battle of wits vs contest

battle of wits and contest both are nouns.

Word	Noun	Adjective	Verb	Adverb
battle of wits	Yes	No	No	No
contest	Yes	No	Yes	No

As nouns, contest is a hypernym of battle of wits; that is, contest is a word with a broader meaning than battle of wits:

battle of wits: a contest in which intelligence rather than violence is used
contest: a struggle between rivals

battle of wits (noun)	contest (noun)
a contest in which intelligence rather than violence is used	a struggle between rivals
	an occasion on which a winner is selected from among two or more contestants

battle of wits (verb)	contest (verb)
	to make the subject of dispute, contention, or litigation