"Matched wits" is a popular phrase used to describe a battle of intelligence or competition between two people. However, there are several synonyms that can be used to convey the same meaning. "Locked horns" is a phrase used to describe a conflict between two determined opponents. "Engaged in verbal sparring" is also an appropriate synonym. Other options include "dueling minds," "mind games," and "intellectual combat." These phrases can be used to describe any situation where two individuals are trying to outsmart each other, whether it be in a debate, game, or social setting.