Cat And Mouse Meaning

(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.

Example:   The Allied and the enemy submarine played a game of cat and mouse, checking up on each other's whereabouts and strategies.
  The hackers played a cat and mouse game with the system administrators: the hackers kept trying new tricks, and the system administrators kept mounting electronic defenses.