[EM] GB 2.2 vs. "The IRV Nightmare"

[John B. Hodges]

(JBH) For reference, I will copy: From Adam Tarr:
>the nightmare scenario of IRV:
>
>10% FarRight>Right>Centrist>Left>FarLeft   [call this group "A""- JBH]
>10% Right>FarRight>Centrist>Left>FarLeft   [B]
>15% Right>Centrist>FarRight>Left>FarLeft   [C]
>16% Centrist>Right>Left>FarRight>FarLeft   [D]
>15% Centrist>Left>Right>FarLeft>FarRight   [E]
>13% Left>Centrist>FarLeft>Right>FarRight   [F]
>11% Left>FarLeft>Centrist>Right>FarRight   [G]
>10% FarLeft>Left>Centrist>Right>FarRight   [H]
>
>Centrist has the most first place votes, the most second place votes, and
>the most third place votes.  Centrist is the only candidate who does not
>appear fourth or fifth on any ballot.  Centrist would win in a landslide
>over any other candidate in a two-way race.  Centrist is quite obviously
>the popular choice by ANY reasonable measure.
>
>In Condorcet, plurality, top two runoff, or really any reasonable method,
>Centrist wins.  But in IRV, Centrist is eliminated before the final runoff,
>and Right wins in a squeaker.

(JBH) I gave commentary on this example August 3. The example simply 
assumes the existence of a large center party, and shows how it can 
be squeezed out under IRV. IMHO it shows that large center parties 
are unlikely to form in a country that uses IRV as its primary voting 
method.

To remain viable, a pure center party has to stay larger than at 
least one wing, taken as a coalition. It has to poach voters from at 
least one wing; it cannot allow itself passively to be poached upon. 
In trying to poach from either wing it may lose some voters to the 
opposite wing; probably the equilibrium solution involves two parties 
glaring at each other over the 50-yard line, each fighting also to 
guard their rear from additional parties. "Wing" parties may win if 
Center-wing and Wing together make a majority but Center-wing allows 
itself to get too centrist. There will always be at least two 
distinctly different parties.

The effect of IRV, in contrast to Plurality, is to add new sets of 
"swing" voters, between center-right and far-right, and between 
center-left and far-left. Center-left and Center-right may still win 
most (or all) of the elections, but they cannot just fight over the 
center swing voters and ignore their "base" voters; they have to keep 
the wing parties from growing too large, so they have to work to 
appeal to "swing" voters on their wings as well. They have to 
actively co-opt the issues of the wing, and the wing has an organized 
party to present those issues to the voters.
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Today, for exercise, I consider what would result from using 
Generalized Bucklin 2.2 with this example. Assume 1000 voters. #seats 
= 1, Droop quota = 500

First Round: 100 FarRight, 250 Right, 310 Centrist, 240 Left, 100 
FarLeft. Nobody exceeds Winning Threshold with first-rank votes. 
Second-rank votes added to the tallies:
200 FarRight, 510 Right, 590 Centrist, 490 Left, 210 FarLeft. Two 
candidates exceed WT, seat awarded to candidate with the largest 
tally.

Centrist wins.
---------------------------

For more exercise, assume the same set of voters, with four seats to 
be filled instead of one. 1000 ballots, S=4, WT= DQ= 200

First Round: Four candidates exceed the WT, seat awarded to Centrist. 
Tallies of groups D and E are multiplied by R= (310-200)/(310) = 
0.3548387, so now
A = 100 ballots  B= 100  C= 150  D= 56.774192  E= 53.225805 F= 130 G= 
110  H= 100

Second Round. #ballots= 800  #seats= 3  WT= Droop Quota= 200
First-rank tallies are 100 FarRight, 306.774192 Right, 293.225805 
Left, 100 FarLeft
Two candidates exceed the WT, Right is awarded a seat. Tallies of 
groups B, C, and D are multiplied by R= 0.3480546, so now A= 100, B= 
34.80546,  C= 52.20819,  D= 19.760518, E= 53.225805,  F= 130,  G= 
110,  H= 100

Third Round. #ballots= 600  #seats= 2  WT= Droop Quota= 200
First-rank tallies are 187.01365 FarRight,  312.98632 Left,  100 FarLeft
Left is awarded a seat. Tallies of groups D, E, F, and G are 
multiplied by R= 0.3609944
So now A= 100,  B= 34.80546,  C= 52.20819,  D= 7.1334363,  E= 
19.214217,  F= 46.929272,  G= 39.709384,  H= 100

Fourth Round.  #ballots= 400  #seats= 1  WT= DQ= 200
First-rank tallies are 194.14708 FarRight,  205.85287 Farleft

FarLeft wins! Interesting. Group D was part of the original Center 
support, they contributed to the victories of Center, Right, and 
Left, so they got reduced three times. Group E, by contrast, voted 
for Left on the second round, so did NOT get reduced on that round. E 
favors Farleft over FarRight, so on the last round, they contributed 
the winning margin to FarLeft.

I also went through this example using the alternative Winning 
Threshold that I proposed, WT = HQ - (1/S)(HQ-DQ). The result was the 
same, though Group E was no longer decisive by itself.
---------------------------

For comparison, What happens under Single Transferable Vote, with 
four seats to be filled?

Surprise! EXACTLY the same as the above. Because all victories are 
with first-rank votes only, and surplus votes are transferred in the 
same way under the two systems, and there is always a winner after 
the surplus transfers, NOBODY NEEDS TO BE ELIMINATED under STV in 
this example. The difference between GB 2.2 and STV lies in what is 
done if there IS no winner with first-rank votes and transfers alone. 
GB looks to the next-ranked votes, STV eliminates the last-place 
candidate to generate more transfer votes.
---------------------------

SO: in this "IRV Nightmare" scenario, GB 2.2 performs better in the 
single-seat case, and the same in the 4-seat case.

Is Centrist, Right, Left, Farleft the "correct" answer in the four-seat case?
If not, does this show that GB 2.2 fails monotonicity, or some other 
desirable criterion?
Comments invited.
-- 
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John B. Hodges, jbhodges@  @usit.net
Do Justice, Love Mercy, and Be Irreverent.