[EM] Manipulability definitions

[Kristofer Munsterhjelm]

I've been trying to find methods that are maximally resistant to
strategy, but I'm not sure how manipulability should be defined in the
case of ties.

My definition for non-tied elections is that a method, call it X, is
manipulable in election eA if X elects a candidate (A), but there exists
a way for the voters who prefer B to A to alter their ballots so that,
in the post-altering election (eB), B wins. The manipulability of X for
a certain number of voters and candidates is just the mean
manipulability for X over every election.

If A wins with certainty in eA, and B wins with certainty in eB, there's
no problem; X's manipulability in election eA is 1.

But if A wins in eA with positive probability (but not certainty),
and/or B wins in eB with positive probability, but not certainty, then
what should the manipulability measure be?

Suppose A's win probability in eA is 0.6, and B's win probability in eA
is 0.4; and in eB, B's win probability is 0.9, and A's win probability
is 0.1. Furthermore, say it's possible for B>A strategists to manipulate
eA and turn it into eB, but no other manipulation from eA is possible.
Then I see three possible ways to quantify the manipulability, plus
another pragmatic one:

i.The manipulability is 1, because there still exists a way to
manipulate eA in favor of some candidate so as to increase the win
probability of that candidate at the expense of some other candidate
with positive win probability in eA;

ii. The manipulability is 0.6, because the manipulation only ever has an
effect if A wins in eA, which happens with probability 0.6; if B wins,
there's no need to do any manipulation, and none has any effect;

iii. The manipulability is 0.9 - 0.6 = 0.3 because that's the benefit to
B by executing the strategy.

(iv. Whatever makes my optimization easier to do. Two of these measures
require integer programming to optimize for, and one can be done by
linear programming alone.)

The first definition could be called "count the number of manipulable
elections", the second "count the probability that the outcome can be
manipulated", and the third "count the expected benefit to manipulators".

What do you think would be the appropriate definition?