The best way to find out why they are competitive is to actually talk to them. I think it is better if you do it in two steps:
- Have a team meeting with everybody. Tell them what you observed, and that you are unhappy with the situation. Also admit to them that you do not understand why these things happen, and that there might be a good reason for it. Let them explain whatever they want / need, BEFORE you try to fix anything. Based on the feedback, you might want to try asking them to find a solution which works both for them and for you.
- If step 1 fails (no feedback, or unusable feedback), then organize F2F meetings with each one of them. In this way, they will be released from the pressure of being judged by their colleagues.
The ground rules:
- make it clear from the beginning that the discussion will not be taken into consideration for job evaluations. It will be as if it did not happen;
- get as much information as possible BEFORE you start fixing anything;
- personally, you cannot fix anything; if the team does not fix their issues, no other effort will help; forcing things will result in people leaving the boat;
- the issue might take time to get solved, even if the process starts immediately;
A good read in your case is to read more about teams and teams development.
The rules of feedback are essential also.
@Meg made a very useful comment:
If anyone came to the workplace SE with the question, "My boss asked for a meeting and said that it wouldn't be taken into consideration for later evaluations, can I actually be candid?" I would wager that most of the top answers would be along the lines of -ITS A TRAP-, so don't be surprised if team members are hesitant to give feedback.
That is very true, that a relevant number of managers promise one thing and do another. However, some of them are not evil. How do we know which manager is evil and which one is not? Only experience will tell. And yes, I had the experience when managers said that the feedback was more like a team-building / training session, only to find out that all the information collected appeared on the personal evaluations at the end of the year.
If a manager asks for feedback without specifying the purpose of the asking, then there is a good chance that the feedback shall be used during evaluation. But the test comes when he promises not to use the feedback. If the indeed keeps his promise, then you know you can build trust with that manager.
Of course, it is always a double-edged sword. It is always the question of who was the first, the chicken or the egg. But somebody must take a leap of faith to try to make the world better.
Again of course, as a team member, do not spill out everything from the beginning . But if the manager keeps being the nice guy, you can more and more open, for the mutual benefit.