[EM] Unranked IRV versus Approval - divergent winners exist!

[Tom Ruen]

Well! Unranked-IRV does diverge from Approval voting, although perhaps only
barely!

Being fundamentally lazy, I did a computer search! :)

For a simple survey, I searched for ballots {A,B,C,AB,BC,AC} with counts
0..9 for each. (10^6=1 million elections)

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Example #1: (21 voters)
Approval (unranked) ballots: A=1, B=3, C=5, AB=7, BC=1, AC=4

Approval: A=12, B=11, C=10
A wins with 57% approval. (B loses with 52.4% approval)

Unranked IRV:
Round 1: A=6.5, B=7, C=7.5
Eliminate A
Round 2: B=10.5, C=9.5, NOTA=1
B wins with 50%.
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Example #2: (31 voters) (Minimized split votes (AB+BC+AC+1)/(A+B+C+1) with
divergent solutions)
Approval (unranked) ballots: A=6, B=8, C=9, AB=5, BC=0, AC=3

Approval: A=14, B=13, C=12
A wins with 45% approval. (B loses with 42% approval)

Unranked IRV:
Round 1: A=10, B=10.5, C=10.5
Eliminate A
Round 2: B=13, C=12, NOTA=6
B wins with 42%.
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Example #3: (13 voters) (Minimized voters (A+B+C+AB+BC+AC) with divergent
solutions)
Approval (unranked) ballots: A=0, B=2, C=3, AB=5, BC=0, AC=3

Approval: A=8, B=7, C=6
A wins with 61.5% approval. (B loses with 53.8% approval)

Unranked IRV:
Round 1: A=4, B=4.5, C=4.5
Eliminate A
Round 2: B=7, C=6, NOTA=0
B wins with 53.8%.

These are all clearly transitional cases. Perhaps good divergence examples
exist that are a little more decisive over elimination order. These are
enough for me for now.

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How often do these methods diverge statistically?

Searching all elections with {A, B, C, AB, BC, AC} ballots, each with ballot
frequencies between 0 and 9: (10^6 elections)
907932 elections agreed. (90.7%)
87526 elections had ambiguous ties. (8.7%)
4542 elections diverged (0.45%)

For ballot type counts from 0 to 12:  (13^6 elections)
4447603 agreed (92.1%)
347364 ambiguous (7.2%)
31842 diverged (0.66%)

For ballot type counts from 0 to 19: (20^6 elections)
60150214 agreed (94.0%)
3223602 ambiguous (5.0%)
626184 diverged (1.0%)

For ballot type counts from 0 to 29: (1/6 of 30^6 elections) (ordered
A>=B>=C)
126671362 agreed (94.5%)
5663600 ambiguous (4.2%)
1585038 diverged (1.2%)

Brute force is insufficient to tell us what fraction may diverge for
elections with many more voters. This progression suggests that as voters
increase and the ambiguous proportion is reduced, the divergent cases rise,
so perhaps there is a limit around 2-3% divergence for larger elections?
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Finally, I'll ask a bigger question than my original one (Unranked-IRV or
Plurality):

If we are limited to unranked ballots, for single seat, single ballot
elections, which method would you prefer: Plurality, Unranked-IRV or
Approval voting? (or some other method?)

Tom Ruen