play one off against another

play one off against another (maneuver opposing groups in order to benefit oneself) — натравливать друг на друга, восстанавливать друг против друга; разжигать соперничество; ≈столкнуть кого-л. лбами;

Example 1: He thought the only way to keep his top men on their toes was to play one off against the other: to hint now and again someone else was after their job....

Example 2: This is a play about an old, scruffy vagrant who comes into a house and tries to manipulate two brothers and play one off against the other.

Example 3: Of course the roles might be reversed, but either way the end result is Mom and Dad fighting and the child figuring out that she can get whatever she wants if she just plays one off against the other in the right way.

see also
[play both ends against the middle]
[drive a wedge between sb]

[run with the hare and hunt with the hounds]
[и нашим и вашим]