------------------------------------------
Some say the world will end in fire,
Some say in ice.
>From what I've tasted of desire
I hold with those who favour fire.
But if it had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice.
Robert Lee Frost

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