[EM] Getting support levels and a social ordering from DAC/DSC

[Kristofer Munsterhjelm]

> Suppose X is the winner of DAC/DSC by the ordinary rules. Since X is the 
> winner, and not Y, that means that the only non-X coalitions we 
> encountered were ones that would have emptied the eligibles set. Suppose 
> the coalition that reduced the eligibles set to X had support s_X. Then 
> there exists some subset of coalitions, all with support greater than or 
> equal to s_X, that reduced the eligibles set to X. However, there does 
> not, for any other candidate Y, exist a subset of coalitions of support 
> greater than s_X that reduces the eligibles set to Y; if there were, 
> then Y would have been the winner, not X. So any other candidate Y has 
> support level at most s_X. Hence X is also a winner using the 
> per-candidate procedure above.
> 
> If there's no way of tiebreaking equal support coalitions that would 
> cause the DAC/DSC winner by ordinary rules to be Y, then there exists no 
> subset of coalitions of support >= s_X that reduces the eligibles set to 
> Y, and X is the unique winner using the per-candidate procedure above.

Now that I think about it, that could well be wrong. Consider something like

100: ABC
80: BC
20: A
19: B
10: C

The standard winner is B (intersect ABC, intersect BC, skip A, intersect 
B), but the support-level check will falsely say A has support 20 here 
against B's 19.

That's unfortunate. Is there any way to salvage the reasoning? It 
doesn't seem obvious what A's score should be in the social ordering.