Playing their game - Idioms by The Free Dictionary
play (someone's) game (redirected from playing their game)
play (someone's) game
To engage or participate in something according to the design, intentions, or constrictions created by someone else. I know the application process seems nonsensical and unintuitive, but you just have to play their game and do the paperwork the way they ask. I'll play your game for now, but if I find out you're not holding up your side of the bargain, I'm going to the police!
play someone's game advance another's plans, whether intentionally or not.
play somebody’s ˈgame,
play the same ˈgame (as somebody) (also play somebody at their own ˈgame) use the same methods as a competitor, an opponent, an enemy, etc: Safeway started cutting their food prices, so Asda decided to play them at their own game by cutting prices even more.