To answer your question simply: It depends on the individual and the group.
Less simply:
Whether video or P&P, people game the way they live. Some people play cooperatively, some people don't.
I am of the opinion that D&D plays best when all of the players are of the cooperative variety. Both in and out of character.
When it comes to the DM, I think is is all about making sure it is challenging, but not overwhelming. Using Gygaxian (1HP short of TPK = good) logic for major or final bosses, and the DM's dirty little secret (you're on the players side) logic the rest of the time.
I will admit that there is a lot of potential in the concept of in-party conflict, where team members compete with on another. But in my experience, anything more than the Legolas vs Gimli kill-count quickly ruins the game, instead of enriching it. Some players can't even take the kill-count type of competition in good fun.