[user5994461]

Counter example: Got offer from two companies, the second a few weeks later.

Preferred the second but their offer was slightly lower. They could match easily (big company) but they didn't want to. I got calls by HR and by the manager to convince me to take the job but they wouldn't bulge on the offer.

Well. Went to work for the first company. I still got a call from HR after I started working at the company, to try to get me. I was upfront, I like your company and your project more, I want to work for you, all you have to do is to match £XYZ salary and I can start working for you on the XYth (notice period). They still refused to match the offer.