[EM] Unranked IRV versus Approval - where do winners diverge?

[Tom Ruen]

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