Which preposition to use with outdo

in Occurrences 27%

He then proffered to kiss the King's hand, but this sign of vassalage was likewise declined; whereupon, not to be outdone in magnanimity, he leaned forward and kissed the right arm of Ferdinand.

at Occurrences 1%

Though we had passed over not less than four-and-twenty miles of very rugged ground, and had a Highland dance on the top of Dun Can, the highest mountain in the island, we returned in the evening not at all fatigued, and piqued ourselves at not being outdone at the nightly ball by our less active friends, who had remained at home.

by Occurrences 1%

And Geissler, was he to be outdone by an engineer on a copper mine?

on Occurrences 1%

Was Samuel to be outdone on his own one-time field of action by an old ladies' darling?

Which preposition to use with  outdo