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

[David Catchpole]

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 
> Schulze in the head-to-head comparison of whether voters would 
> 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 
> 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)
> 
> 

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Some say in ice.
>From what I've tasted of desire
I hold with those who favour fire.
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To say that for destruction ice
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And would suffice.
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