I have found the opposite to be true.

I have been on both sides of the table over the years: as the interviewer and as the applicant. I've also done consulting contract negotiation on both sides of the table.

I believe the party who makes the first bid is at a disadvantage. If you are hiring and you bid first, the applicant will almost always ask for ~5% more. However, if you get them to bid first you may find their number was 25% less than you were prepared to pay. This results in significant savings and gives you budget room for granting future raises to keep the employee happy. Same general principle applies to consulting contract negotiations.

As always, there is a possibility that the parties can't come to terms if expectations don't overlap, but that's ubiquitous in negotiation.