Green’s Dictionary of Slang

(US) to confuse, manipulate or brainwash someone by psychological means.

1964		R. Abrahams Deep Down In The Jungle 141: It is an attempt to ‘one-up’ or ‘out-psych’ the opposing players.
1973		J. Wambaugh Onion Field 397: ‘I would perhaps suggest that Mr. Powell had reasons, and he can certainly out-psyche the psychiatrist’.