I have never really bought into the idea of being afraid of competition. I think most peoples goals in life are to "work smart and not hard", I dont mean that people want to be lazy but I mean people would rather find the smartest way to accomplish their goals rather then the most difficult.
Put it into relation with a job. Say you have a path and want to be a CEO of a company and you are at company X right now with 5 people that would probably be "ahead of you" for that job that are all around your age, meaning that your chances are pretty slim, assuming you are all the same type of worker.
So then company Y offers you a position as asst CEO to a current CEO that is say, 60 years old. Does it make you scared of competition if you take that offer because it increases your odds of getting your goal?
You are right on the money. The notion of "afraid of competition" is really rather odd. You go where you are needed, can slot in and contribute, help the team, and meet your goals. If at University X there is one position "open" and the team has three guys waiting in line for the position (already on the roster) and two alongside you are being recruited --- and at University Y, there are two "openings" at your position, two guys waiting in line and one other is recruited along with you --- so long as the two universities are comparable in caliber of football/education... of course you would hedge towards University Y. That is not being afraid of competition... it is just smart and reasonable.