Nvidia didn’t want to do business with Sony for the PS4
While AMD is really happy with their successful business partnerships with Sony and Nintendo (and potentially Microsoft) for providing the major components of their respective home consoles, Nvidia has another outlook on the situation altogether. Speaking to GameSpot, the Senior VP of content and technology at Nvidia, Tony Tamasi, said that, "Having been through the original Xbox and PS3, we understand the economics of [console development] and the tradeoffs."
"I'm sure there was a negotiation that went on," said Tamasi, "and we came to the conclusion that we didn't want to do the business at the price those guys were willing to pay."
Nvidia's position on working with console manufacturers is that the opportunity cost would be putting another part of their business on hold. They're certainly in a position to know this having worked with Microsoft on the original XBOX and then with Sony for the PS3.
"In the end, you only have so many engineers and so much capability, and
if you're going to go off and do chips for Sony or Microsoft, then
that's probably a chip that you're not doing for some other portion of
your business. And at least in the case of Sony and Nvidia, in terms of
PS4, AMD has the business and Nvidia doesn't. We'll see how that plays
out from a business perspective I guess. It's clearly not a technology