Cat and mouse
Encyclopedia
Cat and mouse, often expressed as cat-and-mouse game, is an English-language idiom
Idiom
Idiom is an expression, word, or phrase that has a figurative meaning that is comprehended in regard to a common use of that expression that is separate from the literal meaning or definition of the words of which it is made...

 dating back to 1675 that means "a contrived action involving constant pursuit, near captures, and repeated escapes." 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 cat
Cat
The cat , also known as the domestic cat or housecat to distinguish it from other felids and felines, is a small, usually furry, domesticated, carnivorous mammal that is valued by humans for its companionship and for its ability to hunt vermin and household pests...

s, which often appear to "play" with prey by releasing it after capture. This behavior is due to an instinctive imperative to ensure that the prey is weak enough to be killed without endangering the cat.

In colloquial usage it has often been generalized (or corrupted) to mean simply that the advantage constantly shifts between the contestants, leading to an impasse
Impasse
A bargaining impasse occurs when the two sides negotiating an agreement are unable to reach an agreement and become deadlocked. An impasse is almost invariably mutually harmful, either as a result of direct action which may be taken such as a strike in employment negotiation or sanctions/military...

 or de facto
De facto
De facto is a Latin expression that means "concerning fact." In law, it often means "in practice but not necessarily ordained by law" or "in practice or actuality, but not officially established." It is commonly used in contrast to de jure when referring to matters of law, governance, or...

stalemate.
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