The phrase you’re looking for is dog-eat-dog, meaning ruthless or competitive. Doggy dog, when used in the phrase doggy-dog world, is a mondegreen. It results from a mishearing of dog-eat-dog.
In all its uses—and it’s usually used as a phrasal adjective—dog-eat-dog should have hyphens, and there’s no need for quotation marks.
Examples
Here are some examples of dog-eat-dog used well:
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]
