[EM] Fwd: Some chance for consensus revisited: Most simple solution

[Raph Frank]

On Tue, Feb 10, 2009 at 10:53 PM, Jobst Heitzig <heitzig-j at web.de> wrote:
> For example, consider our example in which the true utilities are
>   55: A(100) > C(70) >> B(0)
>   45: B(100) > C(70) >> A(0).
> With the suggested method, the resulting approval score of C will not be 100
> (which would be desirable and would be the result under D2MAC, e.g.) but
> would rather be only 56. This is because if it would be larger than 56, any
> voter who approves of C will have an incentive to not approve of C as this
> would transfer some probability from C to her favourite.

I don't think that is correct.

Assuming that only 1 of B's faction compromises, the votes would be

55: A*, C
1: B*, C
44: B*

C wins round 1 and C's pile becomes
55: A* + C
1: B*+C

The 50% rule kicks in as the A* ballots make up more than 50%.  54 of
the A*+C ballots are removed from the pile to balance it.

This is converted to
1: A*+C
1: B*+C

The remaining ballots are
54: A*+C
44: B*

These just form 2 piles as there is no common candidates between the 2
sets of ballots.  The result is
A: 54%
B: 44%
C: 2%

The utility for faction B is:
0.54*0 + 0.44*100 + 0.02*70 = 45.4

Assume that 2 of them compromise, the votes become

55: A*, C
2: B*, C
43: B*

C's pile becomes
55: A* + C
2 B*+C

This is converted to
2: A*+C
2: B*+C

Remaining
53: A*+C
43: B*

These just form 2 piles, so the result is
A: 53%
B: 43%
C: 4%

The utility for faction B is:
0.53*0 + 0.43*100 + 0.04*70 = 45.8

This is an improvement for faction B.  Thus there is an incentive to
compromise for the B faction.

In general, assume that x of them compromise, the votes become

55: A*, C
x: B*, C
45-x: B*

C wins with 55+x approvals.

C's pile becomes
55: A* + C
x B*+C

This is converted to
x: A*+C
x: B*+C

Remaining
55-x: A*+C
45-x: B*

These just form 2 piles, so the result is
A: 55-x%
B: 45-x%
C: 2*x%

The utility for faction B is:
(55-x)*0 + (45-x)*100 + (2*x)*70 = 4500 - 100*x + 140*x = 4500 +40*x

Thus, it is in the interests of the B faction to set x as high as possible.

The effect is that all voters in both factions will compromise.

Ofc, that doesn't result in 100% probability though.
C: 90%
A: 10%

Also, the last 10 voters from A could decide not to also approve C
without any effect.