My wife and I have nine children, and one of them has a little bit of that symptom. He is 10 years old, and is a good competitive distance runner - 19:47 in the 5K on the road, 5:42 mile on the track, district champion in the 1500 meters in his age group. He is extremely competitive and is constantly trying to run his siblings into the ground in training runs that are supposed to be easy.
What I found is that such a competitive drive is both a strength and a weakness, and with proper approach the strengths can be magnified while the weaknesses mitigated. I have seen my son through his competitiveness find a gear at the end of a race that I did not think was there, and that makes all of the headaches of trying to tell to him to stop creating a race out of an easy run every 100 meters of it over the course of 4 miles worth it.
What I do with my children is look at what they appear to be capable of and then give them a reasonable challenge. Find the most desired reasonable reward that they would like, and make a deal with them that they are getting it once they have met the challenge. In the process they will fail a number of times before they succeed. Each time they fail, analyze with them why, and come up with a plan to do better next time.
Our goals are usually running times. For a competitive child this redirects the focus from beating an opponent who may or may not be good, to beating the clock which is constant and predictable. He learns that if he practices sound training and racing principles he will have good results. Competitors are his friends, not his foes - they help him run a faster time.
You should not expect perfection with regard to toning the excessive competitive drive down, but with a solid consistent effort and a lot of patience you should be able to make it manageable.