My family can be competitive. I'm not always sure about how I feel about it. I like competition because it provides a measurable objective that I can contribute to. I like competition because it can get people to do their best. It can get people to engage. But competition can easily turn sour. As soon as competition becomes divisive or puts people down, then I think competition has gotten out of hand.
There isn't a lot that jumps to mind about competition specifically in Scripture. There's more about how a person, whether in competition or not, should live. That said, there must have been a competitive streak within some of the disciples. They started asking about who was the greatest among them? They wanted to know how the stood in comparison to the others? Where were they in the ladder? Jesus doesn't answer their question directly. He provides no rankings. He says though that the greatest among them is the servant (passage). Effectively, he says: "So you want to win? To win in the Kingdom is to get down and serve. It's to put others before yourself."
Given Jesus' definition of greatness, I have a hard time imagining him being domineering as we see among people today. He would probably have a parable or some insightful and/or cutting questions about the heart to ask like: "Would you rather win or ensure that everyone has a good time? Would you rather get a victory, or ensure that everyone feels encouraged and empowered? Do you genuinely applaud and praise your opponent when he or she does well? Do you let 'love your enemies' apply to competition?"
Optimizing for winning is quite easy. You get as many good people on you team and have the best one's play the most amount of time. Challenge and real success to me comes in taking a team, weak links and all, and working together in a way where everyone's strengths are utilized, and everyone feels encouraged to do their best.
I know there has been times for me where I have put competition above people. I fortunately have a sensitive wife who picks up on these things quicker than I do. She's helped me learn how I can do this. I'm getting better, and I hope that competition would never get in the way of loving people. Should they ever be at odds, I want love lo reign. Competition can take the back seat...