The phrase "battle of wits" refers to an intellectual competition between two opponents. Some synonyms for this phrase include "intellectual contest," "mind game," and "cerebral match." Another synonym is "brilliant exchange," which describes a conversation or argument characterized by quick thinking and clever responses. Similarly, "wit showdown" suggests a tense and competitive atmosphere, while "brain duel" emphasizes the mental agility required of each participant. Other possibilities include "mental joust," "cognitive clash," and "wit war." Regardless of the specific phrasing, these synonyms all convey the idea of a sophisticated and demanding competition between individuals with sharp minds and quick tongues.