Cat and mouse

Cat and mouse, often expressed as cat-and-mouse game, is an English-language idiom that means "a contrived action involving constant pursuit, near captures, and repeated escapes."[1] The "cat" is unable to secure a definitive victory over the "mouse", who, despite not being able to defeat the cat, is able to avoid capture. In extreme cases, the idiom may imply that the contest is never-ending. The term is derived from the hunting behavior of domestic cats, which often appear to "play" with prey by releasing it after capture. This behavior may arise from an instinctive imperative to ensure that the prey is weak enough to be killed without endangering the cat.[2]

Look up cat and mouse in Wiktionary, the free dictionary.

Felis catus versus rattus rattus; domestic cats often capture and release their prey multiple times before the target of the hunt stops moving for good.

In colloquial usage, it has often been generalized to mean the advantage constantly shifts between the contestants, leading to an impasse or de facto stalemate. In classical game theory, cat and mouse classifies as a "copycat" archetype whereby there exists no equilibrium, and most importantly, no endgame, its two protagonists, Dot and Ditto, running amok in their game space to infinity, with no endpoint to their game anywhere in sight attributable to a defective reward system; conflicting incentives.[3] Theoretical active reading on "cat and mouse," or "Dot and Ditto" is ominous in its implication; multiple protagonists, each armed a passive strategy, can remain theoretically locked in total perpetual war indefinitely, wholly unable to rise to a plateau sufficient to intellectualize their plight.

The term has also been used to refer to the game hide-and-seek.