[EM] Unranked IRV versus Approval - where do winners diverge?
Tom Ruen
tomruen at itascacg.com
Wed Mar 28 00:40:54 PST 2001
What if Unranked-IRV is equivalent to Approval (for single seat elections)?!
My intuition offers a sweet reply, although I may be wrong.
It can't really be! But where is a counter-example? I would think if it is
false, a counter example would exist for 3 candidates.
Given 6 types of approval ballots: {A, B, C, AB, BC, AC} with counts of
{a,b,c,x,y,z}, find an example where Approval and Unranked-IRV produce
different results.
Approval:
Ta(A)=a+x+z
Ta(B)=b+x+y
Ta(C)=c+y+z
Assume Ta(A)>Ta(B) and Ta(A)>Ta(C) (A wins)
Unranked-IRV:
Round 1:
T1(A)=a+(x+z)/2
T1(B)=b+(x+y)/2
T1(C)=c+(y+z)/2
Assume T1(C)<T1(A) and T1(C)<T1(B) (C weakest)
Round 2:
T2(A)=a+x/2+z
T2(B)=b+x/2+y
Find {a,b,c,x,y,z} such that Ta(A)>Ta(B) [A wins] and T2(A)<T2(B) [B wins].
If these methods are equivalent, then I'm silly, but it would mean Approval
is much sneakier than I thought!
Are there any quick minds out there to help me prove or disprove this
hypothesis? :)
Tom Ruen
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