[EM] How to convert a set of CR ballots to a set of Approval ballots

[Kevin Venzke]

Forest,

I implemented the new method (picking the least viable rank and merging it with
the less viable neighbor) and did some trials.

First, I should say that the Condorcet cycle-breaker I have implemented is:
"Successively drop the weakest defeats (as measured by WV) until a Smith set
member is undefeated."  I'm not sure what the proper name for this is, but I'll
call it "SD."

To my great surprise (should I have been?), I could not generate any three-candidate,
three-rank scenarios where MPCR did not elect the CW when one existed:

With 3 factions (random sizes) and 3 candidates and ranks:
matches        10474         100%  (this is total trials)
sdindec        36            .34  (SD was indecisive)
(I should've checked that Borda and MPCR were decisive, but I didn't.)
mpcr<>cw       0             0   (was a CW, MPCR didn't pick him)
mpcr=cw        9935          94.85  (was a CW, MPCR picked him)
unanimous      220           2.1 (MPCR, Black, SD agreed)
mpcr=borda     0             0   (MPCR and Black agreed against SD)
mpcr=sd        112           1.06 (MPCR and SD agreed against Black)
borda=sd       171           1.63 (Black and SD agreed against MPCR)
no matches     0             0  (Black, SD, MPCR all disagreed)

I got the same results with 5 and 8 factions.

With 4 candidates and ranks (5 factions), some "errors" appeared:
matches        4594
sdindec        5             .1
mpcr<>cw       7             .15  (.175%, excluding no-CW cases)
mpcr=cw        3971          86.43
unanimous      203           4.41
mpcr=borda     23            .5
mpcr=sd        192           4.17
borda=sd       165           3.59
no matches     28            .6

Five factions and 7 candidates:
matches        3563
sdindec        67            1.88
mpcr<>cw       27            .75    (1.08%)
mpcr=cw        2469          69.29
unanimous      250           7.01
mpcr=borda     66            1.85
mpcr=sd        306           8.58
borda=sd       254           7.12
no matches     124           3.48

15 candidates:
matches        431
sdindec        20            4.64
mpcr<>cw       6             1.39   (2.79%)
mpcr=cw        209           48.49
unanimous      33            7.65
mpcr=borda     9             2.08
mpcr=sd        41            9.51
borda=sd       57            13.22
no matches     56            12.99


I went through and picked out some appalling results from the 4-rank batch.
Here's one which would have been OK if the "merge the pair minimizing the sum
of viability" rule had been used.

This is pretty nasty because it elects the Condorcet and Borda Loser, and
ignores a majority favorite:

 56 22% CDBA
 92 37% CABD
 100 40% BDAC
 248

Borda order is B>C>D>A
RP order is C>B>D>A

:[viability order] [new grouping as result]  etc...
:BCDA C,D,BA  CA,B,D  B,DA,C
:CBAD C,DBA  CA,BD  BDA,C
:ACBD   (MPCR's result)

Using the earlier rule, the MPCR part goes:
:BCDA CD,B,A  CA,B,D  B,DA,C
:BCAD CD,BA  CA,BD  B,DAC
:CBAD

But in general it doesn't seem to me that one variant is clearly better than
the other.  (I did 9-candidate trials on both methods and found that the new
version was twice as likely to miss a CW, but it might've been a fluke.)

I hope no errors crept into this message.

Kevin Venzke
stepjak at yahoo.fr


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