“She had provided him with a son who could play tennis nearly as well as a professional, dance quite as well as a gigolo, and hold his own at bridge with any of the experts;”—APPEARANCE AND REALITY by W. Somerset Maugham
What does the son hold at “bridge”?
Bridge is a card game.
To hold his own position among the experts?
It basically means that the son is good enough at Bridge to be able to play at an expert level.
I hope that helps make it a bit more clear.