Dog-eat-dog is an idiomatic adjective meaning ruthless or competitive. Doggy dog, when used in the phrase doggy-dog world, is an eggcorn resulting from a mishearing of dog-eat-dog.
Examples
Business is business, and Hollywood is as dog-eat-dog as every other industry. [Reel Loop]
We are locked in the hermetic, dog-eat-dog world of the court, where every smile is false. [Guardian]
They think they can do anything they want, that it’s a dog-eat-dog market and all these sophisticated buyers know that. [CNN Money]
In the dog-eat-dog world of pickup truck one-upmanship, you certainly can beat a truckmaker while it’s down. [Chicago Tribune]
Well, I am sure doggy dog world came more for a song that Snoop Dog put out in 93. It is a classic rap album and dog eats dog world just didn’t flow as well as a doggy dog world in the song, so we have the birth of a confused nation. :)
http://en.wikipedia.org/wiki/Doggy_Dogg_World