Another instalment on my research-life stories.
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The Talmud says: “competition/envy among scholars increases wisdom” (kinat sofrim tarbe chochma). Good or bad, competition is here to stay. Nevertheless, one of the strengths of our community is in its collaborative nature. This is good for science, but in my eyes also makes our life so much better. A recent example is a research project with Guy Rothblum. For a few weeks, we met quite regularly and every meeting went more or less as follows: First, we would go over the solution from the previous meeting and find a bug. Then we would work together on a new and improved solution. This sounds frustrating (and would probably have been frustrating if I worked alone), but instead it was a great joy. We got to solve this problem again and again, and in the process enjoy each other’s creativity and company. Unfortunately, our current solution seems quite robust, so our fun ritual ended.
My best example for turning competition into collaboration is in my long-term collaboration with Salil Vadhan. It started when Ran Raz and I had a modest result on Randomness Extractors (following the breakthrough work of Luca Trevisan). We then learned that Salil had the same result, and already managed to write it down. Salil invited us to join (and I’m sure he was a bit sad to lose his first single-authored paper), on the other hand, Ran and I decided to decline and give up on the result altogether (and I was sad to lose a paper at this early stage of my career). In retrospect, losing that result would have been quite inconsequential, and similarly for Salil. But what did turn out to be extremely significant was what happened next. The three of us started collaborating together, leading to a stronger paper and then an additional collaboration, and before long Salil and I established not only a long-term research collaboration but also a great friendship. The unfortunate accident turned out to be most fortunate after all! Not all collaborations end up so fruitful, but I almost never regretted a collaboration (DBLP gives me 74 coauthors so this is a large sample). I hope that the set of collaborators that regret working with me is equally small.
So let’s all choose collaboration over competition and happily ride into the sunset. Right? Well, not so fast. Collaboration and competition are not mutually exclusive. Turns out, we cannot shut down our egos even when we enter a collaboration. While I strongly believe that the contributions to a collaboration cannot be attributed to any one of the contributors, we all like to feel that we contributed our fair share and that we demonstrated our worth (to others and more importantly to ourselves). An over-competitive collaboration can be destructive, but in moderation it could indeed be that competition among scholars does increase wisdom.
Windows On Theory 
Hear, hear! The “misfortune” of you and Ran getting the same result as me was a pivotal event in my research life, and the collaboration and friendship that followed has been a source of much happiness for me. I often tell this story to students who are distressed when they find themselves in a similar situation. Research is much more fun when done with others, and, like you say, healthy “competition” among collaborators can help draw the best of out each person (much like having a training partner in sports).