The company's latest cell phone has no 'rival/rivals'. [=the company's newest cell phone is much better than all others available]
The company's paper is the 'rival' of any in the world. [=it is as good as any paper in the world]
He told a 'rival' version of the story. [=a version that disagrees with another version]
(..) I shall never reveal the best of you, Severus. Can you risk your life every day to protect the boy? ... He possesses no measurable talent, his arrogance rivals even that of his father and he seems to relish in his fame... - My father was a great man! - Your father was a swine! (..)2011 Harry Potter and the Deathly Hallows: Part 2