What the Lakers did was very smart. They worked behind the scenes with AR and got his word he was planning to stay in LA. They then let the entire NBA know they would match any offer he got. Some teams who might have offered high just to make the Lakers go high in matching were thrown off when LA said they would wait until the last minute to match any offer, so the other team would have that money tied up, and couldn't go after other free agents.
It was a perfect plan and only worked because AR made it known he wasn't really interested in leaving LA. Teams like the Spurs & Rockets, who were expected to offer very high, were scared off because of the fact LA & AR seemed to be working together from day 1.