  In the context of computational social choice, we study voting methods that
assign a set of winners to each profile of voter preferences. A voting method
satisfies the property of positive involvement (PI) if for any election in
which a candidate x would be among the winners, adding another voter to the
election who ranks x first does not cause x to lose. Surprisingly, a number of
standard voting methods violate this natural property. In this paper, we
investigate different ways of measuring the extent to which a voting method
violates PI, using computer simulations. We consider the probability (under
different probability models for preferences) of PI violations in randomly
drawn profiles vs. profile-coalition pairs (involving coalitions of different
sizes). We argue that in order to choose between a voting method that satisfies
PI and one that does not, we should consider the probability of PI violation
conditional on the voting methods choosing different winners. We should also
relativize the probability of PI violation to what we call voter potency, the
probability that a voter causes a candidate to lose. Although absolute
frequencies of PI violations may be low, after this conditioning and
relativization, we see that under certain voting methods that violate PI, much
of a voter's potency is turned against them - in particular, against their
desire to see their favorite candidate elected.
