If the assumption is that employers tend to quote a higher initial number than prospective employees, then the anchoring effect supports the claim that the first person to quote a number loses.

What's that assumption based on? I would assume the opposite, that the employer negotiates in their self-interests, and tend to quote lower numbers.

Right, but even then, you're in a position to counter their initial offer and move up from there. If you name the number first, then it only has one direction to move (and it's not the good direction).

That rationale contradicts negotiation approaches that I've read about in a few places, specifically the idea of "anchoring." (Other commenters have also mentioned this) I'm not sure how much data there is to support either approach.

https://en.m.wikipedia.org/wiki/Anchoring