Egol's explained the true scope of what you're asking, but to give a specific response to your specific question:
No, just having one more incoming link to your page than your competitor has to his will never guarantee you'll outrank him.
Search engine algorithms use hundreds of signals when ranking a page, and the number of incoming links to that page is only one measure.(And it's more than just wht else is on the page, like code, keyword use etc)
Even if we just restrict ourselves to considering the links to a single page, additional issues like the authority of the sources of the incoming links plays a huge part. Not all links are of equal value.
But more importantly, things like the authority of the whole domain have huge impact as well. That's why SEOMoz goes out of it's way to compute scores for domain authority as well as page authority. A weak page on a strong site will frequently outrank a strong page on a weaker site. That's one of the big frustrations/challenges of nearly all small site owners.
Bottom line, neither Page nor Domain LRD metrics are sufficient in themselves for assessing the work ahead of you.
Make sense?
Paul