I guess I should stop using analogies then.
The point isn't whether the players are competing with the DM. The point is that there's two people playing a game and one person can just screw over the other whenever they feel like it. Painting that in a competitive setting hits closer to home for a lot of people since they're more likely to have experienced that themselves. It wasn't meant to be indicative of how I perceive a good player/DM relationship.
I'm sorry, I had no idea it would confuse so many people so badly.