Consider two independent sequences of travelers arriving at
opposite ends of a one-lane shared pathway. Each traveler
attempts to traverse the entire pathway to the opposite
end. An attempt fails if the traveler collides with an
opposing traveler. In a collision, both opposing travelers are
annihilated. We study the probability that a traveler manages
to traverse the entire length of the one-lane shared pathway
unobstructed. The dynamics of the travelers include the
possibility of acting as bodyguards and "running interference"
for a more recent arrival traveling in the same
direction. This model was developed to address some questions
in the theory of crystal growth. It may have possible
applications in Particle Physics as well as to traffic at a
one-lane bridge. This paper develops some properties of the
model while focusing on the probability that a traveler
crosses the entire pathway without interference.