[EM] Bullet voting/truncation in Condorcet elections (was Re: NPV vs Condorcet)

[Bob Richard]

Kristofer Munsterhjelm wrote:

 > If we consider the votes as bullet votes, then we can expand to:

 > 45: Able > Baker = Charlie
 > 40: Baker > Able = Charlie
 > 15: Charlie > Able = Baker

 > which produces the matrix you gave above.

        Able     Baker    Charlie
        -------  -------  -------
Able        --       45       45
Baker       40       --       40
Charlie     15       15       --

OK, I was wrong when I said the cross-diagonal cells have to add up to 
100. This way of accounting for tied rankings dictates otherwise.

Suppose, instead, we treat tied rankings as a half a vote for each 
candidate:

22.5: Able > Bake > Charlie
22.5: Able > Charlie > Baker
20.0: Baker > Able > Charlie
20.0: Baker > Charlie > Able
 7.5: Charlie > Able > Baker
 7.5: Charlie > Baker > Able
 
        Able     Baker    Charlie
        -------  -------  -------
Able        --     52.5     65.0
Baker     47.5       --     62.5
Charlie   35.0     37.5       --

In another post in this thread, Raph Frank describes a third way of 
representing tied rankings using proportions. Using the example above 
instead of his example:

32.73: Able > Baker > Charlie
12.27: Able > Charlie > Baker
30.00: Baker > Able > Charlie
10.00: Baker > Charlie > Able
 7.94: Charlie > Able > Baker
 7.06: Charlie > Baker > Able

        Able     Baker    Charlie
        -------  -------  -------
Able        --    52.94    75.00
Baker    47.06       --    72.73
Charlie  25.00    27.27       --

In this example, Able is the Condorcet winner in all three matrices. 
Several questions:

(1) Is this true in general, for all possible profiles? If there's a 
Condorcet winner, is it always the same candidate no matter how you 
treat tied rankings?

(2) Are there profiles containing cycles for which different 
Condorcet-completion methods would give different winners depending on 
how the tied rankings are represented?

(3) Going back to Dave Ketchum's original proposal that different voting 
methods can be used in different subjurisdictions (e.g. states in the 
case of NPV) and the matrices added together, could the method of 
representing tied rankings ever affect the outcome in the jurisdiction 
as a whole? I haven't tried to work this out, but intuitively it seems 
to me that the answer is yes.

(4) I gather that Kristofer's procedure is the one most frequently used 
in discussions of Condorcet. Is that true, and what is the history or 
reasoning behind this?

Thanks,
Bob


Kristofer Munsterhjelm wrote:
> Bob Richard wrote:
>> I'm obviously missing something really, really basic here. Can 
>> someone explain to me what it is?
>>
>>  > Take it from the FPTP count and recount it
>>  > into the N*N array by Condorcet rules ...
>>
>> I still have no idea what this means. Here's an example:
>>
>> Plurality result:
>>   Able: 45
>>   Baker: 40
>>   Charlie: 15
>>
>> Here's a (very naive) NxN matrix (fixed-width font required):
>>
>>         Able     Baker    Charlie
>>         -------  -------  -------
>> Able        --       45       45
>> Baker       40       --       40
>> Charlie     15       15       --
>>
>> But it's not a Condorcet count because we have, for example, no idea 
>> how many of the Able voters prefer Baker to Charlie and how many 
>> prefer Charlie to Baker. As a result, the pairs of cells above and 
>> below the diagonal don't add up to 100. I still don't see how we can 
>> "recount it into the NxN matrix by Condorcet rules".
>>
>> Someone please show me the NxN matrix that Dave Ketchum would use to 
>> combine these votes with the other votes that had been cast on ranked 
>> ballots.
>
> If we consider the votes as bullet votes, then we can expand to:
>
> 45: Able > Baker = Charlie
> 40: Baker > Able = Charlie
> 15: Charlie > Able = Baker
>
> which produces the matrix you gave above.
>
> That's the "consider bullet voters" idea. The other one is to count 
> the plurality vote locally, so you get:
>
> 100: Able > Baker > Charlie
>
> which gives
>
>    A   B   C
> A  0 100 100
> B  0   0 100
> C  0   0   0
>
> and which could be used for any voting system. I think the first idea 
> is better, though.

-- 
Bob Richard
Marin Ranked Voting
P.O. Box 235
Kentfield, CA 94914-0235
415-256-9393
http://www.marinrankedvoting.org