[EM] The general form of Quick Runoff

[Kevin Venzke]

Hi Chris,

--- En date de : Lun 24.5.10, C.Benham <cbenhamau at yahoo.com.au> a écrit :
>>But in contrast to e.g. SPST, a small center candidate can't be so easily
>>sunk by the presence of a stronger but futile extremist candidate. So
>>I do see improvement here.
>
>I'd be interested in seeing a 3-candidate example where it gives a >different result from SPST.

That's an easy one:

40 A>?
35 B
25 C>B

SPST elects A (B and C needed to unite) and QR elects B or C depending
upon the second preference of the A voters.

This works vs DSC also (DSC and SPST are quite similar with three
candidates).

Above I was talking more about a four-candidate scenario which would
look something like this:

40 A>C
35 B>C
15 D>B
10 C>B

Here SPST only elects A: Strongest pair is {BD} as C is disqualified, and
A has far more FPs than this (so-called) coalition.

QR at least picks B, like IRV. In IRV B mops up C/D votes and in QR B
defeats A and does not lose to D.

DSC picks C, which is quite good: {BC} is the strongest coalition and
{AC} places second.

MMPO should be a BC tie.

So why not just use DSC. Here are some numbers from my method comparer.
They are three-candidate elections with up to nine ballot types (which
depends on the scenario). The first row will be QR2's similarities to
the other methods (as percentage of scenarios where there was any
disagreement among these methods). Then the remaining rows will show
the criterion violations (Condorcet, MD, Plurality, and electing a
candidate with a majority defeat).

QR2 is the version that uses majorities all the time. In QR1 the final
contest is an elimination/transfer with no majority requirement.


scen 1 (all 9 ballot types, no size bias):
qr2 67.4  48.6  35.1  37.8  69.2  94.6  0

fpp 75.056  3.172  0  37.456
irv 11.376  .204  0  7.104
mmpo 8.14  0  .172  .832
schwv 0  0  0  .832
dsc 73.156  2.128  0  34.764
qr1 47.876  .524  .036  8.04
qr2 49.012  .492  0  5.512

Here QR is between IRV and DSC in Condorcet-efficiency: DSC is about as
bad as FPP. QR fails MD more often than IRV but not as often as DSC or
(worst) FPP. I won't worry about the Plurality column since only MMPO
and QR1 fail it. Finally look at elections of majority-defeated candidates
and see that QR2 was better than IRV and much better than DSC which was
almost as bad as FPP.

Morals of the story: DSC is terrible with truncation and QR is good
wrt majority defeats even when it is mediocre at Condorcet.


scen 2 (same but no wrap: no more A>C C>A ballots):
qr2 64.6  58.7  31.1  37.8  68.4  96.5  0

fpp 89.27  15.328  0  44.9
irv 25.212  1.304  0  9.944
mmpo 7.044  0  .616  0
schwv 0  0  0  0
dsc 84.128  9.668  0  38.04
qr1 55.26  3.632  .024  9.54
qr2 57.776  3.632  0  9.54

This scenario takes place on a 1D spectrum, but still no center squeeze.
The situation is roughly the same.


scen 3 (ends truncation too, so only AB BA BC CB):
qr2 24.8  87.6  75.1  75.1  50.1  100  0

fpp 100  24.664  0  100
irv 37.176  0  0  37.176
mmpo 0  0  0  0
schwv 0  0  0  0
dsc 49.948  0  0  49.948
qr1 24.812  0  0  24.812
qr2 24.812  0  0  24.812

Now QR is the most Condorcet-efficient method besides MMPO and WV. IRV
is *still* better than DSC. We have to do one more thing for DSC to
prevail.


scen 4 (same but adds spread: B ballot types are shrunk in quantity):
qr2 52.8  84.1  47.1  47.1  52.7  100  0

fpp 100  18.112  0  100
irv 68.732  0  0  68.732
mmpo 0  0  0  0
schwv 0  0  0  0
dsc 50.5119  0  0  50.5119
qr1 52.88  0  0  52.88
qr2 52.88  0  0  52.88

Now DSC is slightly better than QR and by a bit more, IRV. If I advocate
DSC or SPST then this is the scenario I'm hoping for.


scen 5 (adds truncation ballot types back to this):
qr2 86.6  74.3  8.3  37.1  83.9  98.5  0

fpp 60.664  10.16  0  32.316
irv 34.524  2.308  0  20.756
mmpo 36.14  0  6.004  0
schwv 0  0  0  0
dsc 57.644  7.164  0  27.312
qr1 48.716  3.336  .008  18.94
qr2 49.28  3.336  0  18.94

In other words this scenario retains center-squeeze and one-dimensionality
but now voters may truncate. DSC goes back to being the worst and IRV
the best, even better than MMPO at Condorcet (which is suffering from 
Plurality failures, I would assume).


scen 6 (no truncation, no spread: AB AC BA BC CA CB types):
qr2 16.6  78.9  70.2  70.2  33.3  100  0

fpp 78.156  5.816  0  100
irv 15.88  0  0  37.724
mmpo 0  0  0  21.844
schwv 0  0  0  21.844
dsc 53.124  0  0  74.968
qr1 9.972  0  0  31.816
qr2 9.972  0  0  31.816

This is a scenario (symmetric on two-dimensions or perhaps non-partisan)
that concerns me about DSC and SPST: Even though nobody is truncating,
it's pretty awful with Condorcet and electing candidates with majority
losses. QR is the best except for MMPO. Whenever DSC does terribly here I
believe the result in practice would be FPP-style voting.


scen 7 (contrived: four of the ballot types used at random):
qr2 37.6  79.9  54.7  55.8  53.6  94.7  0

fpp 62.844  56.788  0  94.58
irv 20.356  18.652  0  49.064
mmpo 4.372  0  5.636  8.916
schwv 0  0  0  8.916
dsc 45.48  37.796  0  75.532
qr1 16.124  20.444  2.156  40.692
qr2 17.808  18.6  0  38.848

And here too QR is the best at Condorcet and majority losses, except for
MMPO, ahead of IRV by a little.

Kevin