Answer
Jim, the lighter fellow, will lose.
Work Step by Step
Consider Bob, Jim, and the rope as a system. Because there is no net external force on the system and the system is initially stationary, the center of mass (CM) does not move. The CM starts out closer to Bob (because he is more massive), i.e., the CM is on Bob’s side of the midline.
The CM doesn’t move, so as the men pull, they will eventually meet at the center of mass. Since the CM is on Bob’s side, Jim will have crossed the midline to get there.
This problem may also be analyzed using Newton’s third and second laws.
