[EM] Symmetric ICT reformulation and exploration

[Forest Simmons]

Here's my reformulation of Symmetric ICT:

Ballots are ranked preference style with equal rankings allowed, especially
at top and bottom, with bottom as default.

Let the notation [phi(X,Y)] mean the number of ballots on which the formula
phi(X,Y) is satisfied, so that, for example, [X=Y=Top] is the number of
ballots on which X and Y are ranked equal Top.

Definition:
X strongly beats Y
       if and only if
[X>Y] - [Y>X] > max(0, [X=Y=Top] - [X=Y=Bottom])

Step 1.  Eliminate all strongly beaten candidates unless that would
eliminate all candidates.
Step 2.  Among the candidates remaining after step 1, elect the candidate
ranked Top on the greatest number of ballots.

Is that an accurate restatement of Symmetric ICT?

If it is I would like to make one suggestion to make it even more
symmetrical, namely ...

Step 2'.  Among the candidates remaining after step 1, elect the candidate
X with the greatest value of  [X=Top] - [X=Bottom].

I don't think it will worsen the Later-no-help failure.

Of course, then we would have to change the name to something like

IC(TmB), Improved Condorcet (Top minus Bottom).

Does this comply with CD?

45 C
30 A>B
25 B>A

Candidate A is elected as the only unbeaten candidate.

If the B>A faction defects by truncating A ...

45 C
30 A>B
25 B

then no candidate is unbeaten.  The respective Top minus Bottom scores for
the respective candidates A, B, and C, are

30 - 70 = -40
25 - 45 = -20, and
45 - 55 = -10,

so C is elected.

However, if we reduce C's faction slightly and increase B's faction, we can
produce a CD failure:

41 C
30 A>B
29 B

The Top - Bottom scores are
30 - 70 = -40
29 - 41 = -12
41 - 59 = -18,
so the B faction benefits from its defection.

So the ICT version appears to be better even though it is not quite as
symmetrical in its definition as IC(T-B).
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