There is a place, near where I used to live, that looked like this - you see it, right there? It's a bowling green. Not the bowling you and I would do, the bowling that belongs to another age. Mostly the elderly.
There were, in fact, two near me - high amusement, I can tell you, since we came to the conclusion that one had decided it was a rival for the other. And that said other had no idea that it existed. That this perceived rivalry would fuel them entirely, even though the other lived in blissful ignorance of their existance.
It's one of those things which seems funny, but the more you think about it (alright, the more I think about it) the more upsetting it gets. We set ourselves these challenges, these goals - we decide on these rivalries, but are they real? Is there anything there? If I reach the finish line - actually, that's a bad example (well, it is for me). If I get a higher grade (better) than him, is he going to care? Of course not. He's thinking about that other guy, did he beat him, or he's not thinking about me at all and I'm not even on his radar.
The things we do to get noticed.
Of course the real solution is not to create any rivalry, perceived or otherwise, the real solution is just to please yourself, work to goals that mean something to you, not the rest of the world (...well, that depends on your job, but still).
The real solution is also a fairytale.