[EM] Why Pab > Pbc

[MIKE OSSIPOFF]

We're interested in the probability that voting B over C will change
the winner from C to B. That's what I mean by Pbc, for the purposes of
this problem.

Examining the ways that voting for A can change Pbc shows that voting
for A can only make it less likely that voting B over C will change the
winner from C to B:

Say B & C are tied before we vote for B, making B win. Say our vote
for A promotes A to the vote total of the tied B & C. Without that A
vote, and without our vote for B, it's 50% that C wins. And so
it's 50% that our vote for B over A changes the winner from C to B.

But say we vote for A & thereby promote A to the same vote total as
that initial BC tie. Now, without our vote for B over C, it's 1/3
that C wins. And so it's only 1/3 that our vote for B over C changes
the winner from C to B.

Say that, instead, our vote for A promotes A from the level at which
B & C are tied before I vote for B over C, to the level 1 vote above
that. In that case, the winner before I casts my B vote is A, and
there's zero probability that doing so changes the winner from C to B.
It might change it from A to B, but not from C to B.

Or maybe A & B are one vote below C. My vote for B changes that to a
BC tie. I change the winner from C to B with a probability of 1/2.

Say, instead that I've voted for A too, so now I'm changing an AC
tie into an ABC tie. There's 1/2 probability that C was the winner
before I voted B over C, and there's only 1/3 probability that B is
the winner after I vote B over C.

This can be summarized by pointing out that wherever I make A's vote
total arrive at, to make some tie, it has one effect: It reduces the
likelihood that some particular other member of that tie wins. Changing
the winner from C to B requires that first C win and then B win.
Making A arrive where one of those is can only have the effect of making
it less likely for one of those to win, either before or after I
vote B over C, or both.

And of course if my promotion of A makes it the initial winner, then
too I'm not changing the winner from C to anything, because C isn't
the winner before I vote for B. A is the winner before I vote for B.

So the only thing that A can do when it arrives is reduce the
probability of B or C winning, and that reduces Pbc overall.

Mike Ossipoff



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