[EM] De-Cloned Kendall-tau Example

[Forest Simmons]

El lun., 7 de mar. de 2022 9:22 p. m., Forest Simmons <
forest.simmons21 at gmail.com> escribió:

> I appreciate very much Kevin Venzke's recent tour de force towards
> classification of methods by their criteria compliances ... finding seven
> 3-candidate, 4-faction profiles that distinguish all of the well known
> methods by their basic compliances.
>
> I see this as an opportunity to put the de-cloned Kendall-tau metric
> through its paces.
>
> First a simple example with factions suitable for hand computation:
>
> 4ABC3+3BCA+2CAB
>
> We need favorite and anti-favorite counts for the de-cloning:
>
> The respective favorite counts f, are 4, 3, and 2. The respective a
> anti-favorite counts f', are 3, 2, and 4.
>
> The respective costs of the basic decloned order swaps are ...
> AB to BA 4×2 from f(A)×f'(B)
> BA to AB 3×3 from f(B)×f'(A)
> BC to CB 3×4 from f(B)×f'(C)
> CB to BC 2×2 from f(C)×f'(B)
> CA to AC 2×3 from f(C)× f'(A)
> AC to CA 4×4 from f(A)×f'(C)
>
> The respective costs from faction ...
> ABC to BCA 8+16
> BCA to ABC 6+9
> [Round trip 39]
> BCA to CAB 12+9
> CAB to BCA 8+4
> [Round trip 33]
> CAB to ABC 6+4
> ABC to CAB 12+16
> [Round trip 38]
>

At this point we can insert the respective decloned Kemeny-Young costs:

Converting all ballots to ABC:
3×15+2×10=65
Converting all ballots to BCA:
4×24+2×12=120
Converting all ballots to CAB:
4×28+3×21=175

So the de-cloned K-Y finish order is ABC, the same as clone dependent K-Y
... not particularly interesting.

This method ignores the preference information implicit in the geometrc
proximities ... blissfully taking the ballot preferences at face value
despite the geometric evidence to the contrary.


So the distant pair removal method is a more interesting application of
de-cloned Kendall-tau.

>
> Our method is to remove pairs of ballots that are as far apart as possible
> until only one faction remains:
>
> ABC and BCA are the furthest apart ballots with a round trip distance of
> 39.
>
> Removing 3  ballots from each of these factions leaves ...
>
> ABC+2CAB
>
> After removing one ballot from each of these two remaining factions, we
> see that the winning faction is CAB.
>
> So this is one way to use the de-cloned Kendall-tau distance.
>
> To do Kevin's seven examples I will need some triple A batteries for my
> calculator.
>
> The most interesting part for me is that the geometrically derived
> preferences would be ...
>
> The 4 members of the ABC faction should geometrically prefer the CAB
> faction over the BCA faction because they are closer to it ... distance 38
> verses 39.
>
> The 3 members of the BCA faction should prefer the CAB faction over the
> ABC faction ... distance 33 versus 39.
>
> The two member CAB faction should prefer BCA to ABC .... distance 33 vs 39.
>
> So geometric preferences are
>
> 2 CAB>BCA>ABC
> 3 BCA>CAB>ABC
> 4 ABC>CAB>BCA
>
> The CAB faction is the Condorcet faction.
>
> In other words sincere preferences seem to be
>
> 4 A>C>B
> 3 B>C>A
> 2 C>B>A
>
> It looks like insincere or mistaken burials of C by A and A by C.
>
> Any other thoughts?
>
>
>
>
>
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