Sure, I'd like to get the maximum salary the employer could possibly afford, as same as the employer would like to pay the minimum salary I could possibly accept but I just don't see how being the second one to call a number resolves this. It's like believing that if you put an ask order only if there is already a bid at that price and vice versa then and only then you will be extracting the whole value from the market.
The only situation, I can imagine, where this work is when your maximum ask is below the firm's minimum bid, which, provided that you already have a job and understand few things about your industry, is very unlikely to happen. Even people out of college are pretty informed about the market situation as they talk to their friends who are getting offers too. And, even if this happened, you just got literally low-balled anyways, you took the lowest bid they could make!