outdo

out|do /'aʊtd'uː/ (outdoes outdoing outdid outdone)
1 [VERB] V n
If you outdo someone, you are a lot more successful than they are at a particular activity.
Both sides have tried to outdo each other to show how tough they can be.
2 [PHRASE] PHR with cl
You use not to be outdone to introduce an action which someone takes in response to a previous action.
She wore a lovely tiara but the groom, not to be outdone, had on a very smart embroidered waistcoat.