argue (someone/something) down

1. To successfully sway or influence someone who has an opposing viewpoint. They opposed this bill for so long that I'm amazed we were finally able to argue them down.

2. To successfully sway someone to lower the price of something. At a flea market, you can always try to argue the prices down. A: "He's asking $20 for it." B: "Try to argue him down to $10."