[EM] Claim justification: Symmetrical ICT, unimproved Condorcet, ZLNHe.

[Michael Ossipoff]

Let me tell some justification of my claim that Symmetrical ICT meets
Strong ZLNHe, and that traditional unimproved Condorcet (TUC) fails
ordinary ZLNHe:

Why Symmetrical ICT (SITC) meets Strong ZLNHe:

In SICT, bottom-voting X and Y (typically done by not ranking them)
counts as a vote for some (either) one of {X,Y} beating the other.
Whichever one could be made to beat the other, your bottom-voting of X
and Y counts toward that pairwise beaten-ness. In the event that
SICT's beat-condition rule says that both X and Y beat eachother, then
SICT says that the one that beats the other is the one ranked over the
other on more ballots than vice-versa.

So, as I said, not ranking X and Y counts as a vote fdor some (either)
one of {X,Y} beating the other. Whichever one could be made to beat
the other.

You're voting for X>Y, and you're voting for Y>X.

Now, what if you rank X, but not Y? You're then only voting for X>Y.
You're voting for one pairwise defeat among {X,Y}, instead of two.
You're thereby increasing the probability that the winner will come
from {X,Y}.

Sure, suppose you knew for a fact that X was going to be beaten by
someone other than Y, and that Y is the candidate who could be
unbeaten and win. Then voting X>Y is much more important than voting
Y>X. There wouldn't be much point in voting Y>X, because X is already
going to be beaten by someone else. Furthermore, there'd be an
advantage to voting X>Y, instead of not ranking either: By helping the
number of X>Y ballots be greater than the number of Y>X ballots you're
voting for X being the one that beats the other, in the event of both
beating eachother according to SICT's beat-condition rule.

So, if you knew that, then it would be better to rank X and not Y. But
ZLNHe and Strong ZLNHe are about a 0-info election.

So what applies to the situation is what I said earlier, supporting
the conclusion that ranking one of {X,Y} must increase the probability
that the winner will come from {X,Y}.

Why traditional unimproved Condorcet (TUC) fails ordinary ZLNHe:

In a large official public election, pairwise ties are vanishingly
unlikely and rare. One of {X,Y} is going to beat the other.

But, even aside from that, if you vote X over Y, you're voting for
X>Y, but you're also voting against Y>X.

For both of those reasons, as regards beatenness of X and Y, you gain
nothing and lose nothing by voting X over Y.

But, as regards the votes-against, and the margin of defeat, in an X>Y
defeat, you're increasing it when you vote X>Y (while at the same
time, voting for X to beat Y).

Therefore, by voting X>Y, you're decreasing the probability that the
winner will come from {X,Y}.

[end of demonstration that SICT meets Strong ZLNHe and that TUC fails
ordinary ZLNHe]

You might say, how is Strong ZLNHe better than ordinary ZLNHe? Well,
suppose that there were a little not-too-reliable information
suggesting something about beatenness of X and Y by other candidates.
That could tend to make a tendency for some strategic incentive to
rank one and not the other. But the stronger benefit of not ranking
either, in a Strong ZLNHe complying method does more to outweigh that
tendency.

Mike Ossipoff