> You claim that if the employer calls the number first, the final number will magically be higher than if the candidate called the number first.No, I claim that, on net, you will end up with a better final deal if you do not give any number until after you have an offer and you are ready to respond to it. See further comments below.
> In your model, if employer said 200 first and you, using your strategy responded with 250, and while the employer is thinking or negotiating further, you've got an offer for 260 from another company
My strategy would be to not respond with a number to any offer until I have all of the offers I expect to get on the table (either that or those companies have told me definitely that they are not interested in going any further, so I can cross them off my list of possibles). My only response to an offer before that point would be to say "Ok, thanks very much for your offer, let me do some thinking about it and get back to you".
> I think you are personalizing things too much, nobody really cares.
It's not a matter of "personalizing", it's a matter of recognizing that, just like you, the company is trying to get the best deal they can. I'm not saying they will always try to negotiate you down if you give them a number before they've made you an offer. They might, as you say, decide it's not worth it to do that and just write you an offer at that number. But if they do do that, you are almost certainly leaving money on the table that you could have gotten if you had not given them a number and waited for them to make an offer.
And if, in that scenario, you then come back and tell them you want more because someone else made you a higher offer, you're, as I said before, reneging on what you already told them, and you're now saddling them with additional work they didn't even think they would need to do--they thought it was just a matter of writing you an offer at the number you said and you accepting it, boom, done. If you expected that you might get a higher offer, you shouldn't have given them the lower number in the first place.
> The HM has a vacancy that needs to be closed ASAP
Yes, but "ASAP" doesn't have to mean "as soon as you give them a number that's within their range". They might prefer that, but that doesn't mean you should. If they are genuinely interested in hiring you, they will be willing to negotiate if they have to. And it will be to your advantage to make them have to by not giving them a number up front.