cat and mouse
Definition from Wiktionary, the free dictionary
See also cat-and-mouse
- (idiomatic) A relationship in which two parties closely monitor and challenge one another in a suspicious or self-protective manner, often because each party is attempting to gain an advantage over the other.
- The Allied and the enemy submarine played a game of cat and mouse on checking up on each other's whereabouts and supposed strategic plans.
- The hackers played a cat and mouse game with the computer's system administrators: The hackers kept trying new tricks, and the system administrators kept mounting electronic defenses to prevent damage and catch the hackers.
Usage notes 
- Often preceded by the verb play.
two keeping check on one another