If you were in a customer service position, how would you deal with someone who does not want to accept the answers you give them?

Get a manager. I'm not paid enough to think about that. But i don't work in customer service. I went for a customer service job once but the interviewer decided I was better in sales.

If I was a manager and I didn't have the power to placate them I'd probably refer them to a complaint department or ombudsman. People usually run out of steam anyway. Which reminds me I need to see what's happening with all the complaints I've put in about stuff and I have vouchers to use lol